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De.next(Fe)},error:function(Fe){v=!0,B(),p=jr(ne,o,Fe),De.error(Fe)},complete:function(){h=!0,B(),p=jr(ne,a),De.complete()}}),$(T).subscribe(u))})(c)}}function jr(e,t){for(var r=[],n=2;ne.next(document)),e}function K(e,t=document){return Array.from(t.querySelectorAll(e))}function V(e,t=document){let r=ce(e,t);if(typeof r=="undefined")throw new ReferenceError(`Missing element: expected "${e}" to be present`);return r}function ce(e,t=document){return t.querySelector(e)||void 0}function _e(){return document.activeElement instanceof HTMLElement&&document.activeElement||void 0}function rr(e){return L(b(document.body,"focusin"),b(document.body,"focusout")).pipe(He(1),l(()=>{let t=_e();return typeof t!="undefined"?e.contains(t):!1}),N(e===_e()),G())}function Je(e){return{x:e.offsetLeft,y:e.offsetTop}}function Yn(e){return L(b(window,"load"),b(window,"resize")).pipe(Re(0,we),l(()=>Je(e)),N(Je(e)))}function nr(e){return{x:e.scrollLeft,y:e.scrollTop}}function dt(e){return L(b(e,"scroll"),b(window,"resize")).pipe(Re(0,we),l(()=>nr(e)),N(nr(e)))}var Bn=function(){if(typeof Map!="undefined")return Map;function e(t,r){var n=-1;return t.some(function(o,i){return o[0]===r?(n=i,!0):!1}),n}return function(){function t(){this.__entries__=[]}return Object.defineProperty(t.prototype,"size",{get:function(){return this.__entries__.length},enumerable:!0,configurable:!0}),t.prototype.get=function(r){var n=e(this.__entries__,r),o=this.__entries__[n];return o&&o[1]},t.prototype.set=function(r,n){var o=e(this.__entries__,r);~o?this.__entries__[o][1]=n:this.__entries__.push([r,n])},t.prototype.delete=function(r){var n=this.__entries__,o=e(n,r);~o&&n.splice(o,1)},t.prototype.has=function(r){return!!~e(this.__entries__,r)},t.prototype.clear=function(){this.__entries__.splice(0)},t.prototype.forEach=function(r,n){n===void 0&&(n=null);for(var o=0,i=this.__entries__;o0},e.prototype.connect_=function(){!zr||this.connected_||(document.addEventListener("transitionend",this.onTransitionEnd_),window.addEventListener("resize",this.refresh),xa?(this.mutationsObserver_=new MutationObserver(this.refresh),this.mutationsObserver_.observe(document,{attributes:!0,childList:!0,characterData:!0,subtree:!0})):(document.addEventListener("DOMSubtreeModified",this.refresh),this.mutationEventsAdded_=!0),this.connected_=!0)},e.prototype.disconnect_=function(){!zr||!this.connected_||(document.removeEventListener("transitionend",this.onTransitionEnd_),window.removeEventListener("resize",this.refresh),this.mutationsObserver_&&this.mutationsObserver_.disconnect(),this.mutationEventsAdded_&&document.removeEventListener("DOMSubtreeModified",this.refresh),this.mutationsObserver_=null,this.mutationEventsAdded_=!1,this.connected_=!1)},e.prototype.onTransitionEnd_=function(t){var r=t.propertyName,n=r===void 0?"":r,o=ya.some(function(i){return!!~n.indexOf(i)});o&&this.refresh()},e.getInstance=function(){return this.instance_||(this.instance_=new e),this.instance_},e.instance_=null,e}(),Jn=function(e,t){for(var r=0,n=Object.keys(t);r0},e}(),Zn=typeof WeakMap!="undefined"?new WeakMap:new Bn,eo=function(){function e(t){if(!(this instanceof e))throw new TypeError("Cannot call a class as a function.");if(!arguments.length)throw new TypeError("1 argument required, but only 0 present.");var r=Ea.getInstance(),n=new Ra(t,r,this);Zn.set(this,n)}return e}();["observe","unobserve","disconnect"].forEach(function(e){eo.prototype[e]=function(){var t;return(t=Zn.get(this))[e].apply(t,arguments)}});var ka=function(){return typeof or.ResizeObserver!="undefined"?or.ResizeObserver:eo}(),to=ka;var ro=new E,Ha=I(()=>H(new to(e=>{for(let t of e)ro.next(t)}))).pipe(x(e=>L(Te,H(e)).pipe(C(()=>e.disconnect()))),J(1));function he(e){return{width:e.offsetWidth,height:e.offsetHeight}}function ge(e){return Ha.pipe(S(t=>t.observe(e)),x(t=>ro.pipe(_(({target:r})=>r===e),C(()=>t.unobserve(e)),l(()=>he(e)))),N(he(e)))}function bt(e){return{width:e.scrollWidth,height:e.scrollHeight}}function sr(e){let t=e.parentElement;for(;t&&(e.scrollWidth<=t.scrollWidth&&e.scrollHeight<=t.scrollHeight);)t=(e=t).parentElement;return t?e:void 0}var no=new E,Pa=I(()=>H(new IntersectionObserver(e=>{for(let t of e)no.next(t)},{threshold:0}))).pipe(x(e=>L(Te,H(e)).pipe(C(()=>e.disconnect()))),J(1));function cr(e){return Pa.pipe(S(t=>t.observe(e)),x(t=>no.pipe(_(({target:r})=>r===e),C(()=>t.unobserve(e)),l(({isIntersecting:r})=>r))))}function oo(e,t=16){return dt(e).pipe(l(({y:r})=>{let n=he(e),o=bt(e);return r>=o.height-n.height-t}),G())}var fr={drawer:V("[data-md-toggle=drawer]"),search:V("[data-md-toggle=search]")};function io(e){return fr[e].checked}function qe(e,t){fr[e].checked!==t&&fr[e].click()}function Ue(e){let t=fr[e];return b(t,"change").pipe(l(()=>t.checked),N(t.checked))}function $a(e,t){switch(e.constructor){case HTMLInputElement:return e.type==="radio"?/^Arrow/.test(t):!0;case HTMLSelectElement:case HTMLTextAreaElement:return!0;default:return e.isContentEditable}}function Ia(){return L(b(window,"compositionstart").pipe(l(()=>!0)),b(window,"compositionend").pipe(l(()=>!1))).pipe(N(!1))}function ao(){let e=b(window,"keydown").pipe(_(t=>!(t.metaKey||t.ctrlKey)),l(t=>({mode:io("search")?"search":"global",type:t.key,claim(){t.preventDefault(),t.stopPropagation()}})),_(({mode:t,type:r})=>{if(t==="global"){let n=_e();if(typeof n!="undefined")return!$a(n,r)}return!0}),ue());return Ia().pipe(x(t=>t?R:e))}function Me(){return new URL(location.href)}function ot(e){location.href=e.href}function so(){return new E}function co(e,t){if(typeof t=="string"||typeof t=="number")e.innerHTML+=t.toString();else if(t instanceof Node)e.appendChild(t);else if(Array.isArray(t))for(let r of t)co(e,r)}function M(e,t,...r){let n=document.createElement(e);if(t)for(let o of Object.keys(t))typeof t[o]!="undefined"&&(typeof t[o]!="boolean"?n.setAttribute(o,t[o]):n.setAttribute(o,""));for(let o of r)co(n,o);return n}function ur(e){if(e>999){let t=+((e-950)%1e3>99);return`${((e+1e-6)/1e3).toFixed(t)}k`}else return e.toString()}function fo(){return location.hash.substring(1)}function uo(e){let t=M("a",{href:e});t.addEventListener("click",r=>r.stopPropagation()),t.click()}function Fa(){return b(window,"hashchange").pipe(l(fo),N(fo()),_(e=>e.length>0),J(1))}function po(){return Fa().pipe(l(e=>ce(`[id="${e}"]`)),_(e=>typeof e!="undefined"))}function Nr(e){let t=matchMedia(e);return er(r=>t.addListener(()=>r(t.matches))).pipe(N(t.matches))}function lo(){let e=matchMedia("print");return L(b(window,"beforeprint").pipe(l(()=>!0)),b(window,"afterprint").pipe(l(()=>!1))).pipe(N(e.matches))}function qr(e,t){return e.pipe(x(r=>r?t():R))}function pr(e,t={credentials:"same-origin"}){return pe(fetch(`${e}`,t)).pipe(fe(()=>R),x(r=>r.status!==200?Tt(()=>new Error(r.statusText)):H(r)))}function We(e,t){return pr(e,t).pipe(x(r=>r.json()),J(1))}function mo(e,t){let r=new DOMParser;return pr(e,t).pipe(x(n=>n.text()),l(n=>r.parseFromString(n,"text/xml")),J(1))}function lr(e){let t=M("script",{src:e});return I(()=>(document.head.appendChild(t),L(b(t,"load"),b(t,"error").pipe(x(()=>Tt(()=>new ReferenceError(`Invalid script: ${e}`))))).pipe(l(()=>{}),C(()=>document.head.removeChild(t)),Oe(1))))}function ho(){return{x:Math.max(0,scrollX),y:Math.max(0,scrollY)}}function bo(){return L(b(window,"scroll",{passive:!0}),b(window,"resize",{passive:!0})).pipe(l(ho),N(ho()))}function vo(){return{width:innerWidth,height:innerHeight}}function go(){return b(window,"resize",{passive:!0}).pipe(l(vo),N(vo()))}function yo(){return Q([bo(),go()]).pipe(l(([e,t])=>({offset:e,size:t})),J(1))}function mr(e,{viewport$:t,header$:r}){let n=t.pipe(X("size")),o=Q([n,r]).pipe(l(()=>Je(e)));return Q([r,t,o]).pipe(l(([{height:i},{offset:a,size:s},{x:f,y:c}])=>({offset:{x:a.x-f,y:a.y-c+i},size:s})))}(()=>{function e(n,o){parent.postMessage(n,o||"*")}function t(...n){return n.reduce((o,i)=>o.then(()=>new Promise(a=>{let s=document.createElement("script");s.src=i,s.onload=a,document.body.appendChild(s)})),Promise.resolve())}var r=class{constructor(n){this.url=n,this.onerror=null,this.onmessage=null,this.onmessageerror=null,this.m=a=>{a.source===this.w&&(a.stopImmediatePropagation(),this.dispatchEvent(new MessageEvent("message",{data:a.data})),this.onmessage&&this.onmessage(a))},this.e=(a,s,f,c,u)=>{if(s===this.url.toString()){let p=new ErrorEvent("error",{message:a,filename:s,lineno:f,colno:c,error:u});this.dispatchEvent(p),this.onerror&&this.onerror(p)}};let o=new EventTarget;this.addEventListener=o.addEventListener.bind(o),this.removeEventListener=o.removeEventListener.bind(o),this.dispatchEvent=o.dispatchEvent.bind(o);let i=document.createElement("iframe");i.width=i.height=i.frameBorder="0",document.body.appendChild(this.iframe=i),this.w.document.open(),this.w.document.write(` + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + 跳转至 + + +
+
+ +
+ + + + +
+ + +
+ +
+ + + + + + +
+
+ + + +
+
+
+ + + + +
+
+
+ + + +
+
+
+ + + +
+
+
+ + + +
+
+ + + + + + + + + + + + + +

4.1. 数组

+

「数组 Array」是一种将 相同类型元素 存储在 连续内存空间 的数据结构,将元素在数组中的位置称为元素的「索引 Index」。

+

array_definition

+

Fig. 数组定义与存储方式

+ +
+

Note

+

观察上图,我们发现 数组首元素的索引为 \(0\) 。你可能会想,这并不符合日常习惯,首个元素的索引为什么不是 \(1\) 呢,这不是更加自然吗?我认同你的想法,但请先记住这个设定,后面讲内存地址计算时,我会尝试解答这个问题。

+
+

数组有多种初始化写法。根据实际需要,选代码最短的那一种就好。

+
+
+
+
array.java
/* 初始化数组 */
+int[] arr = new int[5]; // { 0, 0, 0, 0, 0 }
+int[] nums = { 1, 3, 2, 5, 4 };
+
+
+
+
array.cpp
/* 初始化数组 */
+int* arr = new int[5];
+int* nums = new int[5] { 1, 3, 2, 5, 4 };
+
+
+
+
array.py
""" 初始化数组 """
+arr = [0] * 5  # [ 0, 0, 0, 0, 0 ]
+nums = [1, 3, 2, 5, 4]  
+
+
+
+
array.go
/* 初始化数组 */
+var arr [5]int
+// 在 Go 中,指定长度时([5]int)为数组,不指定长度时([]int)为切片
+// 由于 Go 的数组被设计为在编译期确定长度,因此只能使用常量来指定长度
+// 为了方便实现扩容 extend() 方法,以下将切片(Slice)看作数组(Array)
+nums := []int{1, 3, 2, 5, 4}
+
+
+
+
array.js
/* 初始化数组 */
+var arr = new Array(5).fill(0);
+var nums = [1, 3, 2, 5, 4];
+
+
+
+
array.ts
/* 初始化数组 */
+let arr: number[] = new Array(5).fill(0);
+let nums: number[] = [1, 3, 2, 5, 4];
+
+
+
+
array.c

+
+
+
+
array.cs
/* 初始化数组 */
+int[] arr = new int[5]; // { 0, 0, 0, 0, 0 }
+int[] nums = { 1, 3, 2, 5, 4 };
+
+
+
+
array.swift
/* 初始化数组 */
+let arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]
+let nums = [1, 3, 2, 5, 4]
+
+
+
+
array.zig
// 初始化数组
+var arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }
+var nums = [_]i32{ 1, 3, 2, 5, 4 };
+
+
+
+
+

4.1.1. 数组优点

+

在数组中访问元素非常高效。这是因为在数组中,计算元素的内存地址非常容易。给定数组首个元素的地址、和一个元素的索引,利用以下公式可以直接计算得到该元素的内存地址,从而直接访问此元素。

+

array_memory_location_calculation

+

Fig. 数组元素的内存地址计算

+ +
// 元素内存地址 = 数组内存地址 + 元素长度 * 元素索引
+elementAddr = firtstElementAddr + elementLength * elementIndex
+
+

为什么数组元素索引从 0 开始编号? 根据地址计算公式,索引本质上表示的是内存地址偏移量,首个元素的地址偏移量是 \(0\) ,那么索引是 \(0\) 也就很自然了。

+

访问元素的高效性带来了许多便利。例如,我们可以在 \(O(1)\) 时间内随机获取一个数组中的元素。

+
+
+
+
array.java
/* 随机返回一个数组元素 */
+int randomAccess(int[] nums) {
+    // 在区间 [0, nums.length) 中随机抽取一个数字
+    int randomIndex = ThreadLocalRandom.current().
+                      nextInt(0, nums.length);
+    // 获取并返回随机元素
+    int randomNum = nums[randomIndex];
+    return randomNum;
+}
+
+
+
+
array.cpp
/* 随机返回一个数组元素 */
+int randomAccess(int* nums, int size) {
+    // 在区间 [0, size) 中随机抽取一个数字
+    int randomIndex = rand() % size;
+    // 获取并返回随机元素
+    int randomNum = nums[randomIndex];
+    return randomNum;
+}
+
+
+
+
array.py
""" 随机访问元素 """
+def random_access(nums):
+    # 在区间 [0, len(nums)-1] 中随机抽取一个数字
+    random_index = random.randint(0, len(nums) - 1)
+    # 获取并返回随机元素
+    random_num = nums[random_index]
+    return random_num
+
+
+
+
array.go
/* 随机返回一个数组元素 */
+func randomAccess(nums []int) (randomNum int) {
+    // 在区间 [0, nums.length) 中随机抽取一个数字
+    randomIndex := rand.Intn(len(nums))
+    // 获取并返回随机元素
+    randomNum = nums[randomIndex]
+    return
+}
+
+
+
+
array.js
/* 随机返回一个数组元素 */
+function randomAccess(nums) {
+    // 在区间 [0, nums.length) 中随机抽取一个数字
+    const random_index = Math.floor(Math.random() * nums.length);
+    // 获取并返回随机元素
+    const random_num = nums[random_index];
+    return random_num;
+}
+
+
+
+
array.ts
/* 随机返回一个数组元素 */
+function randomAccess(nums: number[]): number {
+    // 在区间 [0, nums.length) 中随机抽取一个数字
+    const random_index = Math.floor(Math.random() * nums.length);
+    // 获取并返回随机元素
+    const random_num = nums[random_index];
+    return random_num;
+}
+
+
+
+
array.c

+
+
+
+
array.cs
/* 随机返回一个数组元素 */
+int RandomAccess(int[] nums)
+{
+    Random random=new();
+    // 在区间 [0, nums.Length) 中随机抽取一个数字
+    int randomIndex = random.Next(nums.Length);
+    // 获取并返回随机元素
+    int randomNum = nums[randomIndex];
+    return randomNum;
+}
+
+
+
+
array.swift
/* 随机返回一个数组元素 */
+func randomAccess(nums: [Int]) -> Int {
+    // 在区间 [0, nums.count) 中随机抽取一个数字
+    let randomIndex = nums.indices.randomElement()!
+    // 获取并返回随机元素
+    let randomNum = nums[randomIndex]
+    return randomNum
+}
+
+
+
+
array.zig
// 随机返回一个数组元素
+pub fn randomAccess(nums: []i32) i32 {
+    // 在区间 [0, nums.len) 中随机抽取一个整数
+    var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);
+    // 获取并返回随机元素
+    var randomNum = nums[randomIndex];
+    return randomNum;
+}
+
+
+
+
+

4.1.2. 数组缺点

+

数组在初始化后长度不可变。由于系统无法保证数组之后的内存空间是可用的,因此数组长度无法扩展。而若希望扩容数组,则需新建一个数组,然后把原数组元素依次拷贝到新数组,在数组很大的情况下,这是非常耗时的。

+
+
+
+
array.java
/* 扩展数组长度 */
+int[] extend(int[] nums, int enlarge) {
+    // 初始化一个扩展长度后的数组
+    int[] res = new int[nums.length + enlarge];
+    // 将原数组中的所有元素复制到新数组
+    for (int i = 0; i < nums.length; i++) {
+        res[i] = nums[i];
+    }
+    // 返回扩展后的新数组
+    return res;
+}
+
+
+
+
array.cpp
/* 扩展数组长度 */
+int* extend(int* nums, int size, int enlarge) {
+    // 初始化一个扩展长度后的数组
+    int* res = new int[size + enlarge];
+    // 将原数组中的所有元素复制到新数组
+    for (int i = 0; i < size; i++) {
+        res[i] = nums[i];
+    }
+    // 释放内存
+    delete[] nums;
+    // 返回扩展后的新数组
+    return res;
+}
+
+
+
+
array.py
""" 扩展数组长度 """
+# 请注意,Python 的 list 是动态数组,可以直接扩展
+# 为了方便学习,本函数将 list 看作是长度不可变的数组
+def extend(nums, enlarge):
+    # 初始化一个扩展长度后的数组
+    res = [0] * (len(nums) + enlarge)
+    # 将原数组中的所有元素复制到新数组
+    for i in range(len(nums)):
+        res[i] = nums[i]
+    # 返回扩展后的新数组
+    return res
+
+
+
+
array.go
/* 扩展数组长度 */
+func extend(nums []int, enlarge int) []int {
+    // 初始化一个扩展长度后的数组
+    res := make([]int, len(nums)+enlarge)
+    // 将原数组中的所有元素复制到新数组
+    for i, num := range nums {
+        res[i] = num
+    }
+    // 返回扩展后的新数组
+    return res
+}
+
+
+
+
array.js
/* 扩展数组长度 */
+function extend(nums, enlarge) {
+    // 初始化一个扩展长度后的数组
+    const res = new Array(nums.length + enlarge).fill(0);
+    // 将原数组中的所有元素复制到新数组
+    for (let i = 0; i < nums.length; i++) {
+        res[i] = nums[i];
+    }
+    // 返回扩展后的新数组
+    return res;
+}
+
+
+
+
array.ts
/* 扩展数组长度 */
+function extend(nums: number[], enlarge: number): number[] {
+    // 初始化一个扩展长度后的数组
+    const res = new Array(nums.length + enlarge).fill(0);
+    // 将原数组中的所有元素复制到新数组
+    for (let i = 0; i < nums.length; i++) {
+        res[i] = nums[i];
+    }
+    // 返回扩展后的新数组
+    return res;
+}
+
+
+
+
array.c

+
+
+
+
array.cs
/* 扩展数组长度 */
+int[] Extend(int[] nums, int enlarge)
+{
+    // 初始化一个扩展长度后的数组
+    int[] res = new int[nums.Length + enlarge];
+    // 将原数组中的所有元素复制到新数组
+    for (int i = 0; i < nums.Length; i++)
+    {
+        res[i] = nums[i];
+    }
+    // 返回扩展后的新数组
+    return res;
+}
+
+
+
+
array.swift
/* 扩展数组长度 */
+func extend(nums: [Int], enlarge: Int) -> [Int] {
+    // 初始化一个扩展长度后的数组
+    var res = Array(repeating: 0, count: nums.count + enlarge)
+    // 将原数组中的所有元素复制到新数组
+    for i in nums.indices {
+        res[i] = nums[i]
+    }
+    // 返回扩展后的新数组
+    return res
+}
+
+
+
+
array.zig
// 扩展数组长度
+pub fn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {
+    // 初始化一个扩展长度后的数组
+    var res = try mem_allocator.alloc(i32, nums.len + enlarge);
+    std.mem.set(i32, res, 0);
+    // 将原数组中的所有元素复制到新数组
+    std.mem.copy(i32, res, nums);
+    // 返回扩展后的新数组
+    return res;
+}
+
+
+
+
+

数组中插入或删除元素效率低下。假设我们想要在数组中间某位置插入一个元素,由于数组元素在内存中是“紧挨着的”,它们之间没有空间再放任何数据。因此,我们不得不将此索引之后的所有元素都向后移动一位,然后再把元素赋值给该索引。删除元素也是类似,需要把此索引之后的元素都向前移动一位。总体看有以下缺点:

+
    +
  • 时间复杂度高:数组的插入和删除的平均时间复杂度均为 \(O(N)\) ,其中 \(N\) 为数组长度。
  • +
  • 丢失元素:由于数组的长度不可变,因此在插入元素后,超出数组长度范围的元素会被丢失。
  • +
  • 内存浪费:我们一般会初始化一个比较长的数组,只用前面一部分,这样在插入数据时,丢失的末尾元素都是我们不关心的,但这样做同时也会造成内存空间的浪费。
  • +
+

array_insert_remove_element

+

Fig. 在数组中插入与删除元素

+ +
+
+
+
array.java
/* 在数组的索引 index 处插入元素 num */
+void insert(int[] nums, int num, int index) {
+    // 把索引 index 以及之后的所有元素向后移动一位
+    for (int i = nums.length - 1; i > index; i--) {
+        nums[i] = nums[i - 1];
+    }
+    // 将 num 赋给 index 处元素
+    nums[index] = num;
+}
+
+/* 删除索引 index 处元素 */
+void remove(int[] nums, int index) {
+    // 把索引 index 之后的所有元素向前移动一位
+    for (int i = index; i < nums.length - 1; i++) {
+        nums[i] = nums[i + 1];
+    }
+}
+
+
+
+
array.cpp
/* 在数组的索引 index 处插入元素 num */
+void insert(int* nums, int size, int num, int index) {
+    // 把索引 index 以及之后的所有元素向后移动一位
+    for (int i = size - 1; i > index; i--) {
+        nums[i] = nums[i - 1];
+    }
+    // 将 num 赋给 index 处元素
+    nums[index] = num;
+}
+
+/* 删除索引 index 处元素 */
+void remove(int* nums, int size, int index) {
+    // 把索引 index 之后的所有元素向前移动一位
+    for (int i = index; i < size - 1; i++) {
+        nums[i] = nums[i + 1];
+    }
+}
+
+
+
+
array.py
""" 在数组的索引 index 处插入元素 num """
+def insert(nums, num, index):
+    # 把索引 index 以及之后的所有元素向后移动一位
+    for i in range(len(nums) - 1, index, -1):
+        nums[i] = nums[i - 1]
+    # 将 num 赋给 index 处元素
+    nums[index] = num
+
+""" 删除索引 index 处元素 """
+def remove(nums, index):
+    # 把索引 index 之后的所有元素向前移动一位
+    for i in range(index, len(nums) - 1):
+        nums[i] = nums[i + 1]
+
+
+
+
array.go
/* 在数组的索引 index 处插入元素 num */
+func insert(nums []int, num int, index int) {
+    // 把索引 index 以及之后的所有元素向后移动一位
+    for i := len(nums) - 1; i > index; i-- {
+        nums[i] = nums[i-1]
+    }
+    // 将 num 赋给 index 处元素
+    nums[index] = num
+}
+
+/* 删除索引 index 处元素 */
+func remove(nums []int, index int) {
+    // 把索引 index 之后的所有元素向前移动一位
+    for i := index; i < len(nums)-1; i++ {
+        nums[i] = nums[i+1]
+    }
+}
+
+
+
+
array.js
/* 在数组的索引 index 处插入元素 num */
+function insert(nums, num, index) {
+    // 把索引 index 以及之后的所有元素向后移动一位
+    for (let i = nums.length - 1; i > index; i--) {
+        nums[i] = nums[i - 1];
+    }
+    // 将 num 赋给 index 处元素
+    nums[index] = num;
+}
+
+/* 删除索引 index 处元素 */
+function remove(nums, index) {
+    // 把索引 index 之后的所有元素向前移动一位
+    for (let i = index; i < nums.length - 1; i++) {
+        nums[i] = nums[i + 1];
+    }
+}
+
+
+
+
array.ts
/* 在数组的索引 index 处插入元素 num */
+function insert(nums: number[], num: number, index: number): void {
+    // 把索引 index 以及之后的所有元素向后移动一位
+    for (let i = nums.length - 1; i > index; i--) {
+        nums[i] = nums[i - 1];
+    }
+    // 将 num 赋给 index 处元素
+    nums[index] = num;
+}
+
+/* 删除索引 index 处元素 */
+function remove(nums: number[], index: number): void {
+    // 把索引 index 之后的所有元素向前移动一位
+    for (let i = index; i < nums.length - 1; i++) {
+        nums[i] = nums[i + 1];
+    }
+}
+
+
+
+
array.c

+
+
+
+
array.cs
/* 在数组的索引 index 处插入元素 num */
+void Insert(int[] nums, int num, int index)
+{
+    // 把索引 index 以及之后的所有元素向后移动一位
+    for (int i = nums.Length - 1; i > index; i--)
+    {
+        nums[i] = nums[i - 1];
+    }
+    // 将 num 赋给 index 处元素
+    nums[index] = num;
+}
+/* 删除索引 index 处元素 */
+void Remove(int[] nums, int index)
+{
+    // 把索引 index 之后的所有元素向前移动一位
+    for (int i = index; i < nums.Length - 1; i++)
+    {
+        nums[i] = nums[i + 1];
+    }
+}
+
+
+
+
array.swift
/* 在数组的索引 index 处插入元素 num */
+func insert(nums: inout [Int], num: Int, index: Int) {
+    // 把索引 index 以及之后的所有元素向后移动一位
+    for i in sequence(first: nums.count - 1, next: { $0 > index + 1 ? $0 - 1 : nil }) {
+        nums[i] = nums[i - 1]
+    }
+    // 将 num 赋给 index 处元素
+    nums[index] = num
+}
+
+/* 删除索引 index 处元素 */
+func remove(nums: inout [Int], index: Int) {
+    let count = nums.count
+    // 把索引 index 之后的所有元素向前移动一位
+    for i in sequence(first: index, next: { $0 < count - 1 - 1 ? $0 + 1 : nil }) {
+        nums[i] = nums[i + 1]
+    }
+}
+
+
+
+
array.zig
// 在数组的索引 index 处插入元素 num
+pub fn insert(nums: []i32, num: i32, index: usize) void {
+    // 把索引 index 以及之后的所有元素向后移动一位
+    var i = nums.len - 1;
+    while (i > index) : (i -= 1) {
+        nums[i] = nums[i - 1];
+    }
+    // 将 num 赋给 index 处元素
+    nums[index] = num;
+}
+
+// 删除索引 index 处元素
+pub fn remove(nums: []i32, index: usize) void {
+    // 把索引 index 之后的所有元素向前移动一位
+    var i = index;
+    while (i < nums.len - 1) : (i += 1) {
+        nums[i] = nums[i + 1];
+    }
+}
+
+
+
+
+

4.1.3. 数组常用操作

+

数组遍历。以下介绍两种常用的遍历方法。

+
+
+
+
array.java
/* 遍历数组 */
+void traverse(int[] nums) {
+    int count = 0;
+    // 通过索引遍历数组
+    for (int i = 0; i < nums.length; i++) {
+        count++;
+    }
+    // 直接遍历数组
+    for (int num : nums) {
+        count++;
+    }
+}
+
+
+
+
array.cpp
/* 遍历数组 */
+void traverse(int* nums, int size) {
+    int count = 0;
+    // 通过索引遍历数组
+    for (int i = 0; i < size; i++) {
+        count++;
+    }
+}
+
+
+
+
array.py
""" 遍历数组 """
+def traverse(nums):
+    count = 0
+    # 通过索引遍历数组
+    for i in range(len(nums)):
+        count += 1
+    # 直接遍历数组
+    for num in nums:
+        count += 1
+
+
+
+
array.go
/* 遍历数组 */
+func traverse(nums []int) {
+    count := 0
+    // 通过索引遍历数组
+    for i := 0; i < len(nums); i++ {
+        count++
+    }
+    // 直接遍历数组
+    for range nums {
+        count++
+    }
+}
+
+
+
+
array.js
/* 遍历数组 */
+function traverse(nums) {
+    let count = 0;
+    // 通过索引遍历数组
+    for (let i = 0; i < nums.length; i++) {
+        count++;
+    }
+    // 直接遍历数组
+    for (let num of nums) {
+        count += 1;
+    }
+}
+
+
+
+
array.ts
/* 遍历数组 */
+function traverse(nums: number[]): void {
+    let count = 0;
+    // 通过索引遍历数组
+    for (let i = 0; i < nums.length; i++) {
+        count++;
+    }
+    // 直接遍历数组
+    for(let num of nums){
+        count += 1;
+    }
+}
+
+
+
+
array.c

+
+
+
+
array.cs
/* 遍历数组 */
+void Traverse(int[] nums)
+{
+    int count = 0;
+    // 通过索引遍历数组
+    for (int i = 0; i < nums.Length; i++)
+    {
+        count++;
+    }
+    // 直接遍历数组
+    foreach (int num in nums)
+    {
+        count++;
+    }
+}
+
+
+
+
array.swift
/* 遍历数组 */
+func traverse(nums: [Int]) {
+    var count = 0
+    // 通过索引遍历数组
+    for _ in nums.indices {
+        count += 1
+    }
+    // 直接遍历数组
+    for _ in nums {
+        count += 1
+    }
+}
+
+
+
+
array.zig
// 遍历数组
+pub fn traverse(nums: []i32) void {
+    var count: i32 = 0;
+    // 通过索引遍历数组
+    var i: i32 = 0;
+    while (i < nums.len) : (i += 1) {
+        count += 1;
+    }
+    count = 0;
+    // 直接遍历数组
+    for (nums) |_| {
+        count += 1;
+    }
+}
+
+
+
+
+

数组查找。通过遍历数组,查找数组内的指定元素,并输出对应索引。

+
+
+
+
array.java
/* 在数组中查找指定元素 */
+int find(int[] nums, int target) {
+    for (int i = 0; i < nums.length; i++) {
+        if (nums[i] == target)
+            return i;
+    }
+    return -1;
+}
+
+
+
+
array.cpp
/* 在数组中查找指定元素 */
+int find(int* nums, int size, int target) {
+    for (int i = 0; i < size; i++) {
+        if (nums[i] == target)
+            return i;
+    }
+    return -1;
+}
+
+
+
+
array.py
""" 在数组中查找指定元素 """
+def find(nums, target):
+    for i in range(len(nums)):
+        if nums[i] == target:
+            return i
+    return -1
+
+
+
+
array.go
/* 在数组中查找指定元素 */
+func find(nums []int, target int) (index int) {
+    index = -1
+    for i := 0; i < len(nums); i++ {
+        if nums[i] == target {
+            index = i
+            break
+        }
+    }
+    return
+}
+
+
+
+
array.js
/* 在数组中查找指定元素 */
+function find(nums, target) {
+    for (let i = 0; i < nums.length; i++) {
+        if (nums[i] == target) return i;
+    }
+    return -1;
+}
+
+
+
+
array.ts
/* 在数组中查找指定元素 */
+function find(nums: number[], target: number): number {
+    for (let i = 0; i < nums.length; i++) {
+        if (nums[i] === target) {
+            return i;
+        }
+    }
+    return -1;
+}
+
+
+
+
array.c

+
+
+
+
array.cs
/* 在数组中查找指定元素 */
+int Find(int[] nums, int target)
+{
+    for (int i = 0; i < nums.Length; i++)
+    {
+        if (nums[i] == target)
+            return i;
+    }
+    return -1;
+}
+
+
+
+
array.swift
/* 在数组中查找指定元素 */
+func find(nums: [Int], target: Int) -> Int {
+    for i in nums.indices {
+        if nums[i] == target {
+            return i
+        }
+    }
+    return -1
+}
+
+
+
+
array.zig
// 在数组中查找指定元素
+pub fn find(nums: []i32, target: i32) i32 {
+    for (nums) |num, i| {
+        if (num == target) return @intCast(i32, i);
+    }
+    return -1;
+}
+
+
+
+
+

4.1.4. 数组典型应用

+

随机访问。如果我们想要随机抽取一些样本,那么可以用数组存储,并生成一个随机序列,根据索引实现样本的随机抽取。

+

二分查找。例如前文查字典的例子,我们可以将字典中的所有字按照拼音顺序存储在数组中,然后使用与日常查纸质字典相同的“翻开中间,排除一半”的方式,来实现一个查电子字典的算法。

+

深度学习。神经网络中大量使用了向量、矩阵、张量之间的线性代数运算,这些数据都是以数组的形式构建的。数组是神经网络编程中最常使用的数据结构。

+ + + + + +

评论

+ + + + + + +
+
+ + + + +
+ + + +
+ + + +
+
+
+
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4.2. 链表

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引言

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内存空间是所有程序的公共资源,排除已占用的内存,空闲内存往往是散落在内存各处的。我们知道,存储数组需要内存空间连续,当我们需要申请一个很大的数组时,系统不一定存在这么大的连续内存空间。而链表则更加灵活,不需要内存是连续的,只要剩余内存空间大小够用即可。

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「链表 Linked List」是一种线性数据结构,其中每个元素都是单独的对象,各个元素(一般称为结点)之间通过指针连接。由于结点中记录了连接关系,因此链表的存储方式相比于数组更加灵活,系统不必保证内存地址的连续性。

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链表的「结点 Node」包含两项数据,一是结点「值 Value」,二是指向下一结点的「指针 Pointer」(或称「引用 Reference」)。

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linkedlist_definition

+

Fig. 链表定义与存储方式

+ +
+
+
+
/* 链表结点类 */
+class ListNode {
+    int val;        // 结点值
+    ListNode next;  // 指向下一结点的指针(引用)
+    ListNode(int x) { val = x; }  // 构造函数
+}
+
+
+
+
/* 链表结点结构体 */
+struct ListNode {
+    int val;         // 结点值
+    ListNode *next;  // 指向下一结点的指针(引用)
+    ListNode(int x) : val(x), next(nullptr) {}  // 构造函数
+};
+
+
+
+
""" 链表结点类 """ 
+class ListNode:
+    def __init__(self, x):
+        self.val = x      # 结点值
+        self.next = None  # 指向下一结点的指针(引用)
+
+
+
+
/* 链表结点结构体 */
+type ListNode struct {
+    Val  int       // 结点值
+    Next *ListNode // 指向下一结点的指针(引用)
+}
+
+// NewListNode 构造函数,创建一个新的链表
+func NewListNode(val int) *ListNode {
+    return &ListNode{
+        Val:  val,
+        Next: nil,
+    }
+}
+
+
+
+
/* 链表结点结构体 */
+class ListNode {
+    val;
+    next;
+    constructor(val, next) {
+        this.val = (val === undefined ? 0 : val);       // 结点值
+        this.next = (next === undefined ? null : next); // 指向下一结点的引用
+    }
+}
+
+
+
+
/* 链表结点结构体 */
+class ListNode {
+    val: number;
+    next: ListNode | null;
+    constructor(val?: number, next?: ListNode | null) {
+        this.val = val === undefined ? 0 : val;        // 结点值
+        this.next = next === undefined ? null : next;  // 指向下一结点的引用
+    }
+}
+
+
+
+

+
+
+
+
/* 链表结点类 */
+class ListNode
+{
+    int val;         // 结点值
+    ListNode next;   // 指向下一结点的引用
+    ListNode(int x) => val = x;  //构造函数
+}
+
+
+
+
/* 链表结点类 */
+class ListNode {
+    var val: Int // 结点值
+    var next: ListNode? // 指向下一结点的指针(引用)
+
+    init(x: Int) { // 构造函数
+        val = x
+    }
+}
+
+
+
+
// 链表结点类
+pub fn ListNode(comptime T: type) type {
+    return struct {
+        const Self = @This();
+
+        val: T = 0, // 结点值
+        next: ?*Self = null, // 指向下一结点的指针(引用)
+
+        // 构造函数
+        pub fn init(self: *Self, x: i32) void {
+            self.val = x;
+            self.next = null;
+        }
+    };
+}
+
+
+
+
+

尾结点指向什么? 我们一般将链表的最后一个结点称为「尾结点」,其指向的是「空」,在 Java / C++ / Python 中分别记为 null / nullptr / None 。在不引起歧义下,本书都使用 null 来表示空。

+

链表初始化方法。建立链表分为两步,第一步是初始化各个结点对象,第二步是构建引用指向关系。完成后,即可以从链表的首个结点(即头结点)出发,访问其余所有的结点。

+
+

Tip

+

我们通常将头结点当作链表的代称,例如头结点 head 和链表 head 实际上是同义的。

+
+
+
+
+
linked_list.java
/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */
+// 初始化各个结点 
+ListNode n0 = new ListNode(1);
+ListNode n1 = new ListNode(3);
+ListNode n2 = new ListNode(2);
+ListNode n3 = new ListNode(5);
+ListNode n4 = new ListNode(4);
+// 构建引用指向
+n0.next = n1;
+n1.next = n2;
+n2.next = n3;
+n3.next = n4;
+
+
+
+
linked_list.cpp
/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */
+// 初始化各个结点 
+ListNode* n0 = new ListNode(1);
+ListNode* n1 = new ListNode(3);
+ListNode* n2 = new ListNode(2);
+ListNode* n3 = new ListNode(5);
+ListNode* n4 = new ListNode(4);
+// 构建引用指向
+n0->next = n1;
+n1->next = n2;
+n2->next = n3;
+n3->next = n4;
+
+
+
+
linked_list.py
""" 初始化链表 1 -> 3 -> 2 -> 5 -> 4 """
+# 初始化各个结点 
+n0 = ListNode(1)
+n1 = ListNode(3)
+n2 = ListNode(2)
+n3 = ListNode(5)
+n4 = ListNode(4)
+# 构建引用指向
+n0.next = n1
+n1.next = n2
+n2.next = n3
+n3.next = n4
+
+
+
+
linked_list.go
/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */
+// 初始化各个结点
+n0 := NewListNode(1)
+n1 := NewListNode(3)
+n2 := NewListNode(2)
+n3 := NewListNode(5)
+n4 := NewListNode(4)
+
+// 构建引用指向
+n0.Next = n1
+n1.Next = n2
+n2.Next = n3
+n3.Next = n4
+
+
+
+
linked_list.js
/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */
+// 初始化各个结点
+const n0 = new ListNode(1);
+const n1 = new ListNode(3);
+const n2 = new ListNode(2);
+const n3 = new ListNode(5);
+const n4 = new ListNode(4);
+// 构建引用指向
+n0.next = n1;
+n1.next = n2;
+n2.next = n3;
+n3.next = n4;
+
+
+
+
linked_list.ts
/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */
+// 初始化各个结点
+const n0 = new ListNode(1);
+const n1 = new ListNode(3);
+const n2 = new ListNode(2);
+const n3 = new ListNode(5);
+const n4 = new ListNode(4);
+// 构建引用指向
+n0.next = n1;
+n1.next = n2;
+n2.next = n3;
+n3.next = n4;
+
+
+
+
linked_list.c

+
+
+
+
linked_list.cs
/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */
+// 初始化各个结点 
+ListNode n0 = new ListNode(1);
+ListNode n1 = new ListNode(3);
+ListNode n2 = new ListNode(2);
+ListNode n3 = new ListNode(5);
+ListNode n4 = new ListNode(4);
+// 构建引用指向
+n0.next = n1;
+n1.next = n2;
+n2.next = n3;
+n3.next = n4;
+
+
+
+
linked_list.swift
/* 初始化链表 1 -> 3 -> 2 -> 5 -> 4 */
+// 初始化各个结点
+let n0 = ListNode(x: 1)
+let n1 = ListNode(x: 3)
+let n2 = ListNode(x: 2)
+let n3 = ListNode(x: 5)
+let n4 = ListNode(x: 4)
+// 构建引用指向
+n0.next = n1
+n1.next = n2
+n2.next = n3
+n3.next = n4
+
+
+
+
linked_list.zig
// 初始化链表
+// 初始化各个结点 
+var n0 = inc.ListNode(i32){.val = 1};
+var n1 = inc.ListNode(i32){.val = 3};
+var n2 = inc.ListNode(i32){.val = 2};
+var n3 = inc.ListNode(i32){.val = 5};
+var n4 = inc.ListNode(i32){.val = 4};
+// 构建引用指向
+n0.next = &n1;
+n1.next = &n2;
+n2.next = &n3;
+n3.next = &n4;
+
+
+
+
+

4.2.1. 链表优点

+

在链表中,插入与删除结点的操作效率高。例如,如果想在链表中间的两个结点 A , B 之间插入一个新结点 P ,我们只需要改变两个结点指针即可,时间复杂度为 \(O(1)\) ,相比数组的插入操作高效很多。在链表中删除某个结点也很方便,只需要改变一个结点指针即可。

+

linkedlist_insert_remove_node

+

Fig. 在链表中插入与删除结点

+ +
+
+
+
linked_list.java
/* 在链表的结点 n0 之后插入结点 P */
+void insert(ListNode n0, ListNode P) {
+    ListNode n1 = n0.next;
+    n0.next = P;
+    P.next = n1;
+}
+
+/* 删除链表的结点 n0 之后的首个结点 */
+void remove(ListNode n0) {
+    if (n0.next == null)
+        return;
+    // n0 -> P -> n1
+    ListNode P = n0.next;
+    ListNode n1 = P.next;
+    n0.next = n1;
+}
+
+
+
+
linked_list.cpp
/* 在链表的结点 n0 之后插入结点 P */
+void insert(ListNode* n0, ListNode* P) {
+    ListNode* n1 = n0->next;
+    n0->next = P;
+    P->next = n1;
+}
+
+/* 删除链表的结点 n0 之后的首个结点 */
+void remove(ListNode* n0) {
+    if (n0->next == nullptr)
+        return;
+    // n0 -> P -> n1
+    ListNode* P = n0->next;
+    ListNode* n1 = P->next;
+    n0->next = n1;
+    // 释放内存
+    delete P;
+}
+
+
+
+
linked_list.py
""" 在链表的结点 n0 之后插入结点 P """
+def insert(n0, P):
+    n1 = n0.next
+    n0.next = P
+    P.next = n1
+
+""" 删除链表的结点 n0 之后的首个结点 """
+def remove(n0):
+    if not n0.next:
+        return
+    # n0 -> P -> n1
+    P = n0.next
+    n1 = P.next
+    n0.next = n1
+
+
+
+
linked_list.go
/* 在链表的结点 n0 之后插入结点 P */
+func insert(n0 *ListNode, P *ListNode) {
+    n1 := n0.Next
+    n0.Next = P
+    P.Next = n1
+}
+
+/* 删除链表的结点 n0 之后的首个结点 */
+func removeNode(n0 *ListNode) {
+    if n0.Next == nil {
+        return
+    }
+    // n0 -> P -> n1
+    P := n0.Next
+    n1 := P.Next
+    n0.Next = n1
+}
+
+
+
+
linked_list.js
/* 在链表的结点 n0 之后插入结点 P */
+function insert(n0, P) {
+    let n1 = n0.next;
+    n0.next = P;
+    P.next = n1;
+}
+
+/* 删除链表的结点 n0 之后的首个结点 */
+function remove(n0) {
+    if (!n0.next)
+        return;
+    // n0 -> P -> n1
+    let P = n0.next;
+    let n1 = P.next;
+    n0.next = n1;
+}
+
+
+
+
linked_list.ts
/* 在链表的结点 n0 之后插入结点 P */
+function insert(n0: ListNode, P: ListNode): void {
+    const n1 = n0.next;
+    n0.next = P;
+    P.next = n1;
+}
+
+/* 删除链表的结点 n0 之后的首个结点 */
+function remove(n0: ListNode): void {
+    if (!n0.next) {
+        return;
+    }
+    // n0 -> P -> n1
+    const P = n0.next;
+    const n1 = P.next;
+    n0.next = n1;
+}
+
+
+
+
linked_list.c

+
+
+
+
linked_list.cs
// 在链表的结点 n0 之后插入结点 P
+void Insert(ListNode n0, ListNode P)
+{
+    ListNode n1 = n0.next;
+    n0.next = P;
+    P.next = n1;
+}
+
+// 删除链表的结点 n0 之后的首个结点
+void Remove(ListNode n0)
+{
+    if (n0.next == null)
+        return;
+    // n0 -> P -> n1
+    ListNode P = n0.next;
+    ListNode n1 = P.next;
+    n0.next = n1;
+}
+
+
+
+
linked_list.swift
/* 在链表的结点 n0 之后插入结点 P */
+func insert(n0: ListNode, P: ListNode) {
+    let n1 = n0.next
+    n0.next = P
+    P.next = n1
+}
+
+/* 删除链表的结点 n0 之后的首个结点 */
+func remove(n0: ListNode) {
+    if n0.next == nil {
+        return
+    }
+    // n0 -> P -> n1
+    let P = n0.next
+    let n1 = P?.next
+    n0.next = n1
+    P?.next = nil
+}
+
+
+
+
linked_list.zig
// 在链表的结点 n0 之后插入结点 P
+pub fn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {
+    var n1 = n0.?.next;
+    n0.?.next = P;
+    P.?.next = n1;
+}
+
+// 删除链表的结点 n0 之后的首个结点
+pub fn remove(n0: ?*inc.ListNode(i32)) void {
+    if (n0.?.next == null) return;
+    // n0 -> P -> n1
+    var P = n0.?.next;
+    var n1 = P.?.next;
+    n0.?.next = n1;
+}
+
+
+
+
+

4.2.2. 链表缺点

+

链表访问结点效率低。上节提到,数组可以在 \(O(1)\) 时间下访问任意元素,但链表无法直接访问任意结点。这是因为计算机需要从头结点出发,一个一个地向后遍历到目标结点。例如,倘若想要访问链表索引为 index (即第 index + 1 个)的结点,那么需要 index 次访问操作。

+
+
+
+
linked_list.java
/* 访问链表中索引为 index 的结点 */
+ListNode access(ListNode head, int index) {
+    for (int i = 0; i < index; i++) {
+        if (head == null)
+            return null;
+        head = head.next;
+    }
+    return head;
+}
+
+
+
+
linked_list.cpp
/* 访问链表中索引为 index 的结点 */
+ListNode* access(ListNode* head, int index) {
+    for (int i = 0; i < index; i++) {
+        if (head == nullptr)
+            return nullptr;
+        head = head->next;
+    }
+    return head;
+}
+
+
+
+
linked_list.py
""" 访问链表中索引为 index 的结点 """
+def access(head, index):
+    for _ in range(index):
+        if not head:
+            return None
+        head = head.next
+    return head
+
+
+
+
linked_list.go
/* 访问链表中索引为 index 的结点 */
+func access(head *ListNode, index int) *ListNode {
+    for i := 0; i < index; i++ {
+        if head == nil {
+            return nil
+        }
+        head = head.Next
+    }
+    return head
+}
+
+
+
+
linked_list.js
/* 访问链表中索引为 index 的结点 */
+function access(head, index) {
+    for (let i = 0; i < index; i++) {
+        if (!head)
+            return null;
+        head = head.next;
+    }
+    return head;
+}
+
+
+
+
linked_list.ts
/* 访问链表中索引为 index 的结点 */
+function access(head: ListNode | null, index: number): ListNode | null {
+    for (let i = 0; i < index; i++) {
+        if (!head) {
+            return null;
+        }
+        head = head.next;
+    }
+    return head;
+}
+
+
+
+
linked_list.c

+
+
+
+
linked_list.cs
// 访问链表中索引为 index 的结点
+ListNode Access(ListNode head, int index)
+{
+    for (int i = 0; i < index; i++)
+    {
+        if (head == null)
+            return null;
+        head = head.next;
+    }
+    return head;
+}
+
+
+
+
linked_list.swift
/* 访问链表中索引为 index 的结点 */
+func access(head: ListNode, index: Int) -> ListNode? {
+    var head: ListNode? = head
+    for _ in 0 ..< index {
+        if head == nil {
+            return nil
+        }
+        head = head?.next
+    }
+    return head
+}
+
+
+
+
linked_list.zig
// 访问链表中索引为 index 的结点
+pub fn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {
+    var head = node;
+    var i: i32 = 0;
+    while (i < index) : (i += 1) {
+        head = head.?.next;
+        if (head == null) return null;
+    }
+    return head;
+}
+
+
+
+
+

链表的内存占用多。链表以结点为单位,每个结点除了保存值外,还需额外保存指针(引用)。这意味着同样数据量下,链表比数组需要占用更多内存空间。

+

4.2.3. 链表常用操作

+

遍历链表查找。遍历链表,查找链表内值为 target 的结点,输出结点在链表中的索引。

+
+
+
+
linked_list.java
/* 在链表中查找值为 target 的首个结点 */
+int find(ListNode head, int target) {
+    int index = 0;
+    while (head != null) {
+        if (head.val == target)
+            return index;
+        head = head.next;
+        index++;
+    }
+    return -1;
+}
+
+
+
+
linked_list.cpp
/* 在链表中查找值为 target 的首个结点 */
+int find(ListNode* head, int target) {
+    int index = 0;
+    while (head != nullptr) {
+        if (head->val == target)
+            return index;
+        head = head->next;
+        index++;
+    }
+    return -1;
+}
+
+
+
+
linked_list.py
""" 在链表中查找值为 target 的首个结点 """
+def find(head, target):
+    index = 0
+    while head:
+        if head.val == target:
+            return index
+        head = head.next
+        index += 1
+    return -1
+
+
+
+
linked_list.go
/* 在链表中查找值为 target 的首个结点 */
+func find(head *ListNode, target int) int {
+    index := 0
+    for head != nil {
+        if head.Val == target {
+            return index
+        }
+        head = head.Next
+        index++
+    }
+    return -1
+}
+
+
+
+
linked_list.js
/* 在链表中查找值为 target 的首个结点 */
+function find(head, target) {
+    let index = 0;
+    while (head !== null) {
+        if (head.val === target) {
+            return index;
+        }
+        head = head.next;
+        index += 1;
+    }
+    return -1;
+}
+
+
+
+
linked_list.ts
/* 在链表中查找值为 target 的首个结点 */
+function find(head: ListNode | null, target: number): number {
+    let index = 0;
+    while (head !== null) {
+        if (head.val === target) {
+            return index;
+        }
+        head = head.next;
+        index += 1;
+    }
+    return -1;
+}
+
+
+
+
linked_list.c

+
+
+
+
linked_list.cs
// 在链表中查找值为 target 的首个结点
+int Find(ListNode head, int target)
+{
+    int index = 0;
+    while (head != null)
+    {
+        if (head.val == target)
+            return index;
+        head = head.next;
+        index++;
+    }
+    return -1;
+}
+
+
+
+
linked_list.swift
/* 在链表中查找值为 target 的首个结点 */
+func find(head: ListNode, target: Int) -> Int {
+    var head: ListNode? = head
+    var index = 0
+    while head != nil {
+        if head?.val == target {
+            return index
+        }
+        head = head?.next
+        index += 1
+    }
+    return -1
+}
+
+
+
+
linked_list.zig
// 在链表中查找值为 target 的首个结点
+pub fn find(node: ?*inc.ListNode(i32), target: i32) i32 {
+    var head = node;
+    var index: i32 = 0;
+    while (head != null) {
+        if (head.?.val == target) return index;
+        head = head.?.next;
+        index += 1;
+    }
+    return -1;
+}
+
+
+
+
+

4.2.4. 常见链表类型

+

单向链表。即上述介绍的普通链表。单向链表的结点有「值」和指向下一结点的「指针(引用)」两项数据。我们将首个结点称为头结点,尾结点指向 null

+

环形链表。如果我们令单向链表的尾结点指向头结点(即首尾相接),则得到一个环形链表。在环形链表中,我们可以将任意结点看作是头结点。

+

双向链表。单向链表仅记录了一个方向的指针(引用),在双向链表的结点定义中,同时有指向下一结点(后继结点)和上一结点(前驱结点)的「指针(引用)」。双向链表相对于单向链表更加灵活,即可以朝两个方向遍历链表,但也需要占用更多的内存空间。

+
+
+
+
/* 双向链表结点类 */
+class ListNode {
+    int val;        // 结点值
+    ListNode next;  // 指向后继结点的指针(引用)
+    ListNode prev;  // 指向前驱结点的指针(引用)
+    ListNode(int x) { val = x; }  // 构造函数
+}
+
+
+
+
/* 链表结点结构体 */
+struct ListNode {
+    int val;         // 结点值
+    ListNode *next;  // 指向后继结点的指针(引用)
+    ListNode *prev;  // 指向前驱结点的指针(引用)
+    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // 构造函数
+};
+
+
+
+
""" 双向链表结点类 """ 
+class ListNode:
+    def __init__(self, x):
+        self.val = x      # 结点值
+        self.next = None  # 指向后继结点的指针(引用)
+        self.prev = None  # 指向前驱结点的指针(引用)
+
+
+
+
/* 双向链表结点结构体 */
+type DoublyListNode struct {
+    Val  int             // 结点值
+    Next *DoublyListNode // 指向后继结点的指针(引用)
+    Prev *DoublyListNode // 指向前驱结点的指针(引用)
+}
+
+// NewDoublyListNode 初始化
+func NewDoublyListNode(val int) *DoublyListNode {
+    return &DoublyListNode{
+        Val:  val,
+        Next: nil,
+        Prev: nil,
+    }
+}
+
+
+
+
/* 双向链表结点类 */
+class ListNode {
+    val;
+    next;
+    prev;
+    constructor(val, next) {
+        this.val = val  ===  undefined ? 0 : val;        // 结点值
+        this.next = next  ===  undefined ? null : next;  // 指向后继结点的指针(引用)
+        this.prev = prev  ===  undefined ? null : prev;  // 指向前驱结点的指针(引用)
+    }
+}
+
+
+
+
/* 双向链表结点类 */
+class ListNode {
+    val: number;
+    next: ListNode | null;
+    prev: ListNode | null;
+    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {
+        this.val = val  ===  undefined ? 0 : val;        // 结点值
+        this.next = next  ===  undefined ? null : next;  // 指向后继结点的指针(引用)
+        this.prev = prev  ===  undefined ? null : prev;  // 指向前驱结点的指针(引用)
+    }
+}
+
+
+
+

+
+
+
+
/* 双向链表结点类 */
+class ListNode {
+    int val;        // 结点值
+    ListNode next;  // 指向后继结点的指针(引用)
+    ListNode prev;  // 指向前驱结点的指针(引用)
+    ListNode(int x) => val = x;  // 构造函数
+}
+
+
+
+
/* 双向链表结点类 */
+class ListNode {
+    var val: Int // 结点值
+    var next: ListNode? // 指向后继结点的指针(引用)
+    var prev: ListNode? // 指向前驱结点的指针(引用)
+
+    init(x: Int) { // 构造函数
+        val = x
+    }
+}
+
+
+
+
// 双向链表结点类
+pub fn ListNode(comptime T: type) type {
+    return struct {
+        const Self = @This();
+
+        val: T = 0, // 结点值
+        next: ?*Self = null, // 指向后继结点的指针(引用)
+        prev: ?*Self = null, // 指向前驱结点的指针(引用)
+
+        // 构造函数
+        pub fn init(self: *Self, x: i32) void {
+            self.val = x;
+            self.next = null;
+            self.prev = null;
+        }
+    };
+}
+
+
+
+
+

linkedlist_common_types

+

Fig. 常见链表类型

+ + + + + +

评论

+ + + + + + +
+
+ + + + +
+ + + +
+ + + +
+
+
+
+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/chapter_array_and_linkedlist/list/index.html b/chapter_array_and_linkedlist/list/index.html new file mode 100644 index 000000000..2c0105cd4 --- /dev/null +++ b/chapter_array_and_linkedlist/list/index.html @@ -0,0 +1,3231 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + 4.3. 列表(List) - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + 跳转至 + + +
+
+ +
+ + + + +
+ + +
+ +
+ + + + + + +
+
+ + + +
+
+
+ + + + +
+
+
+ + + +
+
+
+ + + +
+
+
+ + + +
+
+ + + + + + + + + + + + + +

4.3. 列表

+

由于长度不可变,数组的实用性大大降低。在很多情况下,我们事先并不知道会输入多少数据,这就为数组长度的选择带来了很大困难。长度选小了,需要在添加数据中频繁地扩容数组;长度选大了,又造成内存空间的浪费。

+

为了解决此问题,诞生了一种被称为「列表 List」的数据结构。列表可以被理解为长度可变的数组,因此也常被称为「动态数组 Dynamic Array」。列表基于数组实现,继承了数组的优点,同时还可以在程序运行中实时扩容。在列表中,我们可以自由地添加元素,而不用担心超过容量限制。

+

4.3.1. 列表常用操作

+

初始化列表。我们通常会使用到“无初始值”和“有初始值”的两种初始化方法。

+
+
+
+
list.java
/* 初始化列表 */
+// 无初始值
+List<Integer> list1 = new ArrayList<>();
+// 有初始值(注意数组的元素类型需为 int[] 的包装类 Integer[])
+Integer[] numbers = new Integer[] { 1, 3, 2, 5, 4 };
+List<Integer> list = new ArrayList<>(Arrays.asList(numbers));
+
+
+
+
list.cpp
/* 初始化列表 */
+// 需注意,C++ 中 vector 即是本文描述的 list
+// 无初始值
+vector<int> list1;
+// 有初始值
+vector<int> list = { 1, 3, 2, 5, 4 };
+
+
+
+
list.py
""" 初始化列表 """
+# 无初始值
+list1 = []
+# 有初始值
+list = [1, 3, 2, 5, 4]
+
+
+
+
list_test.go
/* 初始化列表 */
+// 无初始值
+list1 := []int
+// 有初始值
+list := []int{1, 3, 2, 5, 4}
+
+
+
+
list.js
/* 初始化列表 */
+// 无初始值
+const list1 = [];
+// 有初始值
+const list = [1, 3, 2, 5, 4];
+
+
+
+
list.ts
/* 初始化列表 */
+// 无初始值
+const list1: number[] = [];
+// 有初始值
+const list: number[] = [1, 3, 2, 5, 4];
+
+
+
+
list.c

+
+
+
+
list.cs
/* 初始化列表 */
+// 无初始值
+List<int> list1 = new ();
+// 有初始值(注意数组的元素类型需为 int[] 的包装类 Integer[])
+int[] numbers = new int[] { 1, 3, 2, 5, 4 };
+List<int> list = numbers.ToList();
+
+
+
+
list.swift
/* 初始化列表 */
+// 无初始值
+let list1: [Int] = []
+// 有初始值
+var list = [1, 3, 2, 5, 4]
+
+
+
+
list.zig
// 初始化列表
+var list = std.ArrayList(i32).init(std.heap.page_allocator);
+defer list.deinit();
+try list.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 });
+
+
+
+
+

访问与更新元素。列表的底层数据结构是数组,因此可以在 \(O(1)\) 时间内访问与更新元素,效率很高。

+
+
+
+
list.java
/* 访问元素 */
+int num = list.get(1);  // 访问索引 1 处的元素
+
+/* 更新元素 */
+list.set(1, 0);  // 将索引 1 处的元素更新为 0
+
+
+
+
list.cpp
/* 访问元素 */
+int num = list[1];  // 访问索引 1 处的元素
+
+/* 更新元素 */
+list[1] = 0;  // 将索引 1 处的元素更新为 0
+
+
+
+
list.py
""" 访问元素 """
+num = list[1]  # 访问索引 1 处的元素
+
+""" 更新元素 """
+list[1] = 0    # 将索引 1 处的元素更新为 0
+
+
+
+
list_test.go
/* 访问元素 */
+num := list[1]  // 访问索引 1 处的元素
+
+/* 更新元素 */
+list[1] = 0     // 将索引 1 处的元素更新为 0
+
+
+
+
list.js
/* 访问元素 */
+const num = list[1];  // 访问索引 1 处的元素
+
+/* 更新元素 */
+list[1] = 0;  // 将索引 1 处的元素更新为 0
+
+
+
+
list.ts
/* 访问元素 */
+const num: number = list[1];  // 访问索引 1 处的元素
+
+/* 更新元素 */
+list[1] = 0;  // 将索引 1 处的元素更新为 0
+
+
+
+
list.c

+
+
+
+
list.cs
/* 访问元素 */
+int num = list[1];  // 访问索引 1 处的元素
+
+/* 更新元素 */
+list[1] = 0;  // 将索引 1 处的元素更新为 0
+
+
+
+
list.swift
/* 访问元素 */
+let num = list[1] // 访问索引 1 处的元素
+
+/* 更新元素 */
+list[1] = 0 // 将索引 1 处的元素更新为 0
+
+
+
+
list.zig
// 访问元素
+var num = list.items[1]; // 访问索引 1 处的元素
+
+// 更新元素
+list.items[1] = 0; // 将索引 1 处的元素更新为 0  
+
+
+
+
+

在列表中添加、插入、删除元素。相对于数组,列表可以自由地添加与删除元素。在列表尾部添加元素的时间复杂度为 \(O(1)\) ,但是插入与删除元素的效率仍与数组一样低,时间复杂度为 \(O(N)\)

+
+
+
+
list.java
/* 清空列表 */
+list.clear();
+
+/* 尾部添加元素 */
+list.add(1);
+list.add(3);
+list.add(2);
+list.add(5);
+list.add(4);
+
+/* 中间插入元素 */
+list.add(3, 6);  // 在索引 3 处插入数字 6
+
+/* 删除元素 */
+list.remove(3);  // 删除索引 3 处的元素
+
+
+
+
list.cpp
/* 清空列表 */
+list.clear();
+
+/* 尾部添加元素 */
+list.push_back(1);
+list.push_back(3);
+list.push_back(2);
+list.push_back(5);
+list.push_back(4);
+
+/* 中间插入元素 */
+list.insert(list.begin() + 3, 6);  // 在索引 3 处插入数字 6
+
+/* 删除元素 */
+list.erase(list.begin() + 3);      // 删除索引 3 处的元素
+
+
+
+
list.py
""" 清空列表 """
+list.clear()
+
+""" 尾部添加元素 """
+list.append(1)
+list.append(3)
+list.append(2)
+list.append(5)
+list.append(4)
+
+""" 中间插入元素 """
+list.insert(3, 6)  # 在索引 3 处插入数字 6
+
+""" 删除元素 """
+list.pop(3)        # 删除索引 3 处的元素
+
+
+
+
list_test.go
/* 清空列表 */
+list = nil
+
+/* 尾部添加元素 */
+list = append(list, 1)
+list = append(list, 3)
+list = append(list, 2)
+list = append(list, 5)
+list = append(list, 4)
+
+/* 中间插入元素 */
+list = append(list[:3], append([]int{6}, list[3:]...)...) // 在索引 3 处插入数字 6
+
+/* 删除元素 */
+list = append(list[:3], list[4:]...) // 删除索引 3 处的元素
+
+
+
+
list.js
/* 清空列表 */
+list.length = 0;
+
+/* 尾部添加元素 */
+list.push(1);
+list.push(3);
+list.push(2);
+list.push(5);
+list.push(4);
+
+/* 中间插入元素 */
+list.splice(3, 0, 6);
+
+/* 删除元素 */
+list.splice(3, 1);
+
+
+
+
list.ts
/* 清空列表 */
+list.length = 0;
+
+/* 尾部添加元素 */
+list.push(1);
+list.push(3);
+list.push(2);
+list.push(5);
+list.push(4);
+
+/* 中间插入元素 */
+list.splice(3, 0, 6);
+
+/* 删除元素 */
+list.splice(3, 1);
+
+
+
+
list.c

+
+
+
+
list.cs
/* 清空列表 */
+list.Clear();
+
+/* 尾部添加元素 */
+list.Add(1);
+list.Add(3);
+list.Add(2);
+list.Add(5);
+list.Add(4);
+
+/* 中间插入元素 */
+list.Insert(3, 6);
+
+/* 删除元素 */
+list.RemoveAt(3);
+
+
+
+
list.swift
/* 清空列表 */
+list.removeAll()
+
+/* 尾部添加元素 */
+list.append(1)
+list.append(3)
+list.append(2)
+list.append(5)
+list.append(4)
+
+/* 中间插入元素 */
+list.insert(6, at: 3) // 在索引 3 处插入数字 6
+
+/* 删除元素 */
+list.remove(at: 3) // 删除索引 3 处的元素
+
+
+
+
list.zig
// 清空列表
+list.clearRetainingCapacity();
+
+// 尾部添加元素
+try list.append(1);
+try list.append(3);
+try list.append(2);
+try list.append(5);
+try list.append(4);
+
+// 中间插入元素
+try list.insert(3, 6); // 在索引 3 处插入数字 6
+
+// 删除元素
+_ = list.orderedRemove(3); // 删除索引 3 处的元素
+
+
+
+
+

遍历列表。与数组一样,列表可以使用索引遍历,也可以使用 for-each 直接遍历。

+
+
+
+
list.java
/* 通过索引遍历列表 */
+int count = 0;
+for (int i = 0; i < list.size(); i++) {
+    count++;
+}
+
+/* 直接遍历列表元素 */
+count = 0;
+for (int n : list) {
+    count++;
+}
+
+
+
+
list.cpp
/* 通过索引遍历列表 */
+int count = 0;
+for (int i = 0; i < list.size(); i++) {
+    count++;
+}
+
+/* 直接遍历列表元素 */
+count = 0;
+for (int n : list) {
+    count++;
+}
+
+
+
+
list.py
""" 通过索引遍历列表 """
+count = 0
+for i in range(len(list)):
+    count += 1
+
+""" 直接遍历列表元素 """
+count = 0
+for n in list:
+    count += 1
+
+
+
+
list_test.go
/* 通过索引遍历列表 */
+count := 0
+for i := 0; i < len(list); i++ {
+    count++
+}
+
+/* 直接遍历列表元素 */
+count = 0
+for range list {
+    count++
+}
+
+
+
+
list.js
/* 通过索引遍历列表 */
+let count = 0;
+for (let i = 0; i < list.length; i++) {
+    count++;
+}
+
+/* 直接遍历列表元素 */
+count = 0;
+for (const n of list) {
+    count++;
+}
+
+
+
+
list.ts
/* 通过索引遍历列表 */
+let count = 0;
+for (let i = 0; i < list.length; i++) {
+    count++;
+}
+
+/* 直接遍历列表元素 */
+count = 0;
+for (const n of list) {
+    count++;
+}
+
+
+
+
list.c

+
+
+
+
list.cs
/* 通过索引遍历列表 */
+int count = 0;
+for (int i = 0; i < list.Count(); i++)
+{
+    count++;
+}
+
+/* 直接遍历列表元素 */
+count = 0;
+foreach (int n in list)
+{
+    count++;
+}
+
+
+
+
list.swift
/* 通过索引遍历列表 */
+var count = 0
+for _ in list.indices {
+    count += 1
+}
+
+/* 直接遍历列表元素 */
+count = 0
+for _ in list {
+    count += 1
+}
+
+
+
+
list.zig
// 通过索引遍历列表
+var count: i32 = 0;
+var i: i32 = 0;
+while (i < list.items.len) : (i += 1) {
+    count += 1;
+}
+
+// 直接遍历列表元素
+count = 0;
+for (list.items) |_| {
+    count += 1;
+}
+
+
+
+
+

拼接两个列表。再创建一个新列表 list1 ,我们可以将其中一个列表拼接到另一个的尾部。

+
+
+
+
list.java
/* 拼接两个列表 */
+List<Integer> list1 = new ArrayList<>(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));
+list.addAll(list1);  // 将列表 list1 拼接到 list 之后
+
+
+
+
list.cpp
/* 拼接两个列表 */
+vector<int> list1 = { 6, 8, 7, 10, 9 };
+// 将列表 list1 拼接到 list 之后
+list.insert(list.end(), list1.begin(), list1.end());
+
+
+
+
list.py
""" 拼接两个列表 """
+list1 = [6, 8, 7, 10, 9]
+list += list1  # 将列表 list1 拼接到 list 之后
+
+
+
+
list_test.go
/* 拼接两个列表 */
+list1 := []int{6, 8, 7, 10, 9}
+list = append(list, list1...)  // 将列表 list1 拼接到 list 之后
+
+
+
+
list.js
/* 拼接两个列表 */
+const list1 = [6, 8, 7, 10, 9];
+list.push(...list1);  // 将列表 list1 拼接到 list 之后
+
+
+
+
list.ts
/* 拼接两个列表 */
+const list1: number[] = [6, 8, 7, 10, 9];
+list.push(...list1);  // 将列表 list1 拼接到 list 之后
+
+
+
+
list.c

+
+
+
+
list.cs
/* 拼接两个列表 */
+List<int> list1 = new() { 6, 8, 7, 10, 9 };
+list.AddRange(list1);  // 将列表 list1 拼接到 list 之后
+
+
+
+
list.swift
/* 拼接两个列表 */
+let list1 = [6, 8, 7, 10, 9]
+list.append(contentsOf: list1) // 将列表 list1 拼接到 list 之后
+
+
+
+
list.zig
// 拼接两个列表
+var list1 = std.ArrayList(i32).init(std.heap.page_allocator);
+defer list1.deinit();
+try list1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 });
+try list.insertSlice(list.items.len, list1.items); // 将列表 list1 拼接到 list 之后
+
+
+
+
+

排序列表。排序也是常用的方法之一,完成列表排序后,我们就可以使用在数组类算法题中经常考察的「二分查找」和「双指针」算法了。

+
+
+
+
list.java
/* 排序列表 */
+Collections.sort(list);  // 排序后,列表元素从小到大排列
+
+
+
+
list.cpp
/* 排序列表 */
+sort(list.begin(), list.end());  // 排序后,列表元素从小到大排列
+
+
+
+
list.py
""" 排序列表 """
+list.sort()  # 排序后,列表元素从小到大排列
+
+
+
+
list_test.go
/* 排序列表 */
+sort.Ints(list)  // 排序后,列表元素从小到大排列
+
+
+
+
list.js
/* 排序列表 */  
+list.sort((a, b) => a - b);  // 排序后,列表元素从小到大排列
+
+
+
+
list.ts
/* 排序列表 */
+list.sort((a, b) => a - b);  // 排序后,列表元素从小到大排列
+
+
+
+
list.c

+
+
+
+
list.cs
/* 排序列表 */
+list.Sort(); // 排序后,列表元素从小到大排列
+
+
+
+
list.swift
/* 排序列表 */
+list.sort() // 排序后,列表元素从小到大排列
+
+
+
+
list.zig
// 排序列表
+std.sort.sort(i32, list.items, {}, comptime std.sort.asc(i32));
+
+
+
+
+

4.3.2. 列表简易实现 *

+

为了帮助加深对列表的理解,我们在此提供一个列表的简易版本的实现。需要关注三个核心点:

+
    +
  • 初始容量:选取一个合理的数组的初始容量 initialCapacity 。在本示例中,我们选择 10 作为初始容量。
  • +
  • 数量记录:需要声明一个变量 size ,用来记录列表当前有多少个元素,并随着元素插入与删除实时更新。根据此变量,可以定位列表的尾部,以及判断是否需要扩容。
  • +
  • 扩容机制:插入元素有可能导致超出列表容量,此时需要扩容列表,方法是建立一个更大的数组来替换当前数组。需要给定一个扩容倍数 extendRatio ,在本示例中,我们规定每次将数组扩容至之前的 2 倍。
  • +
+

本示例是为了帮助读者对如何实现列表产生直观的认识。实际编程语言中,列表的实现远比以下代码复杂且标准,感兴趣的读者可以查阅源码学习。

+
+
+
+
my_list.java
/* 列表类简易实现 */
+class MyList {
+    private int[] nums;           // 数组(存储列表元素)
+    private int capacity = 10;    // 列表容量
+    private int size = 0;         // 列表长度(即当前元素数量)
+    private int extendRatio = 2;  // 每次列表扩容的倍数
+
+    /* 构造函数 */
+    public MyList() {
+        nums = new int[capacity];
+    }
+
+    /* 获取列表长度(即当前元素数量)*/
+    public int size() {
+        return size;
+    }
+
+    /* 获取列表容量 */
+    public int capacity() {
+        return capacity;
+    }
+
+    /* 访问元素 */
+    public int get(int index) {
+        // 索引如果越界则抛出异常,下同
+        if (index < 0 || index >= size)
+            throw new IndexOutOfBoundsException("索引越界");
+        return nums[index];
+    }
+
+    /* 更新元素 */
+    public void set(int index, int num) {
+        if (index < 0 || index >= size)
+            throw new IndexOutOfBoundsException("索引越界");
+        nums[index] = num;
+    }
+
+    /* 尾部添加元素 */
+    public void add(int num) {
+        // 元素数量超出容量时,触发扩容机制
+        if (size == capacity())
+            extendCapacity();
+        nums[size] = num;
+        // 更新元素数量
+        size++;
+    }
+
+    /* 中间插入元素 */
+    public void insert(int index, int num) {
+        if (index < 0 || index >= size)
+            throw new IndexOutOfBoundsException("索引越界");
+        // 元素数量超出容量时,触发扩容机制
+        if (size == capacity())
+            extendCapacity();
+        // 将索引 index 以及之后的元素都向后移动一位
+        for (int j = size - 1; j >= index; j--) {
+            nums[j + 1] = nums[j];
+        }
+        nums[index] = num;
+        // 更新元素数量
+        size++;
+    }
+
+    /* 删除元素 */
+    public int remove(int index) {
+        if (index < 0 || index >= size)
+            throw new IndexOutOfBoundsException("索引越界");
+        int num = nums[index];
+        // 将索引 index 之后的元素都向前移动一位
+        for (int j = index; j < size - 1; j++) {
+            nums[j] = nums[j + 1];
+        }
+        // 更新元素数量
+        size--;
+        // 返回被删除元素
+        return num;
+    }
+
+    /* 列表扩容 */
+    public void extendCapacity() {
+        // 新建一个长度为 size 的数组,并将原数组拷贝到新数组
+        nums = Arrays.copyOf(nums, capacity() * extendRatio);
+        // 更新列表容量
+        capacity = nums.length;
+    }
+
+    /* 将列表转换为数组 */
+    public int[] toArray() {
+        int size = size();
+        // 仅转换有效长度范围内的列表元素
+        int[] nums = new int[size];
+        for (int i = 0; i < size; i++) {
+            nums[i] = get(i);
+        }
+        return nums;
+    }
+}
+
+
+
+
my_list.cpp
/* 列表类简易实现 */
+class MyList {
+private:
+    int* nums;                // 数组(存储列表元素)
+    int numsCapacity = 10;    // 列表容量
+    int numsSize = 0;         // 列表长度(即当前元素数量)
+    int extendRatio = 2;      // 每次列表扩容的倍数
+
+public:
+    /* 构造函数 */
+    MyList() {
+        nums = new int[numsCapacity];
+    }
+
+    /* 析构函数 */
+    ~MyList() {
+        delete[] nums;
+    }
+
+    /* 获取列表长度(即当前元素数量)*/
+    int size() {
+        return numsSize;
+    }
+
+    /* 获取列表容量 */
+    int capacity() {
+        return numsCapacity;
+    }
+
+    /* 访问元素 */
+    int get(int index) {
+        // 索引如果越界则抛出异常,下同
+        if (index < 0 || index >= size())
+            throw out_of_range("索引越界");
+        return nums[index];
+    }
+
+    /* 更新元素 */
+    void set(int index, int num) {
+        if (index < 0 || index >= size())
+            throw out_of_range("索引越界");
+        nums[index] = num;
+    }
+
+    /* 尾部添加元素 */
+    void add(int num) {
+        // 元素数量超出容量时,触发扩容机制
+        if (size() == capacity())
+            extendCapacity();
+        nums[size()] = num;
+        // 更新元素数量
+        numsSize++;
+    }
+
+    /* 中间插入元素 */
+    void insert(int index, int num) {
+        if (index < 0 || index >= size())
+            throw out_of_range("索引越界");
+        // 元素数量超出容量时,触发扩容机制
+        if (size() == capacity())
+            extendCapacity();
+        // 索引 i 以及之后的元素都向后移动一位
+        for (int j = size() - 1; j >= index; j--) {
+            nums[j + 1] = nums[j];
+        }
+        nums[index] = num;
+        // 更新元素数量
+        numsSize++;
+    }
+
+    /* 删除元素 */
+    int remove(int index) {
+        if (index < 0 || index >= size())
+            throw out_of_range("索引越界");
+        int num = nums[index];
+        // 索引 i 之后的元素都向前移动一位
+        for (int j = index; j < size() - 1; j++) {
+            nums[j] = nums[j + 1];
+        }
+        // 更新元素数量
+        numsSize--;
+        // 返回被删除元素
+        return num;
+    }
+
+    /* 列表扩容 */
+    void extendCapacity() {
+        // 新建一个长度为 size * extendRatio 的数组,并将原数组拷贝到新数组
+        int newCapacity = capacity() * extendRatio;
+        int* tmp = nums;
+        nums = new int[newCapacity];
+        // 将原数组中的所有元素复制到新数组
+        for (int i = 0; i < size(); i++) {
+            nums[i] = tmp[i];
+        }
+        // 释放内存
+        delete[] tmp;
+        numsCapacity = newCapacity;
+    }
+
+    /* 将列表转换为 Vector 用于打印 */
+    vector<int> toVector() {
+        // 仅转换有效长度范围内的列表元素
+        vector<int> vec(size());
+        for (int i = 0; i < size(); i++) {
+            vec[i] = nums[i];
+        }
+        return vec;
+    }
+};
+
+
+
+
my_list.py
""" 列表类简易实现 """
+class MyList:
+    """ 构造函数 """
+    def __init__(self):
+        self.__capacity = 10                 # 列表容量
+        self.__nums = [0] * self.__capacity  # 数组(存储列表元素)
+        self.__size = 0                      # 列表长度(即当前元素数量)
+        self.__extend_ratio = 2              # 每次列表扩容的倍数
+
+    """ 获取列表长度(即当前元素数量) """
+    def size(self):
+        return self.__size
+
+    """ 获取列表容量 """
+    def capacity(self):
+        return self.__capacity
+
+    """ 访问元素 """
+    def get(self, index):
+        # 索引如果越界则抛出异常,下同
+        assert index >= 0 and index < self.__size, "索引越界"
+        return self.__nums[index]
+
+    """ 更新元素 """
+    def set(self, num, index):
+        assert index >= 0 and index < self.__size, "索引越界"
+        self.__nums[index] = num
+
+    """ 中间插入(尾部添加)元素 """
+    def add(self, num, index=-1):
+        assert index >= 0 and index < self.__size, "索引越界"
+        # 若不指定索引 index ,则向数组尾部添加元素
+        if index == -1:
+            index = self.__size
+        # 元素数量超出容量时,触发扩容机制
+        if self.__size == self.capacity():
+            self.extend_capacity()
+        # 索引 i 以及之后的元素都向后移动一位
+        for j in range(self.__size - 1, index - 1, -1):
+            self.__nums[j + 1] = self.__nums[j]
+        self.__nums[index] = num
+        # 更新元素数量
+        self.__size += 1
+
+    """ 删除元素 """
+    def remove(self, index):
+        assert index >= 0 and index < self.__size, "索引越界"
+        num = self.nums[index]
+        # 索引 i 之后的元素都向前移动一位
+        for j in range(index, self.__size - 1):
+            self.__nums[j] = self.__nums[j + 1]
+        # 更新元素数量
+        self.__size -= 1
+        # 返回被删除元素
+        return num
+
+    """ 列表扩容 """
+    def extend_capacity(self):
+        # 新建一个长度为 self.__size 的数组,并将原数组拷贝到新数组
+        self.__nums = self.__nums + [0] * self.capacity() * (self.__extend_ratio - 1)
+        # 更新列表容量
+        self.__capacity = len(self.__nums)
+
+    """ 返回有效长度的列表 """
+    def to_array(self):
+        return self.__nums[:self.__size]
+
+
+
+
my_list.go
/* 列表类简易实现 */
+type myList struct {
+    numsCapacity int
+    nums         []int
+    numsSize     int
+    extendRatio  int
+}
+
+/* 构造函数 */
+func newMyList() *myList {
+    return &myList{
+        numsCapacity: 10,              // 列表容量
+        nums:         make([]int, 10), // 数组(存储列表元素)
+        numsSize:     0,               // 列表长度(即当前元素数量)
+        extendRatio:  2,               // 每次列表扩容的倍数
+    }
+}
+
+/* 获取列表长度(即当前元素数量) */
+func (l *myList) size() int {
+    return l.numsSize
+}
+
+/*  获取列表容量 */
+func (l *myList) capacity() int {
+    return l.numsCapacity
+}
+
+/* 访问元素 */
+func (l *myList) get(index int) int {
+    // 索引如果越界则抛出异常,下同
+    if index < 0 || index >= l.numsSize {
+        panic("索引越界")
+    }
+    return l.nums[index]
+}
+
+/* 更新元素 */
+func (l *myList) set(num, index int) {
+    if index < 0 || index >= l.numsSize {
+        panic("索引越界")
+    }
+    l.nums[index] = num
+}
+
+/* 尾部添加元素 */
+func (l *myList) add(num int) {
+    // 元素数量超出容量时,触发扩容机制
+    if l.numsSize == l.numsCapacity {
+        l.extendCapacity()
+    }
+    l.nums[l.numsSize] = num
+    // 更新元素数量
+    l.numsSize++
+}
+
+/* 中间插入元素 */
+func (l *myList) insert(num, index int) {
+    if index < 0 || index >= l.numsSize {
+        panic("索引越界")
+    }
+    // 元素数量超出容量时,触发扩容机制
+    if l.numsSize == l.numsCapacity {
+        l.extendCapacity()
+    }
+    // 索引 i 以及之后的元素都向后移动一位
+    for j := l.numsSize - 1; j >= index; j-- {
+        l.nums[j+1] = l.nums[j]
+    }
+    l.nums[index] = num
+    // 更新元素数量
+    l.numsSize++
+}
+
+/* 删除元素 */
+func (l *myList) remove(index int) int {
+    if index < 0 || index >= l.numsSize {
+        panic("索引越界")
+    }
+    num := l.nums[index]
+    // 索引 i 之后的元素都向前移动一位
+    for j := index; j < l.numsSize-1; j++ {
+        l.nums[j] = l.nums[j+1]
+    }
+    // 更新元素数量
+    l.numsSize--
+    // 返回被删除元素
+    return num
+}
+
+/* 列表扩容 */
+func (l *myList) extendCapacity() {
+    // 新建一个长度为 self.__size 的数组,并将原数组拷贝到新数组
+    l.nums = append(l.nums, make([]int, l.numsCapacity*(l.extendRatio-1))...)
+    // 更新列表容量
+    l.numsCapacity = len(l.nums)
+}
+
+
+
+
my_list.js
/* 列表类简易实现 */
+class MyList {
+    #nums = new Array(); // 数组(存储列表元素)
+    #capacity = 10; // 列表容量
+    #size = 0; // 列表长度(即当前元素数量)
+    #extendRatio = 2; // 每次列表扩容的倍数
+
+    /* 构造函数 */
+    constructor() {
+        this.#nums = new Array(this.#capacity);
+    }
+
+    /* 获取列表长度(即当前元素数量)*/
+    size() {
+        return this.#size;
+    }
+
+    /* 获取列表容量 */
+    capacity() {
+        return this.#capacity;
+    }
+
+    /* 访问元素 */
+    get(index) {
+        // 索引如果越界则抛出异常,下同
+        if (index < 0 || index >= this.#size)
+            throw new Error('索引越界');
+        return this.#nums[index];
+    }
+
+    /* 更新元素 */
+    set(index, num) {
+        if (index < 0 || index >= this.#size)
+            throw new Error('索引越界');
+        this.#nums[index] = num;
+    }
+
+    /* 尾部添加元素 */
+    add(num) {
+        // 如果长度等于容量,则需要扩容
+        if (this.#size === this.#capacity) {
+            this.extendCapacity();
+        }
+        // 将新元素添加到列表尾部
+        this.#nums[this.#size] = num;
+        this.#size++;
+    }
+
+    /* 中间插入元素 */
+    insert(index, num) {
+        if (index < 0 || index >= this.#size)
+            throw new Error('索引越界');
+        // 元素数量超出容量时,触发扩容机制
+        if (this.#size === this.#capacity) {
+            this.extendCapacity();
+        }
+        // 将索引 index 以及之后的元素都向后移动一位
+        for (let j = this.#size - 1; j >= index; j--) {
+            this.#nums[j + 1] = this.#nums[j];
+        }
+        // 更新元素数量
+        this.#nums[index] = num;
+        this.#size++;
+    }
+
+    /* 删除元素 */
+    remove(index) {
+        if (index < 0 || index >= this.#size)
+            throw new Error('索引越界');
+        let num = this.#nums[index];
+        // 将索引 index 之后的元素都向前移动一位
+        for (let j = index; j < this.#size - 1; j++) {
+            this.#nums[j] = this.#nums[j + 1];
+        }
+        // 更新元素数量
+        this.#size--;
+        // 返回被删除元素
+        return num;
+    }
+
+    /* 列表扩容 */
+    extendCapacity() {
+        // 新建一个长度为 size 的数组,并将原数组拷贝到新数组
+        this.#nums = this.#nums.concat(
+            new Array(this.capacity() * (this.#extendRatio - 1))
+        );
+        // 更新列表容量
+        this.#capacity = this.#nums.length;
+    }
+}
+
+
+
+
my_list.ts
/* 列表类简易实现 */
+class MyList {
+    private nums: Array<number>; // 数组(存储列表元素)
+    private _capacity: number = 10; // 列表容量
+    private _size: number = 0; // 列表长度(即当前元素数量)
+    private extendRatio: number = 2; // 每次列表扩容的倍数
+
+    /* 构造函数 */
+    constructor() {
+        this.nums = new Array(this._capacity);
+    }
+
+    /* 获取列表长度(即当前元素数量)*/
+    public size(): number {
+        return this._size;
+    }
+
+    /* 获取列表容量 */
+    public capacity(): number {
+        return this._capacity;
+    }
+
+    /* 访问元素 */
+    public get(index: number): number {
+        // 索引如果越界则抛出异常,下同
+        if (index < 0 || index >= this._size)
+            throw new Error('索引越界');
+        return this.nums[index];
+    }
+
+    /* 更新元素 */
+    public set(index: number, num: number): void {
+        if (index < 0 || index >= this._size)
+            throw new Error('索引越界');
+        this.nums[index] = num;
+    }
+
+    /* 尾部添加元素 */
+    public add(num: number): void {
+        // 如果长度等于容量,则需要扩容
+        if (this._size === this._capacity)
+            this.extendCapacity();
+        // 将新元素添加到列表尾部
+        this.nums[this._size] = num;
+        this._size++;
+    }
+
+    /* 中间插入元素 */
+    public insert(index: number, num: number): void {
+        if (index < 0 || index >= this._size)
+            throw new Error('索引越界');
+        // 元素数量超出容量时,触发扩容机制
+        if (this._size === this._capacity) {
+            this.extendCapacity();
+        }
+        // 将索引 index 以及之后的元素都向后移动一位
+        for (let j = this._size - 1; j >= index; j--) {
+            this.nums[j + 1] = this.nums[j];
+        }
+        // 更新元素数量
+        this.nums[index] = num;
+        this._size++;
+    }
+
+    /* 删除元素 */
+    public remove(index: number): number {
+        if (index < 0 || index >= this._size)
+            throw new Error('索引越界');
+        let num = this.nums[index];
+        // 将索引 index 之后的元素都向前移动一位
+        for (let j = index; j < this._size - 1; j++) {
+            this.nums[j] = this.nums[j + 1];
+        }
+        // 更新元素数量
+        this._size--;
+        // 返回被删除元素
+        return num;
+    }
+
+    /* 列表扩容 */
+    public extendCapacity(): void {
+        // 新建一个长度为 size 的数组,并将原数组拷贝到新数组
+        this.nums = this.nums.concat(
+            new Array(this.capacity() * (this.extendRatio - 1))
+        );
+        // 更新列表容量
+        this._capacity = this.nums.length;
+    }
+}
+
+
+
+
my_list.c

+
+
+
+
my_list.cs
class MyList
+{
+    private int[] nums;           // 数组(存储列表元素)
+    private int capacity = 10;    // 列表容量
+    private int size = 0;         // 列表长度(即当前元素数量)
+    private int extendRatio = 2;  // 每次列表扩容的倍数
+
+    /* 构造函数 */
+    public MyList()
+    {
+        nums = new int[capacity];
+    }
+
+    /* 获取列表长度(即当前元素数量)*/
+    public int Size()
+    {
+        return size;
+    }
+
+    /* 获取列表容量 */
+    public int Capacity()
+    {
+        return capacity;
+    }
+
+    /* 访问元素 */
+    public int Get(int index)
+    {
+        // 索引如果越界则抛出异常,下同
+        if (index < 0 || index >= size)
+            throw new IndexOutOfRangeException("索引越界");
+        return nums[index];
+    }
+
+    /* 更新元素 */
+    public void Set(int index, int num)
+    {
+        if (index < 0 || index >= size)
+            throw new IndexOutOfRangeException("索引越界");
+        nums[index] = num;
+    }
+
+    /* 尾部添加元素 */
+    public void Add(int num)
+    {
+        // 元素数量超出容量时,触发扩容机制
+        if (size == Capacity())
+            ExtendCapacity();
+        nums[size] = num;
+        // 更新元素数量
+        size++;
+    }
+
+    /* 中间插入元素 */
+    public void Insert(int index, int num)
+    {
+        if (index < 0 || index >= size)
+            throw new IndexOutOfRangeException("索引越界");
+        // 元素数量超出容量时,触发扩容机制
+        if (size == Capacity())
+            ExtendCapacity();
+        // 将索引 index 以及之后的元素都向后移动一位
+        for (int j = size - 1; j >= index; j--)
+        {
+            nums[j + 1] = nums[j];
+        }
+        nums[index] = num;
+        // 更新元素数量
+        size++;
+    }
+
+    /* 删除元素 */
+    public int Remove(int index)
+    {
+        if (index < 0 || index >= size)
+            throw new IndexOutOfRangeException("索引越界");
+        int num = nums[index];
+        // 将索引 index 之后的元素都向前移动一位
+        for (int j = index; j < size - 1; j++)
+        {
+            nums[j] = nums[j + 1];
+        }
+        // 更新元素数量
+        size--;
+        // 返回被删除元素
+        return num;
+    }
+
+    /* 列表扩容 */
+    public void ExtendCapacity()
+    {
+        // 新建一个长度为 size 的数组,并将原数组拷贝到新数组
+        System.Array.Resize(ref nums, Capacity() * extendRatio);
+        // 更新列表容量
+        capacity = nums.Length;
+    }
+}
+
+
+
+
my_list.swift
/* 列表类简易实现 */
+class MyList {
+    private var nums: [Int] // 数组(存储列表元素)
+    private var _capacity = 10 // 列表容量
+    private var _size = 0 // 列表长度(即当前元素数量)
+    private let extendRatio = 2 // 每次列表扩容的倍数
+
+    /* 构造函数 */
+    init() {
+        nums = Array(repeating: 0, count: _capacity)
+    }
+
+    /* 获取列表长度(即当前元素数量)*/
+    func size() -> Int {
+        _size
+    }
+
+    /* 获取列表容量 */
+    func capacity() -> Int {
+        _capacity
+    }
+
+    /* 访问元素 */
+    func get(index: Int) -> Int {
+        // 索引如果越界则抛出错误,下同
+        if index < 0 || index >= _size {
+            fatalError("索引越界")
+        }
+        return nums[index]
+    }
+
+    /* 更新元素 */
+    func set(index: Int, num: Int) {
+        if index < 0 || index >= _size {
+            fatalError("索引越界")
+        }
+        nums[index] = num
+    }
+
+    /* 尾部添加元素 */
+    func add(num: Int) {
+        // 元素数量超出容量时,触发扩容机制
+        if _size == _capacity {
+            extendCapacity()
+        }
+        nums[_size] = num
+        // 更新元素数量
+        _size += 1
+    }
+
+    /* 中间插入元素 */
+    func insert(index: Int, num: Int) {
+        if index < 0 || index >= _size {
+            fatalError("索引越界")
+        }
+        // 元素数量超出容量时,触发扩容机制
+        if _size == _capacity {
+            extendCapacity()
+        }
+        // 将索引 index 以及之后的元素都向后移动一位
+        for j in sequence(first: _size - 1, next: { $0 >= index + 1 ? $0 - 1 : nil }) {
+            nums[j + 1] = nums[j]
+        }
+        nums[index] = num
+        // 更新元素数量
+        _size += 1
+    }
+
+    /* 删除元素 */
+    @discardableResult
+    func remove(index: Int) -> Int {
+        if index < 0 || index >= _size {
+            fatalError("索引越界")
+        }
+        let num = nums[index]
+        // 将索引 index 之后的元素都向前移动一位
+        for j in index ..< (_size - 1) {
+            nums[j] = nums[j + 1]
+        }
+        // 更新元素数量
+        _size -= 1
+        // 返回被删除元素
+        return num
+    }
+
+    /* 列表扩容 */
+    func extendCapacity() {
+        // 新建一个长度为 size 的数组,并将原数组拷贝到新数组
+        nums = nums + Array(repeating: 0, count: _capacity * (extendRatio - 1))
+        // 更新列表容量
+        _capacity = nums.count
+    }
+}
+
+
+
+
my_list.zig
// 列表类简易实现
+pub fn MyList(comptime T: type) type {
+    return struct {
+        const Self = @This();
+
+        nums: []T = undefined,                        // 数组(存储列表元素)
+        numsCapacity: usize = 10,                     // 列表容量
+        numSize: usize = 0,                           // 列表长度(即当前元素数量)
+        extendRatio: usize = 2,                       // 每次列表扩容的倍数
+        mem_arena: ?std.heap.ArenaAllocator = null,
+        mem_allocator: std.mem.Allocator = undefined, // 内存分配器
+
+        // 构造函数(分配内存+初始化列表)
+        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {
+            if (self.mem_arena == null) {
+                self.mem_arena = std.heap.ArenaAllocator.init(allocator);
+                self.mem_allocator = self.mem_arena.?.allocator();
+            }
+            self.nums = try self.mem_allocator.alloc(T, self.numsCapacity);
+            std.mem.set(T, self.nums, @as(T, 0));
+        }
+
+        // 析构函数(释放内存)
+        pub fn deinit(self: *Self) void {
+            if (self.mem_arena == null) return;
+            self.mem_arena.?.deinit();
+        }
+
+        // 获取列表长度(即当前元素数量)
+        pub fn size(self: *Self) usize {
+            return self.numSize;
+        }
+
+        // 获取列表容量
+        pub fn capacity(self: *Self) usize {
+            return self.numsCapacity;
+        }
+
+        // 访问元素
+        pub fn get(self: *Self, index: usize) T {
+            // 索引如果越界则抛出异常,下同
+            if (index < 0 or index >= self.size()) @panic("索引越界");
+            return self.nums[index];
+        }  
+
+        // 更新元素
+        pub fn set(self: *Self, index: usize, num: T) void {
+            // 索引如果越界则抛出异常,下同
+            if (index < 0 or index >= self.size()) @panic("索引越界");
+            self.nums[index] = num;
+        }  
+
+        // 尾部添加元素
+        pub fn add(self: *Self, num: T) !void {
+            // 元素数量超出容量时,触发扩容机制
+            if (self.size() == self.capacity()) try self.extendCapacity();
+            self.nums[self.size()] = num;
+            // 更新元素数量
+            self.numSize += 1;
+        }  
+
+        // 中间插入元素
+        pub fn insert(self: *Self, index: usize, num: T) !void {
+            if (index < 0 or index >= self.size()) @panic("索引越界");
+            // 元素数量超出容量时,触发扩容机制
+            if (self.size() == self.capacity()) try self.extendCapacity();
+            // 索引 i 以及之后的元素都向后移动一位
+            var j = self.size() - 1;
+            while (j >= index) : (j -= 1) {
+                self.nums[j + 1] = self.nums[j];
+            }
+            self.nums[index] = num;
+            // 更新元素数量
+            self.numSize += 1;
+        }
+
+        // 删除元素
+        pub fn remove(self: *Self, index: usize) T {
+            if (index < 0 or index >= self.size()) @panic("索引越界");
+            var num = self.nums[index];
+            // 索引 i 之后的元素都向前移动一位
+            var j = index;
+            while (j < self.size() - 1) : (j += 1) {
+                self.nums[j] = self.nums[j + 1];
+            }
+            // 更新元素数量
+            self.numSize -= 1;
+            // 返回被删除元素
+            return num;
+        }
+
+        // 列表扩容
+        pub fn extendCapacity(self: *Self) !void {
+            // 新建一个长度为 size * extendRatio 的数组,并将原数组拷贝到新数组
+            var newCapacity = self.capacity() * self.extendRatio;
+            var extend = try self.mem_allocator.alloc(T, newCapacity);
+            std.mem.set(T, extend, @as(T, 0));
+            // 将原数组中的所有元素复制到新数组
+            std.mem.copy(T, extend, self.nums);
+            self.nums = extend;
+            // 更新列表容量
+            self.numsCapacity = newCapacity;
+        }
+
+        // 将列表转换为数组
+        pub fn toArray(self: *Self) ![]T {
+            // 仅转换有效长度范围内的列表元素
+            var nums = try self.mem_allocator.alloc(T, self.size());
+            std.mem.set(T, nums, @as(T, 0));
+            for (nums) |*num, i| {
+                num.* = self.get(i);
+            }
+            return nums;
+        }
+    };
+}
+
+
+
+
+ + + + + +

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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/chapter_array_and_linkedlist/summary/index.html b/chapter_array_and_linkedlist/summary/index.html new file mode 100644 index 000000000..60f953477 --- /dev/null +++ b/chapter_array_and_linkedlist/summary/index.html @@ -0,0 +1,1801 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + 4.4. 小结 - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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4.4. 小结

+
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  • 数组和链表是两种基本数据结构,代表了数据在计算机内存中的两种存储方式,即连续空间存储和离散空间存储。两者的优点与缺点呈现出此消彼长的关系。
  • +
  • 数组支持随机访问、内存空间占用小;但插入与删除元素效率低,且初始化后长度不可变。
  • +
  • 链表可通过更改指针实现高效的结点插入与删除,并且可以灵活地修改长度;但结点访问效率低、占用内存多。常见的链表类型有单向链表、循环链表、双向链表。
  • +
  • 列表又称动态数组,是基于数组实现的一种数据结构,其保存了数组的优势,且可以灵活改变长度。列表的出现大大提升了数组的实用性,但副作用是会造成部分内存空间浪费。
  • +
+

4.4.1. 数组 VS 链表

+

Table. 数组与链表特点对比

+ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
数组链表
存储方式连续内存空间离散内存空间
数据结构长度长度不可变长度可变
内存使用率占用内存少、缓存局部性好占用内存多
优势操作随机访问插入、删除
+
+
+

Tip

+

「缓存局部性(Cache locality)」涉及到了计算机操作系统,在本书不做展开介绍,建议有兴趣的同学 Google / Baidu 一下。

+
+

Table. 数组与链表操作时间复杂度

+ +
+ + + + + + + + + + + + + + + + + + + + + + + + + +
操作数组链表
访问元素\(O(1)\)\(O(N)\)
添加元素\(O(N)\)\(O(1)\)
删除元素\(O(N)\)\(O(1)\)
+
+ + + + + +

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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/chapter_computational_complexity/performance_evaluation/index.html b/chapter_computational_complexity/performance_evaluation/index.html new file mode 100644 index 000000000..2e5ddcb4a --- /dev/null +++ b/chapter_computational_complexity/performance_evaluation/index.html @@ -0,0 +1,1817 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + 2.1. 算法效率评估 - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + 跳转至 + + +
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2.1. 算法效率评估

+

2.1.1. 算法评价维度

+

在开始学习算法之前,我们首先要想清楚算法的设计目标是什么,或者说,如何来评判算法的好与坏。整体上看,我们设计算法时追求两个层面的目标。

+
    +
  1. 找到问题解法。算法需要能够在规定的输入范围下,可靠地求得问题的正确解。
  2. +
  3. 寻求最优解法。同一个问题可能存在多种解法,而我们希望算法效率尽可能的高。
  4. +
+

换言之,在可以解决问题的前提下,算法效率则是主要评价维度,包括:

+
    +
  • 时间效率,即算法的运行速度的快慢。
  • +
  • 空间效率,即算法占用的内存空间大小。
  • +
+

数据结构与算法追求“运行速度快、占用内存少”,而如何去评价算法效率则是非常重要的问题,因为只有知道如何评价算法,才能去做算法之间的对比分析,以及优化算法设计。

+

2.1.2. 效率评估方法

+

实际测试

+

假设我们现在有算法 A 和 算法 B ,都能够解决同一问题,现在需要对比两个算法之间的效率。我们能够想到的最直接的方式,就是找一台计算机,把两个算法都完整跑一遍,并监控记录运行时间和内存占用情况。这种评估方式能够反映真实情况,但是也存在很大的硬伤。

+

难以排除测试环境的干扰因素。硬件配置会影响到算法的性能表现。例如,在某台计算机中,算法 A 比算法 B 运行时间更短;但换到另一台配置不同的计算机中,可能会得到相反的测试结果。这意味着我们需要在各种机器上展开测试,而这是不现实的。

+

展开完整测试非常耗费资源。随着输入数据量的大小变化,算法会呈现出不同的效率表现。比如,有可能输入数据量较小时,算法 A 运行时间短于算法 B ,而在输入数据量较大时,测试结果截然相反。因此,若想要达到具有说服力的对比结果,那么需要输入各种体量数据,这样的测试需要占用大量计算资源。

+

理论估算

+

既然实际测试具有很大的局限性,那么我们是否可以仅通过一些计算,就获知算法的效率水平呢?答案是肯定的,我们将此估算方法称为「复杂度分析 Complexity Analysis」或「渐近复杂度分析 Asymptotic Complexity Analysis」。

+

复杂度分析评估随着输入数据量的增长,算法的运行时间和占用空间的增长趋势。根据时间和空间两方面,复杂度可分为「时间复杂度 Time Complexity」和「空间复杂度 Space Complexity」。

+

复杂度分析克服了实际测试方法的弊端。一是独立于测试环境,分析结果适用于所有运行平台。二是可以体现不同数据量下的算法效率,尤其是可以反映大数据量下的算法性能。

+

2.1.3. 复杂度分析重要性

+

复杂度分析给出一把评价算法效率的“标尺”,告诉我们执行某个算法需要多少时间和空间资源,也让我们可以开展不同算法之间的效率对比。

+

计算复杂度是个数学概念,对于初学者可能比较抽象,学习难度相对较高。从这个角度出发,其并不适合作为第一章内容。但是,当我们讨论某个数据结构或者算法的特点时,难以避免需要分析它的运行速度和空间使用情况。因此,在展开学习数据结构与算法之前,建议读者先对计算复杂度建立起初步的了解,并且能够完成简单案例的复杂度分析

+ + + + + +

评论

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+ + + + + + + + + + + + + + + + + + + + + + + + + + 2.3. 空间复杂度 - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + + + 跳转至 + + +
    +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + + + + + +
    +
    + + + + + + + + + + + + + +

    2.3. 空间复杂度

    +

    「空间复杂度 Space Complexity」统计 算法使用内存空间随着数据量变大时的增长趋势。这个概念与时间复杂度很类似。

    +

    2.3.1. 算法相关空间

    +

    算法运行中,使用的内存空间主要有以下几种:

    +
      +
    • 「输入空间」用于存储算法的输入数据;
    • +
    • 「暂存空间」用于存储算法运行中的变量、对象、函数上下文等数据;
    • +
    • 「输出空间」用于存储算法的输出数据;
    • +
    +
    +

    Tip

    +

    通常情况下,空间复杂度统计范围是「暂存空间」+「输出空间」。

    +
    +

    暂存空间可分为三个部分:

    +
      +
    • 「暂存数据」用于保存算法运行中的各种 常量、变量、对象 等。
    • +
    • 「栈帧空间」用于保存调用函数的上下文数据。系统每次调用函数都会在栈的顶部创建一个栈帧,函数返回时,栈帧空间会被释放。
    • +
    • 「指令空间」用于保存编译后的程序指令,在实际统计中一般忽略不计
    • +
    +

    space_types

    +

    Fig. 算法使用的相关空间

    + +
    +
    +
    +
    /* 类 */
    +class Node {
    +    int val;
    +    Node next;
    +    Node(int x) { val = x; }
    +}
    +
    +/* 函数 */
    +int function() {
    +    // do something...
    +    return 0;
    +}
    +
    +int algorithm(int n) {        // 输入数据
    +    final int a = 0;          // 暂存数据(常量)
    +    int b = 0;                // 暂存数据(变量)
    +    Node node = new Node(0);  // 暂存数据(对象)
    +    int c = function();       // 栈帧空间(调用函数)
    +    return a + b + c;         // 输出数据
    +}
    +
    +
    +
    +
    /* 结构体 */
    +struct Node {
    +    int val;
    +    Node *next;
    +    Node(int x) : val(x), next(nullptr) {}
    +};
    +
    +/* 函数 */
    +int func() {
    +    // do something...
    +    return 0;
    +}
    +
    +int algorithm(int n) {        // 输入数据
    +    const int a = 0;          // 暂存数据(常量)
    +    int b = 0;                // 暂存数据(变量)
    +    Node* node = new Node(0);  // 暂存数据(对象)
    +    int c = func();       // 栈帧空间(调用函数)
    +    return a + b + c;         // 输出数据
    +}
    +
    +
    +
    +
    """ 类 """
    +class Node:
    +    def __init__(self, x):
    +        self.val = x      # 结点值
    +        self.next = None  # 指向下一结点的指针(引用)
    +
    +""" 函数 """
    +def function():
    +    # do something...
    +    return 0
    +
    +def algorithm(n):     # 输入数据
    +    b = 0             # 暂存数据(变量)
    +    node = Node(0)    # 暂存数据(对象)
    +    c = function()    # 栈帧空间(调用函数)
    +    return a + b + c  # 输出数据
    +
    +
    +
    +
    /* 结构体 */
    +type node struct {
    +    val  int
    +    next *node
    +}
    +
    +/* 创建 node 结构体  */
    +func newNode(val int) *node {
    +    return &node{val: val}
    +}
    +
    +/* 函数 */
    +func function() int {
    +    // do something...
    +    return 0
    +}
    +
    +func algorithm(n int) int { // 输入数据
    +    const a = 0             // 暂存数据(常量)
    +    b := 0                  // 暂存数据(变量)
    +    newNode(0)              // 暂存数据(对象)
    +    c := function()         // 栈帧空间(调用函数)
    +    return a + b + c        // 输出数据
    +}
    +
    +
    +
    +
    /* 类 */
    +class Node {
    +    val;
    +    next;
    +    constructor(val) {
    +        this.val = val === undefined ? 0 : val; // 结点值
    +        this.next = null;                       // 指向下一结点的引用
    +    }
    +}
    +
    +/* 函数 */
    +function constFunc() {
    +    // do something
    +    return 0;
    +}
    +
    +function algorithm(n) {       // 输入数据
    +    const a = 0;              // 暂存数据(常量)
    +    const b = 0;              // 暂存数据(变量)
    +    const node = new Node(0); // 暂存数据(对象)
    +    const c = constFunc();    // 栈帧空间(调用函数)
    +    return a + b + c;         // 输出数据
    +}
    +
    +
    +
    +
    /* 类 */
    +class Node {
    +    val: number;
    +    next: Node | null;
    +    constructor(val?: number) {
    +        this.val = val === undefined ? 0 : val; // 结点值
    +        this.next = null;                       // 指向下一结点的引用
    +    }
    +}
    +
    +/* 函数 */
    +function constFunc(): number {
    +    // do something
    +    return 0;
    +}
    +
    +function algorithm(n: number): number { // 输入数据
    +    const a = 0;                        // 暂存数据(常量)
    +    const b = 0;                        // 暂存数据(变量)
    +    const node = new Node(0);           // 暂存数据(对象)
    +    const c = constFunc();              // 栈帧空间(调用函数)
    +    return a + b + c;                   // 输出数据
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    /* 类 */
    +class Node
    +{
    +    int val;
    +    Node next;
    +    Node(int x) { val = x; }
    +}
    +
    +/* 函数 */
    +int function()
    +{
    +    // do something...
    +    return 0;
    +}
    +
    +int algorithm(int n)          // 输入数据
    +{
    +    int a = 0;                // 暂存数据(常量)
    +    int b = 0;                // 暂存数据(变量)
    +    Node node = new Node(0);  // 暂存数据(对象)
    +    int c = function();       // 栈帧空间(调用函数)
    +    return a + b + c;         // 输出数据
    +}
    +
    +
    +
    +
    /* 类 */
    +class Node {
    +    var val: Int
    +    var next: Node?
    +
    +    init(x: Int) {
    +        val = x
    +    }
    +}
    +
    +/* 函数 */
    +func function() -> Int {
    +    // do something...
    +    return 0
    +}
    +
    +func algorithm(n: Int) -> Int { // 输入数据
    +    let a = 0 // 暂存数据(常量)
    +    var b = 0 // 暂存数据(变量)
    +    let node = Node(x: 0) // 暂存数据(对象)
    +    let c = function() // 栈帧空间(调用函数)
    +    return a + b + c // 输出数据
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    +

    2.3.2. 推算方法

    +

    空间复杂度的推算方法和时间复杂度总体类似,只是从统计“计算操作数量”变为统计“使用空间大小”。与时间复杂度不同的是,我们一般只关注「最差空间复杂度」。这是因为内存空间是一个硬性要求,我们必须保证在所有输入数据下都有足够的内存空间预留。

    +

    最差空间复杂度中的“最差”有两层含义,分别为输入数据的最差分布、算法运行中的最差时间点。

    +
      +
    • 以最差输入数据为准。当 \(n < 10\) 时,空间复杂度为 \(O(1)\) ;但是当 \(n > 10\) 时,初始化的数组 nums 使用 \(O(n)\) 空间;因此最差空间复杂度为 \(O(n)\)
    • +
    • 以算法运行过程中的峰值内存为准。程序在执行最后一行之前,使用 \(O(1)\) 空间;当初始化数组 nums 时,程序使用 \(O(n)\) 空间;因此最差空间复杂度为 \(O(n)\)
    • +
    +
    +
    +
    +
    void algorithm(int n) {
    +    int a = 0;                   // O(1)
    +    int[] b = new int[10000];    // O(1)
    +    if (n > 10)
    +        int[] nums = new int[n]; // O(n)
    +}
    +
    +
    +
    +
    void algorithm(int n) {
    +    int a = 0;               // O(1)
    +    vector<int> b(10000);    // O(1)
    +    if (n > 10)
    +        vector<int> nums(n); // O(n)
    +}
    +
    +
    +
    +
    def algorithm(n):
    +    a = 0               # O(1)
    +    b = [0] * 10000     # O(1)
    +    if n > 10:
    +        nums = [0] * n  # O(n)
    +
    +
    +
    +
    func algorithm(n int) {
    +    a := 0                      // O(1)
    +    b := make([]int, 10000)     // O(1)
    +    var nums []int
    +    if n > 10 {
    +        nums := make([]int, n)  // O(n)
    +    }
    +    fmt.Println(a, b, nums)
    +}
    +
    +
    +
    +
    function algorithm(n) {
    +    const a = 0;                   // O(1)
    +    const b = new Array(10000);    // O(1)
    +    if (n > 10) {
    +        const nums = new Array(n); // O(n)
    +    }
    +}
    +
    +
    +
    +
    function algorithm(n: number): void {
    +    const a = 0;                   // O(1)
    +    const b = new Array(10000);    // O(1)
    +    if (n > 10) {
    +        const nums = new Array(n); // O(n)
    +    }
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    void algorithm(int n)
    +{
    +    int a = 0;                   // O(1)
    +    int[] b = new int[10000];    // O(1)
    +    if (n > 10)
    +    {
    +        int[] nums = new int[n]; // O(n)
    +    }
    +}
    +
    +
    +
    +
    func algorithm(n: Int) {
    +    let a = 0 // O(1)
    +    let b = Array(repeating: 0, count: 10000) // O(1)
    +    if n > 10 {
    +        let nums = Array(repeating: 0, count: n) // O(n)
    +    }
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    +

    在递归函数中,需要注意统计栈帧空间。例如函数 loop(),在循环中调用了 \(n\)function() ,每轮中的 function() 都返回并释放了栈帧空间,因此空间复杂度仍为 \(O(1)\) 。而递归函数 recur() 在运行中会同时存在 \(n\) 个未返回的 recur() ,从而使用 \(O(n)\) 的栈帧空间。

    +
    +
    +
    +
    int function() {
    +    // do something
    +    return 0;
    +}
    +/* 循环 O(1) */
    +void loop(int n) {
    +    for (int i = 0; i < n; i++) {
    +        function();
    +    }
    +}
    +/* 递归 O(n) */
    +void recur(int n) {
    +    if (n == 1) return;
    +    return recur(n - 1);
    +}
    +
    +
    +
    +
    int func() {
    +    // do something
    +    return 0;
    +}
    +/* 循环 O(1) */
    +void loop(int n) {
    +    for (int i = 0; i < n; i++) {
    +        func();
    +    }
    +}
    +/* 递归 O(n) */
    +void recur(int n) {
    +    if (n == 1) return;
    +    return recur(n - 1);
    +}
    +
    +
    +
    +
    def function():
    +    # do something
    +    return 0
    +
    +""" 循环 O(1) """
    +def loop(n):
    +    for _ in range(n):
    +        function()
    +
    +""" 递归 O(n) """
    +def recur(n):
    +    if n == 1: return
    +    return recur(n - 1)
    +
    +
    +
    +
    func function() int {
    +    // do something
    +    return 0
    +}
    +
    +/* 循环 O(1) */
    +func loop(n int) {
    +    for i := 0; i < n; i++ {
    +        function()
    +    }
    +}
    +
    +/* 递归 O(n) */
    +func recur(n int) {
    +    if n == 1 {
    +        return
    +    }
    +    recur(n - 1)
    +}
    +
    +
    +
    +
    function constFunc() {
    +    // do something
    +    return 0;
    +}
    +/* 循环 O(1) */
    +function loop(n) {
    +    for (let i = 0; i < n; i++) {
    +        constFunc();
    +    }
    +}
    +/* 递归 O(n) */
    +function recur(n) {
    +    if (n === 1) return;
    +    return recur(n - 1);
    +}
    +
    +
    +
    +
    function constFunc(): number {
    +    // do something
    +    return 0;
    +}
    +/* 循环 O(1) */
    +function loop(n: number): void {
    +    for (let i = 0; i < n; i++) {
    +        constFunc();
    +    }
    +}
    +/* 递归 O(n) */
    +function recur(n: number): void {
    +    if (n === 1) return;
    +    return recur(n - 1);
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    int function()
    +{
    +    // do something
    +    return 0;
    +}
    +/* 循环 O(1) */
    +void loop(int n)
    +{
    +    for (int i = 0; i < n; i++)
    +    {
    +        function();
    +    }
    +}
    +/* 递归 O(n) */
    +int recur(int n)
    +{
    +    if (n == 1) return 1;
    +    return recur(n - 1);
    +}
    +
    +
    +
    +
    @discardableResult
    +func function() -> Int {
    +    // do something
    +    return 0
    +}
    +
    +/* 循环 O(1) */
    +func loop(n: Int) {
    +    for _ in 0 ..< n {
    +        function()
    +    }
    +}
    +
    +/* 递归 O(n) */
    +func recur(n: Int) {
    +    if n == 1 {
    +        return
    +    }
    +    recur(n: n - 1)
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    +

    2.3.3. 常见类型

    +

    设输入数据大小为 \(n\) ,常见的空间复杂度类型有(从低到高排列)

    +
    \[ +\begin{aligned} +O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline +\text{常数阶} < \text{对数阶} < \text{线性阶} < \text{平方阶} < \text{指数阶} +\end{aligned} +\]
    +

    space_complexity_common_types

    +

    Fig. 空间复杂度的常见类型

    + +
    +

    Tip

    +

    部分示例代码需要一些前置知识,包括数组、链表、二叉树、递归算法等。如果遇到看不懂的地方无需担心,可以在学习完后面章节后再来复习,现阶段先聚焦在理解空间复杂度含义和推算方法上。

    +
    +

    常数阶 \(O(1)\)

    +

    常数阶常见于数量与输入数据大小 \(n\) 无关的常量、变量、对象。

    +

    需要注意的是,在循环中初始化变量或调用函数而占用的内存,在进入下一循环后就会被释放,即不会累积占用空间,空间复杂度仍为 \(O(1)\)

    +
    +
    +
    +
    space_complexity.java
    /* 常数阶 */
    +void constant(int n) {
    +    // 常量、变量、对象占用 O(1) 空间
    +    final int a = 0;
    +    int b = 0;
    +    int[] nums = new int[10000];
    +    ListNode node = new ListNode(0);
    +    // 循环中的变量占用 O(1) 空间
    +    for (int i = 0; i < n; i++) {
    +        int c = 0;
    +    }
    +    // 循环中的函数占用 O(1) 空间
    +    for (int i = 0; i < n; i++) {
    +        function();
    +    }
    +}
    +
    +
    +
    +
    space_complexity.cpp
    /* 常数阶 */
    +void constant(int n) {
    +    // 常量、变量、对象占用 O(1) 空间
    +    const int a = 0;
    +    int b = 0;
    +    vector<int> nums(10000);
    +    ListNode node(0);
    +    // 循环中的变量占用 O(1) 空间
    +    for (int i = 0; i < n; i++) {
    +        int c = 0;
    +    }
    +    // 循环中的函数占用 O(1) 空间
    +    for (int i = 0; i < n; i++) {
    +        func();
    +    }
    +}
    +
    +
    +
    +
    space_complexity.py
    """ 常数阶 """
    +def constant(n):
    +    # 常量、变量、对象占用 O(1) 空间
    +    a = 0
    +    nums = [0] * 10000
    +    node = ListNode(0)
    +    # 循环中的变量占用 O(1) 空间
    +    for _ in range(n):
    +        c = 0
    +    # 循环中的函数占用 O(1) 空间
    +    for _ in range(n):
    +        function()
    +
    +
    +
    +
    space_complexity.go
    /* 常数阶 */
    +func spaceConstant(n int) {
    +    // 常量、变量、对象占用 O(1) 空间
    +    const a = 0
    +    b := 0
    +    nums := make([]int, 10000)
    +    ListNode := newNode(0)
    +    // 循环中的变量占用 O(1) 空间
    +    var c int
    +    for i := 0; i < n; i++ {
    +        c = 0
    +    }
    +    // 循环中的函数占用 O(1) 空间
    +    for i := 0; i < n; i++ {
    +        function()
    +    }
    +    fmt.Println(a, b, nums, c, ListNode)
    +}
    +
    +
    +
    +
    space_complexity.js
    /* 常数阶 */
    +function constant(n) {
    +    // 常量、变量、对象占用 O(1) 空间
    +    const a = 0;
    +    const b = 0;
    +    const nums = new Array(10000);
    +    const node = new ListNode(0);
    +    // 循环中的变量占用 O(1) 空间
    +    for (let i = 0; i < n; i++) {
    +        const c = 0;
    +    }
    +    // 循环中的函数占用 O(1) 空间
    +    for (let i = 0; i < n; i++) {
    +        constFunc();
    +    }
    +}
    +
    +
    +
    +
    space_complexity.ts
    /* 常数阶 */
    +function constant(n: number): void {
    +    // 常量、变量、对象占用 O(1) 空间
    +    const a = 0;
    +    const b = 0;
    +    const nums = new Array(10000);
    +    const node = new ListNode(0);
    +    // 循环中的变量占用 O(1) 空间
    +    for (let i = 0; i < n; i++) {
    +        const c = 0;
    +    }
    +    // 循环中的函数占用 O(1) 空间
    +    for (let i = 0; i < n; i++) {
    +        constFunc();
    +    }
    +}
    +
    +
    +
    +
    space_complexity.c
    
    +
    +
    +
    +
    space_complexity.cs
    /* 常数阶 */
    +void constant(int n)
    +{
    +    // 常量、变量、对象占用 O(1) 空间
    +    int a = 0;
    +    int b = 0;
    +    int[] nums = new int[10000];
    +    ListNode node = new ListNode(0);
    +    // 循环中的变量占用 O(1) 空间
    +    for (int i = 0; i < n; i++)
    +    {
    +        int c = 0;
    +    }
    +    // 循环中的函数占用 O(1) 空间
    +    for (int i = 0; i < n; i++)
    +    {
    +        function();
    +    }
    +}
    +
    +
    +
    +
    space_complexity.swift
    /* 常数阶 */
    +func constant(n: Int) {
    +    // 常量、变量、对象占用 O(1) 空间
    +    let a = 0
    +    var b = 0
    +    let nums = Array(repeating: 0, count: 10000)
    +    let node = ListNode(x: 0)
    +    // 循环中的变量占用 O(1) 空间
    +    for _ in 0 ..< n {
    +        let c = 0
    +    }
    +    // 循环中的函数占用 O(1) 空间
    +    for _ in 0 ..< n {
    +        function()
    +    }
    +}
    +
    +
    +
    +
    space_complexity.zig
    // 常数阶
    +fn constant(n: i32) void {
    +    // 常量、变量、对象占用 O(1) 空间
    +    const a: i32 = 0;
    +    var b: i32 = 0;
    +    var nums = [_]i32{0}**10000;
    +    var node = inc.ListNode(i32){.val = 0};
    +    var i: i32 = 0;
    +    // 循环中的变量占用 O(1) 空间
    +    while (i < n) : (i += 1) {
    +        var c: i32 = 0;
    +        _ = c;
    +    }
    +    // 循环中的函数占用 O(1) 空间
    +    i = 0;
    +    while (i < n) : (i += 1) {
    +        _ = function();
    +    }
    +    _ = a;
    +    _ = b;
    +    _ = nums;
    +    _ = node;
    +}
    +
    +
    +
    +
    +

    线性阶 \(O(n)\)

    +

    线性阶常见于元素数量与 \(n\) 成正比的数组、链表、栈、队列等。

    +
    +
    +
    +
    space_complexity.java
    /* 线性阶 */
    +void linear(int n) {
    +    // 长度为 n 的数组占用 O(n) 空间
    +    int[] nums = new int[n];
    +    // 长度为 n 的列表占用 O(n) 空间
    +    List<ListNode> nodes = new ArrayList<>();
    +    for (int i = 0; i < n; i++) {
    +        nodes.add(new ListNode(i));
    +    }
    +    // 长度为 n 的哈希表占用 O(n) 空间
    +    Map<Integer, String> map = new HashMap<>();
    +    for (int i = 0; i < n; i++) {
    +        map.put(i, String.valueOf(i));
    +    }
    +}
    +
    +
    +
    +
    space_complexity.cpp
    /* 线性阶 */
    +void linear(int n) {
    +    // 长度为 n 的数组占用 O(n) 空间
    +    vector<int> nums(n);
    +    // 长度为 n 的列表占用 O(n) 空间
    +    vector<ListNode> nodes;
    +    for (int i = 0; i < n; i++) {
    +        nodes.push_back(ListNode(i));
    +    }
    +    // 长度为 n 的哈希表占用 O(n) 空间
    +    unordered_map<int, string> map;
    +    for (int i = 0; i < n; i++) {
    +        map[i] = to_string(i);
    +    }
    +}
    +
    +
    +
    +
    space_complexity.py
    """ 线性阶 """
    +def linear(n):
    +    # 长度为 n 的列表占用 O(n) 空间
    +    nums = [0] * n
    +    # 长度为 n 的哈希表占用 O(n) 空间
    +    mapp = {}
    +    for i in range(n):
    +        mapp[i] = str(i)
    +
    +
    +
    +
    space_complexity.go
    /* 线性阶 */
    +func spaceLinear(n int) {
    +    // 长度为 n 的数组占用 O(n) 空间
    +    _ = make([]int, n)
    +    // 长度为 n 的列表占用 O(n) 空间
    +    var nodes []*node
    +    for i := 0; i < n; i++ {
    +        nodes = append(nodes, newNode(i))
    +    }
    +    // 长度为 n 的哈希表占用 O(n) 空间
    +    m := make(map[int]string, n)
    +    for i := 0; i < n; i++ {
    +        m[i] = strconv.Itoa(i)
    +    }
    +}
    +
    +
    +
    +
    space_complexity.js
    /* 线性阶 */
    +function linear(n) {
    +    // 长度为 n 的数组占用 O(n) 空间
    +    const nums = new Array(n);
    +    // 长度为 n 的列表占用 O(n) 空间
    +    const nodes = [];
    +    for (let i = 0; i < n; i++) {
    +        nodes.push(new ListNode(i));
    +    }
    +    // 长度为 n 的哈希表占用 O(n) 空间
    +    const map = new Map();
    +    for (let i = 0; i < n; i++) {
    +        map.set(i, i.toString());
    +    }
    +}
    +
    +
    +
    +
    space_complexity.ts
    /* 线性阶 */
    +function linear(n: number): void {
    +    // 长度为 n 的数组占用 O(n) 空间
    +    const nums = new Array(n);
    +    // 长度为 n 的列表占用 O(n) 空间
    +    const nodes: ListNode[] = [];
    +    for (let i = 0; i < n; i++) {
    +        nodes.push(new ListNode(i));
    +    }
    +    // 长度为 n 的哈希表占用 O(n) 空间
    +    const map = new Map();
    +    for (let i = 0; i < n; i++) {
    +        map.set(i, i.toString());
    +    }
    +}
    +
    +
    +
    +
    space_complexity.c
    
    +
    +
    +
    +
    space_complexity.cs
    /* 线性阶 */
    +void linear(int n)
    +{
    +    // 长度为 n 的数组占用 O(n) 空间
    +    int[] nums = new int[n];
    +    // 长度为 n 的列表占用 O(n) 空间
    +    List<ListNode> nodes = new();
    +    for (int i = 0; i < n; i++)
    +    {
    +        nodes.Add(new ListNode(i));
    +    }
    +    // 长度为 n 的哈希表占用 O(n) 空间
    +    Dictionary<int, String> map = new();
    +    for (int i = 0; i < n; i++)
    +    {
    +        map.Add(i, i.ToString());
    +    }
    +}
    +
    +
    +
    +
    space_complexity.swift
    /* 线性阶 */
    +func linear(n: Int) {
    +    // 长度为 n 的数组占用 O(n) 空间
    +    let nums = Array(repeating: 0, count: n)
    +    // 长度为 n 的列表占用 O(n) 空间
    +    let nodes = (0 ..< n).map { ListNode(x: $0) }
    +    // 长度为 n 的哈希表占用 O(n) 空间
    +    let map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, "\($0)") })
    +}
    +
    +
    +
    +
    space_complexity.zig
    // 线性阶
    +fn linear(comptime n: i32) !void {
    +    // 长度为 n 的数组占用 O(n) 空间
    +    var nums = [_]i32{0}**n;
    +    // 长度为 n 的列表占用 O(n) 空间
    +    var nodes = std.ArrayList(i32).init(std.heap.page_allocator);
    +    defer nodes.deinit();
    +    var i: i32 = 0;
    +    while (i < n) : (i += 1) {
    +        try nodes.append(i);
    +    }
    +    // 长度为 n 的哈希表占用 O(n) 空间
    +    var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);
    +    defer map.deinit();
    +    var j: i32 = 0;
    +    while (j < n) : (j += 1) {
    +        const string = try std.fmt.allocPrint(std.heap.page_allocator, "{d}", .{j});
    +        defer std.heap.page_allocator.free(string);
    +        try map.put(i, string);
    +    }
    +    _ = nums;
    +}
    +
    +
    +
    +
    +

    以下递归函数会同时存在 \(n\) 个未返回的 algorithm() 函数,使用 \(O(n)\) 大小的栈帧空间。

    +
    +
    +
    +
    space_complexity.java
    /* 线性阶(递归实现) */
    +void linearRecur(int n) {
    +    System.out.println("递归 n = " + n);
    +    if (n == 1) return;
    +    linearRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.cpp
    /* 线性阶(递归实现) */
    +void linearRecur(int n) {
    +    cout << "递归 n = " << n << endl;
    +    if (n == 1) return;
    +    linearRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.py
    """ 线性阶(递归实现) """
    +def linear_recur(n):
    +    print("递归 n =", n)
    +    if n == 1: return
    +    linear_recur(n - 1)
    +
    +
    +
    +
    space_complexity.go
    /* 线性阶(递归实现) */
    +func spaceLinearRecur(n int) {
    +    fmt.Println("递归 n =", n)
    +    if n == 1 {
    +        return
    +    }
    +    spaceLinearRecur(n - 1)
    +}
    +
    +
    +
    +
    space_complexity.js
    /* 线性阶(递归实现) */
    +function linearRecur(n) {
    +    console.log(`递归 n = ${n}`);
    +    if (n === 1) return;
    +    linearRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.ts
    /* 线性阶(递归实现) */
    +function linearRecur(n: number): void {
    +    console.log(`递归 n = ${n}`);
    +    if (n === 1) return;
    +    linearRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.c
    
    +
    +
    +
    +
    space_complexity.cs
    /* 线性阶(递归实现) */
    +void linearRecur(int n)
    +{
    +    Console.WriteLine("递归 n = " + n);
    +    if (n == 1) return;
    +    linearRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.swift
    /* 线性阶(递归实现) */
    +func linearRecur(n: Int) {
    +    print("递归 n = \(n)")
    +    if n == 1 {
    +        return
    +    }
    +    linearRecur(n: n - 1)
    +}
    +
    +
    +
    +
    space_complexity.zig
    // 线性阶(递归实现)
    +fn linearRecur(comptime n: i32) void {
    +    std.debug.print("递归 n = {}\n", .{n});
    +    if (n == 1) return;
    +    linearRecur(n - 1);
    +}
    +
    +
    +
    +
    +

    space_complexity_recursive_linear

    +

    Fig. 递归函数产生的线性阶空间复杂度

    + +

    平方阶 \(O(n^2)\)

    +

    平方阶常见于元素数量与 \(n\) 成平方关系的矩阵、图。

    +
    +
    +
    +
    space_complexity.java
    /* 平方阶 */
    +void quadratic(int n) {
    +    // 矩阵占用 O(n^2) 空间
    +    int[][] numMatrix = new int[n][n];
    +    // 二维列表占用 O(n^2) 空间
    +    List<List<Integer>> numList = new ArrayList<>();
    +    for (int i = 0; i < n; i++) {
    +        List<Integer> tmp = new ArrayList<>();
    +        for (int j = 0; j < n; j++) {
    +            tmp.add(0);
    +        }
    +        numList.add(tmp);
    +    }
    +}
    +
    +
    +
    +
    space_complexity.cpp
    /* 平方阶 */
    +void quadratic(int n) {
    +    // 二维列表占用 O(n^2) 空间
    +    vector<vector<int>> numMatrix;
    +    for (int i = 0; i < n; i++) {
    +        vector<int> tmp;
    +        for (int j = 0; j < n; j++) {
    +            tmp.push_back(0);
    +        }
    +        numMatrix.push_back(tmp);
    +    }
    +}
    +
    +
    +
    +
    space_complexity.py
    """ 平方阶 """
    +def quadratic(n):
    +    # 二维列表占用 O(n^2) 空间
    +    num_matrix = [[0] * n for _ in range(n)]
    +
    +
    +
    +
    space_complexity.go
    /* 平方阶 */
    +func spaceQuadratic(n int) {
    +    // 矩阵占用 O(n^2) 空间
    +    numMatrix := make([][]int, n)
    +    for i := 0; i < n; i++ {
    +        numMatrix[i] = make([]int, n)
    +    }
    +}
    +
    +
    +
    +
    space_complexity.js
    /* 平方阶 */
    +function quadratic(n) {
    +    // 矩阵占用 O(n^2) 空间
    +    const numMatrix = Array(n).fill(null).map(() => Array(n).fill(null));
    +    // 二维列表占用 O(n^2) 空间
    +    const numList = [];
    +    for (let i = 0; i < n; i++) {
    +        const tmp = [];
    +        for (let j = 0; j < n; j++) {
    +            tmp.push(0);
    +        }
    +        numList.push(tmp);
    +    }
    +}
    +
    +
    +
    +
    space_complexity.ts
    /* 平方阶 */
    +function quadratic(n: number): void {
    +    // 矩阵占用 O(n^2) 空间
    +    const numMatrix = Array(n).fill(null).map(() => Array(n).fill(null));
    +    // 二维列表占用 O(n^2) 空间
    +    const numList = [];
    +    for (let i = 0; i < n; i++) {
    +        const tmp = [];
    +        for (let j = 0; j < n; j++) {
    +            tmp.push(0);
    +        }
    +        numList.push(tmp);
    +    }
    +}
    +
    +
    +
    +
    space_complexity.c
    
    +
    +
    +
    +
    space_complexity.cs
    /* 平方阶 */
    +void quadratic(int n)
    +{
    +    // 矩阵占用 O(n^2) 空间
    +    int[,] numMatrix = new int[n, n];
    +    // 二维列表占用 O(n^2) 空间
    +    List<List<int>> numList = new();
    +    for (int i = 0; i < n; i++)
    +    {
    +        List<int> tmp = new();
    +        for (int j = 0; j < n; j++)
    +        {
    +            tmp.Add(0);
    +        }
    +        numList.Add(tmp);
    +    }
    +}
    +
    +
    +
    +
    space_complexity.swift
    /* 平方阶 */
    +func quadratic(n: Int) {
    +    // 二维列表占用 O(n^2) 空间
    +    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)
    +}
    +
    +
    +
    +
    space_complexity.zig
    // 平方阶
    +fn quadratic(n: i32) !void {
    +    // 二维列表占用 O(n^2) 空间
    +    var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);
    +    defer nodes.deinit();
    +    var i: i32 = 0;
    +    while (i < n) : (i += 1) {
    +        var tmp = std.ArrayList(i32).init(std.heap.page_allocator);
    +        defer tmp.deinit();
    +        var j: i32 = 0;
    +        while (j < n) : (j += 1) {
    +            try tmp.append(0);
    +        }
    +        try nodes.append(tmp);
    +    }
    +}
    +
    +
    +
    +
    +

    在以下递归函数中,同时存在 \(n\) 个未返回的 algorithm() ,并且每个函数中都初始化了一个数组,长度分别为 \(n, n-1, n-2, ..., 2, 1\) ,平均长度为 \(\frac{n}{2}\) ,因此总体使用 \(O(n^2)\) 空间。

    +
    +
    +
    +
    space_complexity.java
    /* 平方阶(递归实现) */
    +int quadraticRecur(int n) {
    +    if (n <= 0) return 0;
    +    // 数组 nums 长度为 n, n-1, ..., 2, 1
    +    int[] nums = new int[n];
    +    System.out.println("递归 n = " + n + " 中的 nums 长度 = " + nums.length);
    +    return quadraticRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.cpp
    /* 平方阶(递归实现) */
    +int quadraticRecur(int n) {
    +    if (n <= 0) return 0;
    +    vector<int> nums(n);
    +    cout << "递归 n = " << n << " 中的 nums 长度 = " << nums.size() << endl;
    +    return quadraticRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.py
    """ 平方阶(递归实现) """
    +def quadratic_recur(n):
    +    if n <= 0: return 0
    +    # 数组 nums 长度为 n, n-1, ..., 2, 1
    +    nums = [0] * n
    +    return quadratic_recur(n - 1)
    +
    +
    +
    +
    space_complexity.go
    /* 平方阶(递归实现) */
    +func spaceQuadraticRecur(n int) int {
    +    if n <= 0 {
    +        return 0
    +    }
    +    // 数组 nums 长度为 n, n-1, ..., 2, 1
    +    nums := make([]int, n)
    +    return spaceQuadraticRecur(n - 1)
    +}
    +
    +
    +
    +
    space_complexity.js
    /* 平方阶(递归实现) */
    +function quadraticRecur(n) {
    +    if (n <= 0) return 0;
    +    const nums = new Array(n);
    +    console.log(`递归 n = ${n} 中的 nums 长度 = ${nums.length}`);
    +    return quadraticRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.ts
    /* 平方阶(递归实现) */
    +function quadraticRecur(n: number): number {
    +    if (n <= 0) return 0;
    +    const nums = new Array(n);
    +    console.log(`递归 n = ${n} 中的 nums 长度 = ${nums.length}`);
    +    return quadraticRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.c
    
    +
    +
    +
    +
    space_complexity.cs
    /* 平方阶(递归实现) */
    +int quadraticRecur(int n)
    +{
    +    if (n <= 0) return 0;
    +    // 数组 nums 长度为 n, n-1, ..., 2, 1
    +    int[] nums = new int[n];
    +    return quadraticRecur(n - 1);
    +}
    +
    +
    +
    +
    space_complexity.swift
    /* 平方阶(递归实现) */
    +func quadraticRecur(n: Int) -> Int {
    +    if n <= 0 {
    +        return 0
    +    }
    +    // 数组 nums 长度为 n, n-1, ..., 2, 1
    +    let nums = Array(repeating: 0, count: n)
    +    return quadraticRecur(n: n - 1)
    +}
    +
    +
    +
    +
    space_complexity.zig
    // 平方阶(递归实现)
    +fn quadraticRecur(comptime n: i32) i32 {
    +    if (n <= 0) return 0;
    +    var nums = [_]i32{0}**n;
    +    std.debug.print("递归 n = {} 中的 nums 长度 = {}\n", .{n, nums.len});
    +    return quadraticRecur(n - 1);
    +}
    +
    +
    +
    +
    +

    space_complexity_recursive_quadratic

    +

    Fig. 递归函数产生的平方阶空间复杂度

    + +

    指数阶 \(O(2^n)\)

    +

    指数阶常见于二叉树。高度为 \(n\) 的「满二叉树」的结点数量为 \(2^n - 1\) ,使用 \(O(2^n)\) 空间。

    +
    +
    +
    +
    space_complexity.java
    /* 指数阶(建立满二叉树) */
    +TreeNode buildTree(int n) {
    +    if (n == 0) return null;
    +    TreeNode root = new TreeNode(0);
    +    root.left = buildTree(n - 1);
    +    root.right = buildTree(n - 1);
    +    return root;
    +}
    +
    +
    +
    +
    space_complexity.cpp
    /* 指数阶(建立满二叉树) */
    +TreeNode* buildTree(int n) {
    +    if (n == 0) return nullptr;
    +    TreeNode* root = new TreeNode(0);
    +    root->left = buildTree(n - 1);
    +    root->right = buildTree(n - 1);
    +    return root;
    +}
    +
    +
    +
    +
    space_complexity.py
    """ 指数阶(建立满二叉树) """
    +def build_tree(n):
    +    if n == 0: return None
    +    root = TreeNode(0)
    +    root.left = build_tree(n - 1)
    +    root.right = build_tree(n - 1)
    +    return root
    +
    +
    +
    +
    space_complexity.go
    /* 指数阶(建立满二叉树) */
    +func buildTree(n int) *treeNode {
    +    if n == 0 {
    +        return nil
    +    }
    +    root := newTreeNode(0)
    +    root.left = buildTree(n - 1)
    +    root.right = buildTree(n - 1)
    +    return root
    +}
    +
    +
    +
    +
    space_complexity.js
    /* 指数阶(建立满二叉树) */
    +function buildTree(n) {
    +    if (n === 0) return null;
    +    const root = new TreeNode(0);
    +    root.left = buildTree(n - 1);
    +    root.right = buildTree(n - 1);
    +    return root;
    +}
    +
    +
    +
    +
    space_complexity.ts
    /* 指数阶(建立满二叉树) */
    +function buildTree(n: number): TreeNode | null {
    +    if (n === 0) return null;
    +    const root = new TreeNode(0);
    +    root.left = buildTree(n - 1);
    +    root.right = buildTree(n - 1);
    +    return root;
    +}
    +
    +
    +
    +
    space_complexity.c
    
    +
    +
    +
    +
    space_complexity.cs
    /* 指数阶(建立满二叉树) */
    +TreeNode? buildTree(int n)
    +{
    +    if (n == 0) return null;
    +    TreeNode root = new TreeNode(0);
    +    root.left = buildTree(n - 1);
    +    root.right = buildTree(n - 1);
    +    return root;
    +}
    +
    +
    +
    +
    space_complexity.swift
    /* 指数阶(建立满二叉树) */
    +func buildTree(n: Int) -> TreeNode? {
    +    if n == 0 {
    +        return nil
    +    }
    +    let root = TreeNode(x: 0)
    +    root.left = buildTree(n: n - 1)
    +    root.right = buildTree(n: n - 1)
    +    return root
    +}
    +
    +
    +
    +
    space_complexity.zig
    // 指数阶(建立满二叉树)
    +fn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {
    +    if (n == 0) return null;
    +    const root = try mem_allocator.create(inc.TreeNode(i32));
    +    root.init(0);
    +    root.left = try buildTree(mem_allocator, n - 1);
    +    root.right = try buildTree(mem_allocator, n - 1);
    +    return root;
    +}
    +
    +
    +
    +
    +

    space_complexity_exponential

    +

    Fig. 满二叉树下的指数阶空间复杂度

    + +

    对数阶 \(O(\log n)\)

    +

    对数阶常见于分治算法、数据类型转换等。

    +

    例如「归并排序」,长度为 \(n\) 的数组可以形成高度为 \(\log n\) 的递归树,因此空间复杂度为 \(O(\log n)\)

    +

    再例如「数字转化为字符串」,输入任意正整数 \(n\) ,它的位数为 \(\log_{10} n\) ,即对应字符串长度为 \(\log_{10} n\) ,因此空间复杂度为 \(O(\log_{10} n) = O(\log n)\)

    + + + + + +

    评论

    + + + + + + +
    +
    + + + + +
    + + + +
    + + + +
    +
    +
    +
    + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/chapter_computational_complexity/space_time_tradeoff/index.html b/chapter_computational_complexity/space_time_tradeoff/index.html new file mode 100644 index 000000000..516a79b71 --- /dev/null +++ b/chapter_computational_complexity/space_time_tradeoff/index.html @@ -0,0 +1,2104 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + 2.4. 权衡时间与空间 - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + + + 跳转至 + + +
    +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + + + + + +

    2.4. 权衡时间与空间

    +

    理想情况下,我们希望算法的时间复杂度和空间复杂度都能够达到最优,而实际上,同时优化时间复杂度和空间复杂度是非常困难的。

    +

    降低时间复杂度,往往是以提升空间复杂度为代价的,反之亦然。我们把牺牲内存空间来提升算法运行速度的思路称为「以空间换时间」;反之,称之为「以时间换空间」。选择哪种思路取决于我们更看重哪个方面。

    +

    大多数情况下,时间都是比空间更宝贵的,只要空间复杂度不要太离谱、能接受就行,因此以空间换时间最为常用

    +

    2.4.1. 示例题目 *

    +

    以 LeetCode 全站第一题 两数之和 为例。

    +
    +

    两数之和

    +

    给定一个整数数组 nums 和一个整数目标值 target ,请你在该数组中找出“和”为目标值 target 的那两个整数,并返回它们的数组下标。

    +

    你可以假设每种输入只会对应一个答案。但是,数组中同一个元素在答案里不能重复出现。

    +

    你可以按任意顺序返回答案。

    +
    +

    「暴力枚举」和「辅助哈希表」分别为 空间最优时间最优 的两种解法。本着时间比空间更宝贵的原则,后者是本题的最佳解法。

    +

    方法一:暴力枚举

    +

    时间复杂度 \(O(N^2)\) ,空间复杂度 \(O(1)\) ,属于「时间换空间」。

    +

    虽然仅使用常数大小的额外空间,但运行速度过慢。

    +
    +
    +
    +
    leetcode_two_sum.java
    /* 方法一:暴力枚举 */
    +class SolutionBruteForce {
    +    public int[] twoSum(int[] nums, int target) {
    +        int size = nums.length;
    +        // 两层循环,时间复杂度 O(n^2)
    +        for (int i = 0; i < size - 1; i++) {
    +            for (int j = i + 1; j < size; j++) {
    +                if (nums[i] + nums[j] == target)
    +                    return new int[] { i, j };
    +            }
    +        }
    +        return new int[0];
    +    }
    +}
    +
    +
    +
    +
    leetcode_two_sum.cpp
    /* 方法一:暴力枚举 */
    +class SolutionBruteForce {
    +public:
    +    vector<int> twoSum(vector<int>& nums, int target) {
    +        int size = nums.size();
    +        // 两层循环,时间复杂度 O(n^2)
    +        for (int i = 0; i < size - 1; i++) {
    +            for (int j = i + 1; j < size; j++) {
    +                if (nums[i] + nums[j] == target)
    +                    return { i, j };
    +            }
    +        }
    +        return {};
    +    }
    +};
    +
    +
    +
    +
    leetcode_two_sum.py
    """ 方法一:暴力枚举 """
    +class SolutionBruteForce:
    +    def twoSum(self, nums: List[int], target: int) -> List[int]:
    +        # 两层循环,时间复杂度 O(n^2)
    +        for i in range(len(nums) - 1):
    +            for j in range(i + 1, len(nums)):
    +                if nums[i] + nums[j] == target:
    +                    return i, j
    +        return []
    +
    +
    +
    +
    leetcode_two_sum.go
    func twoSumBruteForce(nums []int, target int) []int {
    +    size := len(nums)
    +    // 两层循环,时间复杂度 O(n^2)
    +    for i := 0; i < size-1; i++ {
    +        for j := i + 1; i < size; j++ {
    +            if nums[i]+nums[j] == target {
    +                return []int{i, j}
    +            }
    +        }
    +    }
    +    return nil
    +}
    +
    +
    +
    +
    leetcode_two_sum.js
    function twoSumBruteForce(nums, target) {
    +    const n = nums.length;
    +    // 两层循环,时间复杂度 O(n^2)
    +    for (let i = 0; i < n; i++) {
    +        for (let j = i + 1; j < n; j++) {
    +            if (nums[i] + nums[j] === target) {
    +                return [i, j];
    +            }
    +        }
    +    }
    +    return [];
    +}
    +
    +
    +
    +
    leetcode_two_sum.ts
    function twoSumBruteForce(nums: number[], target: number): number[] {
    +    const n = nums.length;
    +    // 两层循环,时间复杂度 O(n^2)
    +    for (let i = 0; i < n; i++) {
    +        for (let j = i + 1; j < n; j++) {
    +            if (nums[i] + nums[j] === target) {
    +                return [i, j];
    +            }
    +        }
    +    }
    +    return [];
    +};
    +
    +
    +
    +
    leetcode_two_sum.c
    
    +
    +
    +
    +
    leetcode_two_sum.cs
    class SolutionBruteForce
    +{
    +    public int[] twoSum(int[] nums, int target)
    +    {
    +        int size = nums.Length;
    +        // 两层循环,时间复杂度 O(n^2)
    +        for (int i = 0; i < size - 1; i++)
    +        {
    +            for (int j = i + 1; j < size; j++)
    +            {
    +                if (nums[i] + nums[j] == target)
    +                    return new int[] { i, j };
    +            }
    +        }
    +        return new int[0];
    +    }
    +}
    +
    +
    +
    +
    leetcode_two_sum.swift
    func twoSumBruteForce(nums: [Int], target: Int) -> [Int] {
    +    // 两层循环,时间复杂度 O(n^2)
    +    for i in nums.indices.dropLast() {
    +        for j in nums.indices.dropFirst(i + 1) {
    +            if nums[i] + nums[j] == target {
    +                return [i, j]
    +            }
    +        }
    +    }
    +    return [0]
    +}
    +
    +
    +
    +
    leetcode_two_sum.zig
    const SolutionBruteForce = struct {
    +    pub fn twoSum(self: *SolutionBruteForce, nums: []i32, target: i32) [2]i32 {
    +        _ = self;
    +        var size: usize = nums.len;
    +        var i: usize = 0;
    +        // 两层循环,时间复杂度 O(n^2)
    +        while (i < size - 1) : (i += 1) {
    +            var j = i + 1;
    +            while (j < size) : (j += 1) {
    +                if (nums[i] + nums[j] == target) {
    +                    return [_]i32{@intCast(i32, i), @intCast(i32, j)};
    +                }
    +            }
    +        }
    +        return undefined;
    +    }
    +};
    +
    +
    +
    +
    +

    方法二:辅助哈希表

    +

    时间复杂度 \(O(N)\) ,空间复杂度 \(O(N)\) ,属于「空间换时间」。

    +

    借助辅助哈希表 dic ,通过保存数组元素与索引的映射来提升算法运行速度。

    +
    +
    +
    +
    leetcode_two_sum.java
    /* 方法二:辅助哈希表 */
    +class SolutionHashMap {
    +    public int[] twoSum(int[] nums, int target) {
    +        int size = nums.length;
    +        // 辅助哈希表,空间复杂度 O(n)
    +        Map<Integer, Integer> dic = new HashMap<>();
    +        // 单层循环,时间复杂度 O(n)
    +        for (int i = 0; i < size; i++) {
    +            if (dic.containsKey(target - nums[i])) {
    +                return new int[] { dic.get(target - nums[i]), i };
    +            }
    +            dic.put(nums[i], i);
    +        }
    +        return new int[0];
    +    }
    +}
    +
    +
    +
    +
    leetcode_two_sum.cpp
    /* 方法二:辅助哈希表 */
    +class SolutionHashMap {
    +public:
    +    vector<int> twoSum(vector<int>& nums, int target) {
    +        int size = nums.size();
    +        // 辅助哈希表,空间复杂度 O(n)
    +        unordered_map<int, int> dic;
    +        // 单层循环,时间复杂度 O(n)
    +        for (int i = 0; i < size; i++) {
    +            if (dic.find(target - nums[i]) != dic.end()) {
    +                return { dic[target - nums[i]], i };
    +            }
    +            dic.emplace(nums[i], i);
    +        }
    +        return {};
    +    }
    +};
    +
    +
    +
    +
    leetcode_two_sum.py
    """ 方法二:辅助哈希表 """
    +class SolutionHashMap:
    +    def twoSum(self, nums: List[int], target: int) -> List[int]:
    +        # 辅助哈希表,空间复杂度 O(n)
    +        dic = {}
    +        # 单层循环,时间复杂度 O(n)
    +        for i in range(len(nums)):
    +            if target - nums[i] in dic:
    +                return dic[target - nums[i]], i
    +            dic[nums[i]] = i
    +        return []
    +
    +
    +
    +
    leetcode_two_sum.go
    func twoSumHashTable(nums []int, target int) []int {
    +    // 辅助哈希表,空间复杂度 O(n)
    +    hashTable := map[int]int{}
    +    // 单层循环,时间复杂度 O(n)
    +    for idx, val := range nums {
    +        if preIdx, ok := hashTable[target-val]; ok {
    +            return []int{preIdx, idx}
    +        }
    +        hashTable[val] = idx
    +    }
    +    return nil
    +}
    +
    +
    +
    +
    leetcode_two_sum.js
    function twoSumHashTable(nums, target) {
    +    // 辅助哈希表,空间复杂度 O(n)
    +    let m = {};
    +    // 单层循环,时间复杂度 O(n)
    +    for (let i = 0; i < nums.length; i++) {
    +        if (m[nums[i]] !== undefined) {
    +            return [m[nums[i]], i];
    +        } else {
    +            m[target - nums[i]] = i;
    +        }
    +    }
    +    return [];
    +}
    +
    +
    +
    +
    leetcode_two_sum.ts
    function twoSumHashTable(nums: number[], target: number): number[] {
    +    // 辅助哈希表,空间复杂度 O(n)
    +    let m: Map<number, number> = new Map();
    +    // 单层循环,时间复杂度 O(n)
    +    for (let i = 0; i < nums.length; i++) {
    +        let index = m.get(nums[i]);
    +        if (index !== undefined) {
    +            return [index, i];
    +        } else {
    +            m.set(target - nums[i], i);
    +        }
    +    }
    +    return [];
    +};
    +
    +
    +
    +
    leetcode_two_sum.c
    
    +
    +
    +
    +
    leetcode_two_sum.cs
    class SolutionHashMap
    +{
    +    public int[] twoSum(int[] nums, int target)
    +    {
    +        int size = nums.Length;
    +        // 辅助哈希表,空间复杂度 O(n)
    +        Dictionary<int, int> dic = new();
    +        // 单层循环,时间复杂度 O(n)
    +        for (int i = 0; i < size; i++)
    +        {
    +            if (dic.ContainsKey(target - nums[i]))
    +            {
    +                return new int[] { dic[target - nums[i]], i };
    +            }
    +            dic.Add(nums[i], i);
    +        }
    +        return new int[0];
    +    }
    +}
    +
    +
    +
    +
    leetcode_two_sum.swift
    func twoSumHashTable(nums: [Int], target: Int) -> [Int] {
    +    // 辅助哈希表,空间复杂度 O(n)
    +    var dic: [Int: Int] = [:]
    +    // 单层循环,时间复杂度 O(n)
    +    for i in nums.indices {
    +        if let j = dic[target - nums[i]] {
    +            return [j, i]
    +        }
    +        dic[nums[i]] = i
    +    }
    +    return [0]
    +}
    +
    +
    +
    +
    leetcode_two_sum.zig
    const SolutionHashMap = struct {
    +    pub fn twoSum(self: *SolutionHashMap, nums: []i32, target: i32) ![2]i32 {
    +        _ = self;
    +        var size: usize = nums.len;
    +        // 辅助哈希表,空间复杂度 O(n)
    +        var dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);
    +        defer dic.deinit();
    +        var i: usize = 0;
    +        // 单层循环,时间复杂度 O(n)
    +        while (i < size) : (i += 1) {
    +            if (dic.contains(target - nums[i])) {
    +                return [_]i32{dic.get(target - nums[i]).?, @intCast(i32, i)};
    +            }
    +            try dic.put(nums[i], @intCast(i32, i));
    +        }
    +        return undefined;
    +    }
    +};
    +
    +
    +
    +
    + + + + + +

    评论

    + + + + + + +
    +
    + + + + +
    + + + +
    + + + +
    +
    +
    +
    + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/chapter_computational_complexity/summary/index.html b/chapter_computational_complexity/summary/index.html new file mode 100644 index 000000000..98ce54c71 --- /dev/null +++ b/chapter_computational_complexity/summary/index.html @@ -0,0 +1,1779 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + 2.5. 小结 - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + + + 跳转至 + + +
    +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + + + + + +

    2.5. 小结

    +

    算法效率评估

    +
      +
    • 「时间效率」和「空间效率」是算法性能的两个重要的评价维度。
    • +
    • 我们可以通过「实际测试」来评估算法效率,但难以排除测试环境的干扰,并且非常耗费计算资源。
    • +
    • 「复杂度分析」克服了实际测试的弊端,分析结果适用于所有运行平台,并且可以体现不同数据大小下的算法效率。
    • +
    +

    时间复杂度

    +
      +
    • 「时间复杂度」统计算法运行时间随着数据量变大时的增长趋势,可以有效评估算法效率,但在某些情况下可能失效,比如在输入数据量较小或时间复杂度相同时,无法精确对比算法效率的优劣性。
    • +
    • 「最差时间复杂度」使用大 \(O\) 符号表示,即函数渐近上界,其反映当 \(n\) 趋于正无穷时,\(T(n)\) 处于何种增长级别。
    • +
    • 推算时间复杂度分为两步,首先统计计算操作数量,再判断渐近上界。
    • +
    • 常见时间复杂度从小到大排列有 \(O(1)\) , \(O(\log n)\) , \(O(n)\) , \(O(n \log n)\) , \(O(n^2)\) , \(O(2^n)\) , \(O(n!)\)
    • +
    • 某些算法的时间复杂度不是恒定的,而是与输入数据的分布有关。时间复杂度分为「最差时间复杂度」和「最佳时间复杂度」,后者几乎不用,因为输入数据需要满足苛刻的条件才能达到最佳情况。
    • +
    • 「平均时间复杂度」可以反映在随机数据输入下的算法效率,最贴合实际使用情况下的算法性能。计算平均时间复杂度需要统计输入数据的分布,以及综合后的数学期望。
    • +
    +

    空间复杂度

    +
      +
    • +

      与时间复杂度的定义类似,「空间复杂度」统计算法占用空间随着数据量变大时的增长趋势。

      +
    • +
    • +

      算法运行中相关内存空间可分为输入空间、暂存空间、输出空间。通常情况下,输入空间不计入空间复杂度计算。暂存空间可分为指令空间、数据空间、栈帧空间,其中栈帧空间一般在递归函数中才会影响到空间复杂度。

      +
    • +
    • 我们一般只关心「最差空间复杂度」,即统计算法在「最差输入数据」和「最差运行时间点」下的空间复杂度。
    • +
    • 常见空间复杂度从小到大排列有 \(O(1)\) , \(O(\log n)\) , \(O(n)\) , \(O(n^2)\) , \(O(2^n)\)
    • +
    + + + + + +

    评论

    + + + + + + +
    +
    + + + + +
    + + + +
    + + + +
    +
    +
    +
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zqYOAQ*3QwH9Qe_)vcvVKEQBLpVm!(a{Ka1ww_KSo}KlX8&-wI?u}d zax?b_J745GZe?y;YPoA+5f!cun)LPd4%t`rDG=Yr(S}T6OW1N!@d2Sw`t+S40u*aD zAs!+EHytCwnM_6sCj7uM$g|(;bKdvolIby?558D^aa)(E=ITJ+uNJZwsJqx{77BP@ zFdnZC$IQ!rRNNjgd2!9=n#L7a^-8d`RZw@nLX;vE4k-^#`A*Osahy?vShf!KU+ztfkex +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + + + + + +
    +
    + + + + + + + + + + + + + +

    2.2. 时间复杂度

    +

    2.2.1. 统计算法运行时间

    +

    运行时间能够直观且准确地体现出算法的效率水平。如果我们想要 准确预估一段代码的运行时间 ,该如何做呢?

    +
      +
    1. 首先需要 确定运行平台 ,包括硬件配置、编程语言、系统环境等,这些都会影响到代码的运行效率。
    2. +
    3. 评估 各种计算操作的所需运行时间 ,例如加法操作 + 需要 1 ns ,乘法操作 * 需要 10 ns ,打印操作需要 5 ns 等。
    4. +
    5. 根据代码 统计所有计算操作的数量 ,并将所有操作的执行时间求和,即可得到运行时间。
    6. +
    +

    例如以下代码,输入数据大小为 \(n\) ,根据以上方法,可以得到算法运行时间为 \(6n + 12\) ns 。

    +
    \[ +1 + 1 + 10 + (1 + 5) \times n = 6n + 12 +\]
    +
    +
    +
    +
    // 在某运行平台下
    +void algorithm(int n) {
    +    int a = 2;  // 1 ns
    +    a = a + 1;  // 1 ns
    +    a = a * 2;  // 10 ns
    +    // 循环 n 次
    +    for (int i = 0; i < n; i++) {  // 1 ns ,每轮都要执行 i++
    +        System.out.println(0);     // 5 ns
    +    }
    +}
    +
    +
    +
    +
    // 在某运行平台下
    +void algorithm(int n) {
    +    int a = 2;  // 1 ns
    +    a = a + 1;  // 1 ns
    +    a = a * 2;  // 10 ns
    +    // 循环 n 次
    +    for (int i = 0; i < n; i++) {  // 1 ns ,每轮都要执行 i++
    +        cout << 0 << endl;         // 5 ns
    +    }
    +}
    +
    +
    +
    +
    # 在某运行平台下
    +def algorithm(n):
    +    a = 2      # 1 ns
    +    a = a + 1  # 1 ns
    +    a = a * 2  # 10 ns
    +    # 循环 n 次
    +    for _ in range(n):  # 1 ns
    +        print(0)        # 5 ns
    +
    +
    +
    +
    // 在某运行平台下
    +func algorithm(n int) {
    +    a := 2      // 1 ns
    +    a = a + 1   // 1 ns
    +    a = a * 2   // 10 ns
    +    // 循环 n 次
    +    for i := 0; i < n; i++ {    // 1 ns
    +        fmt.Println(a)          // 5 ns
    +    }
    +}
    +
    +
    +
    +
    // 在某运行平台下
    +function algorithm(n) {
    +    var a = 2; // 1 ns
    +    a = a + 1; // 1 ns
    +    a = a * 2; // 10 ns
    +    // 循环 n 次
    +    for(let i = 0; i < n; i++) { // 1 ns ,每轮都要执行 i++
    +        console.log(0); // 5 ns
    +    }
    +}
    +
    +
    +
    +
    // 在某运行平台下
    +function algorithm(n: number): void {
    +    var a: number = 2; // 1 ns
    +    a = a + 1; // 1 ns
    +    a = a * 2; // 10 ns
    +    // 循环 n 次
    +    for(let i = 0; i < n; i++) { // 1 ns ,每轮都要执行 i++
    +        console.log(0); // 5 ns
    +    }
    +}
    +
    +
    +
    +
    // 在某运行平台下
    +void algorithm(int n) {
    +    int a = 2;  // 1 ns
    +    a = a + 1;  // 1 ns
    +    a = a * 2;  // 10 ns
    +    // 循环 n 次
    +    for (int i = 0; i < n; i++) {   // 1 ns ,每轮都要执行 i++
    +        printf("%d", 0);            // 5 ns
    +    }
    +}
    +
    +
    +
    +
    // 在某运行平台下
    +void algorithm(int n)
    +{
    +    int a = 2;  // 1 ns
    +    a = a + 1;  // 1 ns
    +    a = a * 2;  // 10 ns
    +    // 循环 n 次
    +    for (int i = 0; i < n; i++)
    +    {  // 1 ns ,每轮都要执行 i++
    +        Console.WriteLine(0);     // 5 ns
    +    }
    +}
    +
    +
    +
    +
    // 在某运行平台下
    +func algorithm(_ n: Int) {
    +    var a = 2 // 1 ns
    +    a = a + 1 // 1 ns
    +    a = a * 2 // 10 ns
    +    // 循环 n 次
    +    for _ in 0 ..< n { // 1 ns
    +        print(0) // 5 ns
    +    }
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    +

    但实际上, 统计算法的运行时间既不合理也不现实。首先,我们不希望预估时间和运行平台绑定,毕竟算法需要跑在各式各样的平台之上。其次,我们很难获知每一种操作的运行时间,这为预估过程带来了极大的难度。

    +

    2.2.2. 统计时间增长趋势

    +

    「时间复杂度分析」采取了不同的做法,其统计的不是算法运行时间,而是 算法运行时间随着数据量变大时的增长趋势

    +

    “时间增长趋势”这个概念比较抽象,我们借助一个例子来理解。设输入数据大小为 \(n\) ,给定三个算法 A , B , C

    +
      +
    • 算法 A 只有 \(1\) 个打印操作,算法运行时间不随着 \(n\) 增大而增长。我们称此算法的时间复杂度为「常数阶」。
    • +
    • 算法 B 中的打印操作需要循环 \(n\) 次,算法运行时间随着 \(n\) 增大成线性增长。此算法的时间复杂度被称为「线性阶」。
    • +
    • 算法 C 中的打印操作需要循环 \(1000000\) 次,但运行时间仍与输入数据大小 \(n\) 无关。因此 C 的时间复杂度和 A 相同,仍为「常数阶」。
    • +
    +
    +
    +
    +
    // 算法 A 时间复杂度:常数阶
    +void algorithm_A(int n) {
    +    System.out.println(0);
    +}
    +// 算法 B 时间复杂度:线性阶
    +void algorithm_B(int n) {
    +    for (int i = 0; i < n; i++) {
    +        System.out.println(0);
    +    }
    +}
    +// 算法 C 时间复杂度:常数阶
    +void algorithm_C(int n) {
    +    for (int i = 0; i < 1000000; i++) {
    +        System.out.println(0);
    +    }
    +}
    +
    +
    +
    +
    // 算法 A 时间复杂度:常数阶
    +void algorithm_A(int n) {
    +    cout << 0 << endl;
    +}
    +// 算法 B 时间复杂度:线性阶
    +void algorithm_B(int n) {
    +    for (int i = 0; i < n; i++) {
    +        cout << 0 << endl;
    +    }
    +}
    +// 算法 C 时间复杂度:常数阶
    +void algorithm_C(int n) {
    +    for (int i = 0; i < 1000000; i++) {
    +        cout << 0 << endl;
    +    }
    +}
    +
    +
    +
    +
    # 算法 A 时间复杂度:常数阶
    +def algorithm_A(n):
    +    print(0)
    +# 算法 B 时间复杂度:线性阶
    +def algorithm_B(n):
    +    for _ in range(n):
    +        print(0)
    +# 算法 C 时间复杂度:常数阶
    +def algorithm_C(n):
    +    for _ in range(1000000):
    +        print(0)
    +
    +
    +
    +
    // 算法 A 时间复杂度:常数阶
    +func algorithm_A(n int) {
    +    fmt.Println(0)
    +}
    +// 算法 B 时间复杂度:线性阶
    +func algorithm_B(n int) {
    +    for i := 0; i < n; i++ {
    +        fmt.Println(0)
    +    }
    +}
    +// 算法 C 时间复杂度:常数阶
    +func algorithm_C(n int) {
    +    for i := 0; i < 1000000; i++ {
    +        fmt.Println(0)
    +    }
    +}
    +
    +
    +
    +
    // 算法 A 时间复杂度:常数阶
    +function algorithm_A(n) {
    +    console.log(0);
    +}
    +// 算法 B 时间复杂度:线性阶
    +function algorithm_B(n) {
    +    for (let i = 0; i < n; i++) {
    +        console.log(0);
    +    }
    +}
    +// 算法 C 时间复杂度:常数阶
    +function algorithm_C(n) {
    +    for (let i = 0; i < 1000000; i++) {
    +        console.log(0);
    +    }
    +}
    +
    +
    +
    +
    // 算法 A 时间复杂度:常数阶
    +function algorithm_A(n: number): void {
    +    console.log(0);
    +}
    +// 算法 B 时间复杂度:线性阶
    +function algorithm_B(n: number): void {
    +    for (let i = 0; i < n; i++) {
    +        console.log(0);
    +    }
    +}
    +// 算法 C 时间复杂度:常数阶
    +function algorithm_C(n: number): void {
    +    for (let i = 0; i < 1000000; i++) {
    +        console.log(0);
    +    }
    +}
    +
    +
    +
    +
    // 算法 A 时间复杂度:常数阶
    +void algorithm_A(int n) {
    +    printf("%d", 0);
    +}
    +// 算法 B 时间复杂度:线性阶
    +void algorithm_B(int n) {
    +    for (int i = 0; i < n; i++) {
    +        printf("%d", 0);
    +    }
    +}
    +// 算法 C 时间复杂度:常数阶
    +void algorithm_C(int n) {
    +    for (int i = 0; i < 1000000; i++) {
    +        printf("%d", 0);
    +    }
    +}
    +
    +
    +
    +
    // 算法 A 时间复杂度:常数阶
    +void algorithm_A(int n)
    +{
    +    Console.WriteLine(0);
    +}
    +// 算法 B 时间复杂度:线性阶
    +void algorithm_B(int n)
    +{
    +    for (int i = 0; i < n; i++)
    +    {
    +        Console.WriteLine(0);
    +    }
    +}
    +// 算法 C 时间复杂度:常数阶
    +void algorithm_C(int n)
    +{
    +    for (int i = 0; i < 1000000; i++)
    +    {
    +        Console.WriteLine(0);
    +    }
    +}
    +
    +
    +
    +
    // 算法 A 时间复杂度:常数阶
    +func algorithmA(_ n: Int) {
    +    print(0)
    +}
    +
    +// 算法 B 时间复杂度:线性阶
    +func algorithmB(_ n: Int) {
    +    for _ in 0 ..< n {
    +        print(0)
    +    }
    +}
    +
    +// 算法 C 时间复杂度:常数阶
    +func algorithmC(_ n: Int) {
    +    for _ in 0 ..< 1000000 {
    +        print(0)
    +    }
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    +

    time_complexity_first_example

    +

    Fig. 算法 A, B, C 的时间增长趋势

    + +

    相比直接统计算法运行时间,时间复杂度分析的做法有什么好处呢?以及有什么不足?

    +

    时间复杂度可以有效评估算法效率。算法 B 运行时间的增长是线性的,在 \(n > 1\) 时慢于算法 A ,在 \(n > 1000000\) 时慢于算法 C 。实质上,只要输入数据大小 \(n\) 足够大,复杂度为「常数阶」的算法一定优于「线性阶」的算法,这也正是时间增长趋势的含义。

    +

    时间复杂度的推算方法更加简便。在时间复杂度分析中,我们可以将统计「计算操作的运行时间」简化为统计「计算操作的数量」,这是因为,无论是运行平台还是计算操作类型,都与算法运行时间的增长趋势无关。因而,我们可以简单地将所有计算操作的执行时间统一看作是相同的“单位时间”,这样的简化做法大大降低了估算难度。

    +

    时间复杂度也存在一定的局限性。比如,虽然算法 AC 的时间复杂度相同,但是实际的运行时间有非常大的差别。再比如,虽然算法 BC 的时间复杂度要更高,但在输入数据大小 \(n\) 比较小时,算法 B 是要明显优于算法 C 的。对于以上情况,我们很难仅凭时间复杂度来判定算法效率高低。然而,即使存在这些问题,计算复杂度仍然是评判算法效率的最有效且常用的方法。

    +

    2.2.3. 函数渐近上界

    +

    设算法「计算操作数量」为 \(T(n)\) ,其是一个关于输入数据大小 \(n\) 的函数。例如,以下算法的操作数量为

    +
    \[ +T(n) = 3 + 2n +\]
    +
    +
    +
    +
    void algorithm(int n) {
    +    int a = 1;  // +1
    +    a = a + 1;  // +1
    +    a = a * 2;  // +1
    +    // 循环 n 次
    +    for (int i = 0; i < n; i++) { // +1(每轮都执行 i ++)
    +        System.out.println(0);    // +1
    +    }
    +}
    +
    +
    +
    +
    void algorithm(int n) {
    +    int a = 1;  // +1
    +    a = a + 1;  // +1
    +    a = a * 2;  // +1
    +    // 循环 n 次
    +    for (int i = 0; i < n; i++) { // +1(每轮都执行 i ++)
    +        cout << 0 << endl;    // +1
    +    }
    +}
    +
    +
    +
    +
    def algorithm(n):
    +    a = 1  # +1
    +    a = a + 1  # +1
    +    a = a * 2  # +1
    +    # 循环 n 次
    +    for i in range(n):  # +1
    +        print(0)        # +1
    +
    +
    +
    +
    func algorithm(n int) {
    +    a := 1      // +1
    +    a = a + 1   // +1
    +    a = a * 2   // +1
    +    // 循环 n 次
    +    for i := 0; i < n; i++ {   // +1
    +        fmt.Println(a)         // +1
    +    }
    +}
    +
    +
    +
    +
    function algorithm(n){
    +    var a = 1; // +1
    +    a += 1; // +1
    +    a *= 2; // +1
    +    // 循环 n 次
    +    for(let i = 0; i < n; i++){ // +1(每轮都执行 i ++)
    +        console.log(0); // +1
    +    }
    +
    +}
    +
    +
    +
    +
    function algorithm(n: number): void{
    +    var a: number = 1; // +1
    +    a += 1; // +1
    +    a *= 2; // +1
    +    // 循环 n 次
    +    for(let i = 0; i < n; i++){ // +1(每轮都执行 i ++)
    +        console.log(0); // +1
    +    }
    +
    +}
    +
    +
    +
    +
    void algorithm(int n) {
    +    int a = 1;  // +1
    +    a = a + 1;  // +1
    +    a = a * 2;  // +1
    +    // 循环 n 次
    +    for (int i = 0; i < n; i++) {   // +1(每轮都执行 i ++)
    +        printf("%d", 0);            // +1
    +    }
    +}  
    +
    +
    +
    +
    void algorithm(int n) {
    +    int a = 1;  // +1
    +    a = a + 1;  // +1
    +    a = a * 2;  // +1
    +    // 循环 n 次
    +    for (int i = 0; i < n; i++) { // +1(每轮都执行 i ++)
    +        Console.WriteLine(0);     // +1
    +    }
    +}
    +
    +
    +
    +
    func algorithm(n: Int) {
    +    var a = 1 // +1
    +    a = a + 1 // +1
    +    a = a * 2 // +1
    +    // 循环 n 次
    +    for _ in 0 ..< n { // +1
    +        print(0) // +1
    +    }
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    +

    \(T(n)\) 是个一次函数,说明时间增长趋势是线性的,因此易得时间复杂度是线性阶。

    +

    我们将线性阶的时间复杂度记为 \(O(n)\) ,这个数学符号被称为「大 \(O\) 记号 Big-\(O\) Notation」,代表函数 \(T(n)\) 的「渐近上界 asymptotic upper bound」。

    +

    我们要推算时间复杂度,本质上是在计算「操作数量函数 \(T(n)\) 」的渐近上界。下面我们先来看看函数渐近上界的数学定义。

    +
    +

    函数渐近上界

    +

    若存在正实数 \(c\) 和实数 \(n_0\) ,使得对于所有的 \(n > n_0\) ,均有 +$$ +T(n) \leq c \cdot f(n) +$$ +则可认为 \(f(n)\) 给出了 \(T(n)\) 的一个渐近上界,记为 +$$ +T(n) = O(f(n)) +$$

    +
    +

    asymptotic_upper_bound

    +

    Fig. 函数的渐近上界

    + +

    本质上看,计算渐近上界就是在找一个函数 \(f(n)\)使得在 \(n\) 趋向于无穷大时,\(T(n)\)\(f(n)\) 处于相同的增长级别(仅相差一个常数项 \(c\) 的倍数)

    +
    +

    Tip

    +

    渐近上界的数学味儿有点重,如果你感觉没有完全理解,无需担心,因为在实际使用中我们只需要会推算即可,数学意义可以慢慢领悟。

    +
    +

    2.2.4. 推算方法

    +

    推算出 \(f(n)\) 后,我们就得到时间复杂度 \(O(f(n))\) 。那么,如何来确定渐近上界 \(f(n)\) 呢?总体分为两步,首先「统计操作数量」,然后「判断渐近上界」。

    +

    1) 统计操作数量

    +

    对着代码,从上到下一行一行地计数即可。然而,由于上述 \(c \cdot f(n)\) 中的常数项 \(c\) 可以取任意大小,因此操作数量 \(T(n)\) 中的各种系数、常数项都可以被忽略。根据此原则,可以总结出以下计数偷懒技巧:

    +
      +
    1. 跳过数量与 \(n\) 无关的操作。因为他们都是 \(T(n)\) 中的常数项,对时间复杂度不产生影响。
    2. +
    3. 省略所有系数。例如,循环 \(2n\) 次、\(5n + 1\) 次、……,都可以化简记为 \(n\) 次,因为 \(n\) 前面的系数对时间复杂度也不产生影响。
    4. +
    5. 循环嵌套时使用乘法。总操作数量等于外层循环和内层循环操作数量之积,每一层循环依然可以分别套用上述 1.2. 技巧。
    6. +
    +

    根据以下示例,使用上述技巧前、后的统计结果分别为

    +
    \[ +\begin{aligned} +T(n) & = 2n(n + 1) + (5n + 1) + 2 & \text{完整统计 (-.-|||)} \newline +& = 2n^2 + 7n + 3 \newline +T(n) & = n^2 + n & \text{偷懒统计 (o.O)} +\end{aligned} +\]
    +

    最终,两者都能推出相同的时间复杂度结果,即 \(O(n^2)\)

    +
    +
    +
    +
    void algorithm(int n) {
    +    int a = 1;  // +0(技巧 1)
    +    a = a + n;  // +0(技巧 1)
    +    // +n(技巧 2)
    +    for (int i = 0; i < 5 * n + 1; i++) {
    +        System.out.println(0);
    +    }
    +    // +n*n(技巧 3)
    +    for (int i = 0; i < 2 * n; i++) {
    +        for (int j = 0; j < n + 1; j++) {
    +            System.out.println(0);
    +        }
    +    }
    +}
    +
    +
    +
    +
    void algorithm(int n) {
    +    int a = 1;  // +0(技巧 1)
    +    a = a + n;  // +0(技巧 1)
    +    // +n(技巧 2)
    +    for (int i = 0; i < 5 * n + 1; i++) {
    +        cout << 0 << endl;
    +    }
    +    // +n*n(技巧 3)
    +    for (int i = 0; i < 2 * n; i++) {
    +        for (int j = 0; j < n + 1; j++) {
    +            cout << 0 << endl;
    +        }
    +    }
    +}
    +
    +
    +
    +
    def algorithm(n):
    +    a = 1      # +0(技巧 1)
    +    a = a + n  # +0(技巧 1)
    +    # +n(技巧 2)
    +    for i in range(5 * n + 1):
    +        print(0)
    +    # +n*n(技巧 3)
    +    for i in range(2 * n):
    +        for j in range(n + 1):
    +            print(0)
    +
    +
    +
    +
    func algorithm(n int) {
    +    a := 1      // +0(技巧 1)
    +    a = a + n  // +0(技巧 1)
    +    // +n(技巧 2)
    +    for i := 0; i < 5 * n + 1; i++ {
    +        fmt.Println(0)
    +    }
    +    // +n*n(技巧 3)
    +    for i := 0; i < 2 * n; i++ {
    +        for j := 0; j < n + 1; j++ {
    +            fmt.Println(0)
    +        }
    +    }
    +}
    +
    +
    +
    +
    function algorithm(n) {
    +    let a = 1;  // +0(技巧 1)
    +    a = a + n;  // +0(技巧 1)
    +    // +n(技巧 2)
    +    for (let i = 0; i < 5 * n + 1; i++) {
    +        console.log(0);
    +    }
    +    // +n*n(技巧 3)
    +    for (let i = 0; i < 2 * n; i++) {
    +        for (let j = 0; j < n + 1; j++) {
    +            console.log(0);
    +        }
    +    }
    +}
    +
    +
    +
    +
    function algorithm(n: number): void {
    +    let a = 1;  // +0(技巧 1)
    +    a = a + n;  // +0(技巧 1)
    +    // +n(技巧 2)
    +    for (let i = 0; i < 5 * n + 1; i++) {
    +        console.log(0);
    +    }
    +    // +n*n(技巧 3)
    +    for (let i = 0; i < 2 * n; i++) {
    +        for (let j = 0; j < n + 1; j++) {
    +            console.log(0);
    +        }
    +    }
    +}
    +
    +
    +
    +
    void algorithm(int n) {
    +    int a = 1;  // +0(技巧 1)
    +    a = a + n;  // +0(技巧 1)
    +    // +n(技巧 2)
    +    for (int i = 0; i < 5 * n + 1; i++) {
    +        printf("%d", 0);
    +    }
    +    // +n*n(技巧 3)
    +    for (int i = 0; i < 2 * n; i++) {
    +        for (int j = 0; j < n + 1; j++) {
    +            printf("%d", 0);
    +        }
    +    }
    +}
    +
    +
    +
    +
    void algorithm(int n)
    +{
    +    int a = 1;  // +0(技巧 1)
    +    a = a + n;  // +0(技巧 1)
    +    // +n(技巧 2)
    +    for (int i = 0; i < 5 * n + 1; i++)
    +    {
    +        Console.WriteLine(0);
    +    }
    +    // +n*n(技巧 3)
    +    for (int i = 0; i < 2 * n; i++)
    +    {
    +        for (int j = 0; j < n + 1; j++)
    +        {
    +            Console.WriteLine(0);
    +        }
    +    }
    +}
    +
    +
    +
    +
    func algorithm(n: Int) {
    +    var a = 1 // +0(技巧 1)
    +    a = a + n // +0(技巧 1)
    +    // +n(技巧 2)
    +    for _ in 0 ..< (5 * n + 1) {
    +        print(0)
    +    }
    +    // +n*n(技巧 3)
    +    for _ in 0 ..< (2 * n) {
    +        for _ in 0 ..< (n + 1) {
    +            print(0)
    +        }
    +    }
    +}
    +
    +
    +
    +
    
    +
    +
    +
    +
    +

    2) 判断渐近上界

    +

    时间复杂度由多项式 \(T(n)\) 中最高阶的项来决定。这是因为在 \(n\) 趋于无穷大时,最高阶的项将处于主导作用,其它项的影响都可以被忽略。

    +

    以下表格给出了一些例子,其中有一些夸张的值,是想要向大家强调 系数无法撼动阶数 这一结论。在 \(n\) 趋于无穷大时,这些常数都是“浮云”。

    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    操作数量 \(T(n)\)时间复杂度 \(O(f(n))\)
    \(100000\)\(O(1)\)
    \(3n + 2\)\(O(n)\)
    \(2n^2 + 3n + 2\)\(O(n^2)\)
    \(n^3 + 10000n^2\)\(O(n^3)\)
    \(2^n + 10000n^{10000}\)\(O(2^n)\)
    +
    +

    2.2.5. 常见类型

    +

    设输入数据大小为 \(n\) ,常见的时间复杂度类型有(从低到高排列)

    +
    \[ +\begin{aligned} +O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!) \newline +\text{常数阶} < \text{对数阶} < \text{线性阶} < \text{线性对数阶} < \text{平方阶} < \text{指数阶} < \text{阶乘阶} +\end{aligned} +\]
    +

    time_complexity_common_types

    +

    Fig. 时间复杂度的常见类型

    + +
    +

    Tip

    +

    部分示例代码需要一些前置知识,包括数组、递归算法等。如果遇到看不懂的地方无需担心,可以在学习完后面章节后再来复习,现阶段先聚焦在理解时间复杂度含义和推算方法上。

    +
    +

    常数阶 \(O(1)\)

    +

    常数阶的操作数量与输入数据大小 \(n\) 无关,即不随着 \(n\) 的变化而变化。

    +

    对于以下算法,无论操作数量 size 有多大,只要与数据大小 \(n\) 无关,时间复杂度就仍为 \(O(1)\)

    +
    +
    +
    +
    time_complexity.java
    /* 常数阶 */
    +int constant(int n) {
    +    int count = 0;
    +    int size = 100000;
    +    for (int i = 0; i < size; i++)
    +        count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 常数阶 */
    +int constant(int n) {
    +    int count = 0;
    +    int size = 100000;
    +    for (int i = 0; i < size; i++)
    +        count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 常数阶 """
    +def constant(n):
    +    count = 0
    +    size = 100000
    +    for _ in range(size):
    +        count += 1
    +    return count
    +
    +
    +
    +
    time_complexity.go
    /* 常数阶 */
    +func constant(n int) int {
    +    count := 0
    +    size := 100000
    +    for i := 0; i < size; i++ {
    +        count ++
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 常数阶 */
    +function constant(n) {
    +    let count = 0;
    +    const size = 100000;
    +    for (let i = 0; i < size; i++) count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 常数阶 */
    +function constant(n: number): number {
    +    let count = 0;
    +    const size = 100000;
    +    for (let i = 0; i < size; i++) count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 常数阶 */
    +int constant(int n) {
    +    int count = 0;
    +    int size = 100000;
    +    int i = 0;
    +    for (int i = 0; i < size; i++) {
    +        count ++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 常数阶 */
    +int constant(int n)
    +{
    +    int count = 0;
    +    int size = 100000;
    +    for (int i = 0; i < size; i++)
    +        count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 常数阶 */
    +func constant(n: Int) -> Int {
    +    var count = 0
    +    let size = 100000
    +    for _ in 0 ..< size {
    +        count += 1
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 常数阶
    +fn constant(n: i32) i32 {
    +    _ = n;
    +    var count: i32 = 0;
    +    const size: i32 = 100_000;
    +    var i: i32 = 0;
    +    while(i<size) : (i += 1) {
    +        count += 1;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    +

    线性阶 \(O(n)\)

    +

    线性阶的操作数量相对输入数据大小成线性级别增长。线性阶常出现于单层循环。

    +
    +
    +
    +
    time_complexity.java
    /* 线性阶 */
    +int linear(int n) {
    +    int count = 0;
    +    for (int i = 0; i < n; i++)
    +        count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 线性阶 */
    +int linear(int n) {
    +    int count = 0;
    +    for (int i = 0; i < n; i++)
    +        count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 线性阶 """
    +def linear(n):
    +    count = 0
    +    for _ in range(n):
    +        count += 1
    +    return count
    +
    +
    +
    +
    time_complexity.go
    /* 线性阶 */
    +func linear(n int) int {
    +    count := 0
    +    for i := 0; i < n; i++ {
    +        count++
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 线性阶 */
    +function linear(n) {
    +    let count = 0;
    +    for (let i = 0; i < n; i++) count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 线性阶 */
    +function linear(n: number): number {
    +    let count = 0;
    +    for (let i = 0; i < n; i++) count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 线性阶 */
    +int linear(int n) {
    +    int count = 0;
    +    for (int i = 0; i < n; i++) {
    +        count ++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 线性阶 */
    +int linear(int n)
    +{
    +    int count = 0;
    +    for (int i = 0; i < n; i++)
    +        count++;
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 线性阶 */
    +func linear(n: Int) -> Int {
    +    var count = 0
    +    for _ in 0 ..< n {
    +        count += 1
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 线性阶
    +fn linear(n: i32) i32 {
    +    var count: i32 = 0;
    +    var i: i32 = 0;
    +    while (i < n) : (i += 1) {
    +        count += 1;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    +

    「遍历数组」和「遍历链表」等操作,时间复杂度都为 \(O(n)\) ,其中 \(n\) 为数组或链表的长度。

    +
    +

    Tip

    +

    数据大小 \(n\) 是根据输入数据的类型来确定的。比如,在上述示例中,我们直接将 \(n\) 看作输入数据大小;以下遍历数组示例中,数据大小 \(n\) 为数组的长度。

    +
    +
    +
    +
    +
    time_complexity.java
    /* 线性阶(遍历数组) */
    +int arrayTraversal(int[] nums) {
    +    int count = 0;
    +    // 循环次数与数组长度成正比
    +    for (int num : nums) {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 线性阶(遍历数组) */
    +int arrayTraversal(vector<int>& nums) {
    +    int count = 0;
    +    // 循环次数与数组长度成正比
    +    for (int num : nums) {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 线性阶(遍历数组)"""
    +def array_traversal(nums):
    +    count = 0
    +    # 循环次数与数组长度成正比
    +    for num in nums:
    +        count += 1
    +    return count
    +
    +
    +
    +
    time_complexity.go
    /* 线性阶(遍历数组) */
    +func arrayTraversal(nums []int) int {
    +    count := 0
    +    // 循环次数与数组长度成正比
    +    for range nums {
    +        count++
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 线性阶(遍历数组) */
    +function arrayTraversal(nums) {
    +    let count = 0;
    +    // 循环次数与数组长度成正比
    +    for (let i = 0; i < nums.length; i++) {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 线性阶(遍历数组) */
    +function arrayTraversal(nums: number[]): number {
    +    let count = 0;
    +    // 循环次数与数组长度成正比
    +    for (let i = 0; i < nums.length; i++) {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 线性阶(遍历数组) */
    +int arrayTraversal(int *nums, int n) {
    +    int count = 0;
    +    // 循环次数与数组长度成正比
    +    for (int i = 0; i < n; i++) {
    +        count ++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 线性阶(遍历数组) */
    +int arrayTraversal(int[] nums)
    +{
    +    int count = 0;
    +    // 循环次数与数组长度成正比
    +    foreach(int num in nums)
    +    {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 线性阶(遍历数组) */
    +func arrayTraversal(nums: [Int]) -> Int {
    +    var count = 0
    +    // 循环次数与数组长度成正比
    +    for _ in nums {
    +        count += 1
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 线性阶(遍历数组)
    +fn arrayTraversal(nums: []i32) i32 {
    +    var count: i32 = 0;
    +    // 循环次数与数组长度成正比
    +    for (nums) |_| {
    +        count += 1;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    +

    平方阶 \(O(n^2)\)

    +

    平方阶的操作数量相对输入数据大小成平方级别增长。平方阶常出现于嵌套循环,外层循环和内层循环都为 \(O(n)\) ,总体为 \(O(n^2)\)

    +
    +
    +
    +
    time_complexity.java
    /* 平方阶 */
    +int quadratic(int n) {
    +    int count = 0;
    +    // 循环次数与数组长度成平方关系
    +    for (int i = 0; i < n; i++) {
    +        for (int j = 0; j < n; j++) {
    +            count++;
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 平方阶 */
    +int quadratic(int n) {
    +    int count = 0;
    +    // 循环次数与数组长度成平方关系
    +    for (int i = 0; i < n; i++) {
    +        for (int j = 0; j < n; j++) {
    +            count++;
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 平方阶 """
    +def quadratic(n):
    +    count = 0
    +    # 循环次数与数组长度成平方关系
    +    for i in range(n):
    +        for j in range(n):
    +            count += 1
    +    return count
    +
    +
    +
    +
    time_complexity.go
    /* 平方阶 */
    +func quadratic(n int) int {
    +    count := 0
    +    // 循环次数与数组长度成平方关系
    +    for i := 0; i < n; i++ {
    +        for j := 0; j < n; j++ {
    +            count++
    +        }
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 平方阶 */
    +function quadratic(n) {
    +    let count = 0;
    +    // 循环次数与数组长度成平方关系
    +    for (let i = 0; i < n; i++) {
    +        for (let j = 0; j < n; j++) {
    +            count++;
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 平方阶 */
    +function quadratic(n: number): number {
    +    let count = 0;
    +    // 循环次数与数组长度成平方关系
    +    for (let i = 0; i < n; i++) {
    +        for (let j = 0; j < n; j++) {
    +            count++;
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 平方阶 */
    +int quadratic(int n) {
    +    int count = 0;
    +    // 循环次数与数组长度成平方关系
    +    for (int i = 0; i < n; i++) {
    +        for (int j = 0; j < n; j++) {
    +            count ++;
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 平方阶 */
    +int quadratic(int n)
    +{
    +    int count = 0;
    +    // 循环次数与数组长度成平方关系
    +    for (int i = 0; i < n; i++)
    +    {
    +        for (int j = 0; j < n; j++)
    +        {
    +            count++;
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 平方阶 */
    +func quadratic(n: Int) -> Int {
    +    var count = 0
    +    // 循环次数与数组长度成平方关系
    +    for _ in 0 ..< n {
    +        for _ in 0 ..< n {
    +            count += 1
    +        }
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 平方阶
    +fn quadratic(n: i32) i32 {
    +    var count: i32 = 0;
    +    var i: i32 = 0;
    +    // 循环次数与数组长度成平方关系
    +    while (i < n) : (i += 1) {
    +        var j: i32 = 0;
    +        while (j < n) : (j += 1) {
    +            count += 1;
    +        }
    +    }
    +    return count;
    +}   
    +
    +
    +
    +
    +

    time_complexity_constant_linear_quadratic

    +

    Fig. 常数阶、线性阶、平方阶的时间复杂度

    + +

    以「冒泡排序」为例,外层循环 \(n - 1\) 次,内层循环 \(n-1, n-2, \cdots, 2, 1\) 次,平均为 \(\frac{n}{2}\) 次,因此时间复杂度为 \(O(n^2)\)

    +
    \[ +O((n - 1) \frac{n}{2}) = O(n^2) +\]
    +
    +
    +
    +
    time_complexity.java
    /* 平方阶(冒泡排序) */
    +int bubbleSort(int[] nums) {
    +    int count = 0;  // 计数器
    +    // 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    for (int i = nums.length - 1; i > 0; i--) {
    +        // 内循环:冒泡操作
    +        for (int j = 0; j < i; j++) {
    +            if (nums[j] > nums[j + 1]) {
    +                // 交换 nums[j] 与 nums[j + 1]
    +                int tmp = nums[j];
    +                nums[j] = nums[j + 1];
    +                nums[j + 1] = tmp;
    +                count += 3;  // 元素交换包含 3 个单元操作
    +            }
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 平方阶(冒泡排序) */
    +int bubbleSort(vector<int>& nums) {
    +    int count = 0;  // 计数器
    +    // 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    for (int i = nums.size() - 1; i > 0; i--) {
    +        // 内循环:冒泡操作
    +        for (int j = 0; j < i; j++) {
    +            if (nums[j] > nums[j + 1]) {
    +                // 交换 nums[j] 与 nums[j + 1]
    +                int tmp = nums[j];
    +                nums[j] = nums[j + 1];
    +                nums[j + 1] = tmp;
    +                count += 3;  // 元素交换包含 3 个单元操作
    +            }
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 平方阶(冒泡排序)"""
    +def bubble_sort(nums):
    +    count = 0  # 计数器
    +    # 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    for i in range(len(nums) - 1, 0, -1):
    +        # 内循环:冒泡操作
    +        for j in range(i):
    +            if nums[j] > nums[j + 1]:
    +                # 交换 nums[j] 与 nums[j + 1]
    +                tmp = nums[j]
    +                nums[j] = nums[j + 1]
    +                nums[j + 1] = tmp
    +                count += 3  # 元素交换包含 3 个单元操作
    +    return count
    +
    +
    +
    +
    time_complexity.go
    /* 平方阶(冒泡排序) */
    +func bubbleSort(nums []int) int {
    +    count := 0 // 计数器
    +    // 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    for i := len(nums) - 1; i > 0; i-- {
    +        // 内循环:冒泡操作
    +        for j := 0; j < i; j++ {
    +            if nums[j] > nums[j+1] {
    +                // 交换 nums[j] 与 nums[j + 1]
    +                tmp := nums[j]
    +                nums[j] = nums[j+1]
    +                nums[j+1] = tmp
    +                count += 3 // 元素交换包含 3 个单元操作
    +            }
    +        }
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 平方阶(冒泡排序) */
    +function bubbleSort(nums) {
    +    let count = 0; // 计数器
    +    // 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    for (let i = nums.length - 1; i > 0; i--) {
    +        // 内循环:冒泡操作
    +        for (let j = 0; j < i; j++) {
    +            if (nums[j] > nums[j + 1]) {
    +                // 交换 nums[j] 与 nums[j + 1]
    +                let tmp = nums[j];
    +                nums[j] = nums[j + 1];
    +                nums[j + 1] = tmp;
    +                count += 3; // 元素交换包含 3 个单元操作
    +            }
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 平方阶(冒泡排序) */
    +function bubbleSort(nums: number[]): number {
    +    let count = 0; // 计数器
    +    // 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    for (let i = nums.length - 1; i > 0; i--) {
    +        // 内循环:冒泡操作
    +        for (let j = 0; j < i; j++) {
    +            if (nums[j] > nums[j + 1]) {
    +                // 交换 nums[j] 与 nums[j + 1]
    +                let tmp = nums[j];
    +                nums[j] = nums[j + 1];
    +                nums[j + 1] = tmp;
    +                count += 3; // 元素交换包含 3 个单元操作
    +            }
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 平方阶(冒泡排序) */
    +int bubbleSort(int *nums, int n) {
    +    int count = 0;  // 计数器 
    +    // 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    for (int i = n - 1; i > 0; i--) {
    +        // 内循环:冒泡操作
    +        for (int j = 0; j < i; j++) {
    +            if (nums[j] > nums [j + 1]) 
    +            {
    +                // 交换 nums[j] 与 nums[j + 1]
    +                int tmp = nums[j];
    +                nums[j] = nums[j + 1];
    +                nums[j + 1] = tmp;
    +                count += 3;  // 元素交换包含 3 个单元操作
    +            }
    +        }
    +
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 平方阶(冒泡排序) */
    +int bubbleSort(int[] nums)
    +{
    +    int count = 0;  // 计数器
    +                    // 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    for (int i = nums.Length - 1; i > 0; i--)
    +    {
    +        // 内循环:冒泡操作
    +        for (int j = 0; j < i; j++)
    +        {
    +            if (nums[j] > nums[j + 1])
    +            {
    +                // 交换 nums[j] 与 nums[j + 1]
    +                int tmp = nums[j];
    +                nums[j] = nums[j + 1];
    +                nums[j + 1] = tmp;
    +                count += 3;  // 元素交换包含 3 个单元操作
    +            }
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 平方阶(冒泡排序) */
    +func bubbleSort(nums: inout [Int]) -> Int {
    +    var count = 0 // 计数器
    +    // 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    for i in sequence(first: nums.count - 1, next: { $0 > 0 + 1 ? $0 - 1 : nil }) {
    +        // 内循环:冒泡操作
    +        for j in 0 ..< i {
    +            if nums[j] > nums[j + 1] {
    +                // 交换 nums[j] 与 nums[j + 1]
    +                let tmp = nums[j]
    +                nums[j] = nums[j + 1]
    +                nums[j + 1] = tmp
    +                count += 3 // 元素交换包含 3 个单元操作
    +            }
    +        }
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 平方阶(冒泡排序)
    +fn bubbleSort(nums: []i32) i32 {
    +    var count: i32 = 0;  // 计数器 
    +    // 外循环:待排序元素数量为 n-1, n-2, ..., 1
    +    var i: i32 = @intCast(i32, nums.len ) - 1;
    +    while (i > 0) : (i -= 1) {
    +        var j: usize = 0;
    +        // 内循环:冒泡操作
    +        while (j < i) : (j += 1) {
    +            if (nums[j] > nums[j + 1]) {
    +                // 交换 nums[j] 与 nums[j + 1]
    +                var tmp = nums[j];
    +                nums[j] = nums[j + 1];
    +                nums[j + 1] = tmp;
    +                count += 3;  // 元素交换包含 3 个单元操作
    +            }
    +        }
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    +

    指数阶 \(O(2^n)\)

    +
    +

    Note

    +

    生物学科中的“细胞分裂”即是指数阶增长:初始状态为 \(1\) 个细胞,分裂一轮后为 \(2\) 个,分裂两轮后为 \(4\) 个,……,分裂 \(n\) 轮后有 \(2^n\) 个细胞。

    +
    +

    指数阶增长得非常快,在实际应用中一般是不能被接受的。若一个问题使用「暴力枚举」求解的时间复杂度是 \(O(2^n)\) ,那么一般都需要使用「动态规划」或「贪心算法」等算法来求解。

    +
    +
    +
    +
    time_complexity.java
    /* 指数阶(循环实现) */
    +int exponential(int n) {
    +    int count = 0, base = 1;
    +    // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    for (int i = 0; i < n; i++) {
    +        for (int j = 0; j < base; j++) {
    +            count++;
    +        }
    +        base *= 2;
    +    }
    +    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 指数阶(循环实现) */
    +int exponential(int n) {
    +    int count = 0, base = 1;
    +    // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    for (int i = 0; i < n; i++) {
    +        for (int j = 0; j < base; j++) {
    +            count++;
    +        }
    +        base *= 2;
    +    }
    +    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 指数阶(循环实现)"""
    +def exponential(n):
    +    count, base = 0, 1
    +    # cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    for _ in range(n):
    +        for _ in range(base):
    +            count += 1
    +        base *= 2
    +    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count
    +
    +
    +
    +
    time_complexity.go
    /* 指数阶(循环实现)*/
    +func exponential(n int) int {
    +    count, base := 0, 1
    +    // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    for i := 0; i < n; i++ {
    +        for j := 0; j < base; j++ {
    +            count++
    +        }
    +        base *= 2
    +    }
    +    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 指数阶(循环实现) */
    +function exponential(n) {
    +    let count = 0,
    +        base = 1;
    +    // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    for (let i = 0; i < n; i++) {
    +        for (let j = 0; j < base; j++) {
    +            count++;
    +        }
    +        base *= 2;
    +    }
    +    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 指数阶(循环实现) */
    +function exponential(n: number): number {
    +    let count = 0,
    +        base = 1;
    +    // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    for (let i = 0; i < n; i++) {
    +        for (let j = 0; j < base; j++) {
    +            count++;
    +        }
    +        base *= 2;
    +    }
    +    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 指数阶(循环实现) */
    +int exponential(int n) {
    +    int count = 0;
    +    int bas = 1;
    +    // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    for (int i = 0; i < n; i++) {
    +        for (int j = 0; j < bas; j++) {
    +            count++;
    +        }
    +        bas *= 2;
    +    }
    +    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 指数阶(循环实现) */
    +int exponential(int n)
    +{
    +    int count = 0, bas = 1;
    +    // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    for (int i = 0; i < n; i++)
    +    {
    +        for (int j = 0; j < bas; j++)
    +        {
    +            count++;
    +        }
    +        bas *= 2;
    +    }
    +    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 指数阶(循环实现) */
    +func exponential(n: Int) -> Int {
    +    var count = 0
    +    var base = 1
    +    // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    for _ in 0 ..< n {
    +        for _ in 0 ..< base {
    +            count += 1
    +        }
    +        base *= 2
    +    }
    +    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 指数阶(循环实现)
    +fn exponential(n: i32) i32{
    +    var count: i32 = 0;
    +    var bas: i32 = 1;
    +    var i: i32 = 0;
    +    // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
    +    while (i < n) : (i += 1) {
    +        var j: i32 = 0;
    +        while (j < bas) : (j += 1) {
    +            count += 1;
    +        }
    +        bas *= 2;
    +    }
    +    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
    +    return count;
    +}       
    +
    +
    +
    +
    +

    time_complexity_exponential

    +

    Fig. 指数阶的时间复杂度

    + +

    在实际算法中,指数阶常出现于递归函数。例如以下代码,不断地一分为二,分裂 \(n\) 次后停止。

    +
    +
    +
    +
    time_complexity.java
    /* 指数阶(递归实现) */
    +int expRecur(int n) {
    +    if (n == 1) return 1;
    +    return expRecur(n - 1) + expRecur(n - 1) + 1;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 指数阶(递归实现) */
    +int expRecur(int n) {
    +    if (n == 1) return 1;
    +    return expRecur(n - 1) + expRecur(n - 1) + 1;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 指数阶(递归实现)"""
    +def exp_recur(n):
    +    if n == 1: return 1
    +    return exp_recur(n - 1) + exp_recur(n - 1) + 1
    +
    +
    +
    +
    time_complexity.go
    /* 指数阶(递归实现)*/
    +func expRecur(n int) int {
    +    if n == 1 {
    +        return 1
    +    }
    +    return expRecur(n-1) + expRecur(n-1) + 1
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 指数阶(递归实现) */
    +function expRecur(n) {
    +    if (n == 1) return 1;
    +    return expRecur(n - 1) + expRecur(n - 1) + 1;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 指数阶(递归实现) */
    +function expRecur(n: number): number {
    +    if (n == 1) return 1;
    +    return expRecur(n - 1) + expRecur(n - 1) + 1;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 指数阶(递归实现) */
    +int expRecur(int n) {
    +    if (n == 1) return 1;
    +    return expRecur(n - 1) + expRecur(n - 1) + 1;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 指数阶(递归实现) */
    +int expRecur(int n)
    +{
    +    if (n == 1) return 1;
    +    return expRecur(n - 1) + expRecur(n - 1) + 1;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 指数阶(递归实现) */
    +func expRecur(n: Int) -> Int {
    +    if n == 1 {
    +        return 1
    +    }
    +    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 指数阶(递归实现)
    +fn expRecur(n: i32) i32{
    +    if (n == 1) return 1;
    +    return expRecur(n - 1) + expRecur(n - 1) + 1;
    +}
    +
    +
    +
    +
    +

    对数阶 \(O(\log n)\)

    +

    对数阶与指数阶正好相反,后者反映“每轮增加到两倍的情况”,而前者反映“每轮缩减到一半的情况”。对数阶仅次于常数阶,时间增长得很慢,是理想的时间复杂度。

    +

    对数阶常出现于「二分查找」和「分治算法」中,体现“一分为多”、“化繁为简”的算法思想。

    +

    设输入数据大小为 \(n\) ,由于每轮缩减到一半,因此循环次数是 \(\log_2 n\) ,即 \(2^n\) 的反函数。

    +
    +
    +
    +
    time_complexity.java
    /* 对数阶(循环实现) */
    +int logarithmic(float n) {
    +    int count = 0;
    +    while (n > 1) {
    +        n = n / 2;
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 对数阶(循环实现) */
    +int logarithmic(float n) {
    +    int count = 0;
    +    while (n > 1) {
    +        n = n / 2;
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 对数阶(循环实现)"""
    +def logarithmic(n):
    +    count = 0
    +    while n > 1:
    +        n = n / 2
    +        count += 1
    +    return count
    +
    +
    +
    +
    time_complexity.go
    /* 对数阶(循环实现)*/
    +func logarithmic(n float64) int {
    +    count := 0
    +    for n > 1 {
    +        n = n / 2
    +        count++
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 对数阶(循环实现) */
    +function logarithmic(n) {
    +    let count = 0;
    +    while (n > 1) {
    +        n = n / 2;
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 对数阶(循环实现) */
    +function logarithmic(n: number): number {
    +    let count = 0;
    +    while (n > 1) {
    +        n = n / 2;
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 对数阶(循环实现) */
    +int logarithmic(float n) {
    +    int count = 0;
    +    while (n > 1) {
    +        n = n / 2;
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 对数阶(循环实现) */
    +int logarithmic(float n)
    +{
    +    int count = 0;
    +    while (n > 1)
    +    {
    +        n = n / 2;
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 对数阶(循环实现) */
    +func logarithmic(n: Int) -> Int {
    +    var count = 0
    +    var n = n
    +    while n > 1 {
    +        n = n / 2
    +        count += 1
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 对数阶(循环实现)
    +fn logarithmic(n: f32) i32
    +{
    +    var count: i32 = 0;
    +    var n_var = n;
    +    while (n_var > 1)
    +    {
    +        n_var = n_var / 2;
    +        count +=1;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    +

    time_complexity_logarithmic

    +

    Fig. 对数阶的时间复杂度

    + +

    与指数阶类似,对数阶也常出现于递归函数。以下代码形成了一个高度为 \(\log_2 n\) 的递归树。

    +
    +
    +
    +
    time_complexity.java
    /* 对数阶(递归实现) */
    +int logRecur(float n) {
    +    if (n <= 1) return 0;
    +    return logRecur(n / 2) + 1;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 对数阶(递归实现) */
    +int logRecur(float n) {
    +    if (n <= 1) return 0;
    +    return logRecur(n / 2) + 1;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 对数阶(递归实现)"""
    +def log_recur(n):
    +    if n <= 1: return 0
    +    return log_recur(n / 2) + 1
    +
    +
    +
    +
    time_complexity.go
    /* 对数阶(递归实现)*/
    +func logRecur(n float64) int {
    +    if n <= 1 {
    +        return 0
    +    }
    +    return logRecur(n/2) + 1
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 对数阶(递归实现) */
    +function logRecur(n) {
    +    if (n <= 1) return 0;
    +    return logRecur(n / 2) + 1;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 对数阶(递归实现) */
    +function logRecur(n: number): number {
    +    if (n <= 1) return 0;
    +    return logRecur(n / 2) + 1;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 对数阶(递归实现) */
    +int logRecur(float n) {
    +    if (n <= 1) return 0;
    +    return logRecur(n / 2) + 1;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 对数阶(递归实现) */
    +int logRecur(float n)
    +{
    +    if (n <= 1) return 0;
    +    return logRecur(n / 2) + 1;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 对数阶(递归实现) */
    +func logRecur(n: Int) -> Int {
    +    if n <= 1 {
    +        return 0
    +    }
    +    return logRecur(n: n / 2) + 1
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 对数阶(递归实现)
    +fn logRecur(n: f32) i32
    +{
    +    if (n <= 1) return 0;
    +    return logRecur(n / 2) + 1;
    +}
    +
    +
    +
    +
    +

    线性对数阶 \(O(n \log n)\)

    +

    线性对数阶常出现于嵌套循环中,两层循环的时间复杂度分别为 \(O(\log n)\)\(O(n)\)

    +

    主流排序算法的时间复杂度都是 \(O(n \log n )\) ,例如快速排序、归并排序、堆排序等。

    +
    +
    +
    +
    time_complexity.java
    /* 线性对数阶 */
    +int linearLogRecur(float n) {
    +    if (n <= 1) return 1;
    +    int count = linearLogRecur(n / 2) + 
    +                linearLogRecur(n / 2);
    +    for (int i = 0; i < n; i++) {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 线性对数阶 */
    +int linearLogRecur(float n) {
    +    if (n <= 1) return 1;
    +    int count = linearLogRecur(n / 2) + 
    +                linearLogRecur(n / 2);
    +    for (int i = 0; i < n; i++) {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 线性对数阶 """
    +def linear_log_recur(n):
    +    if n <= 1: return 1
    +    count = linear_log_recur(n // 2) + \
    +            linear_log_recur(n // 2)
    +    for _ in range(n):
    +        count += 1
    +    return count
    +
    +
    +
    +
    time_complexity.go
    /* 线性对数阶 */
    +func linearLogRecur(n float64) int {
    +    if n <= 1 {
    +        return 1
    +    }
    +    count := linearLogRecur(n/2) +
    +        linearLogRecur(n/2)
    +    for i := 0.0; i < n; i++ {
    +        count++
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 线性对数阶 */
    +function linearLogRecur(n) {
    +    if (n <= 1) return 1;
    +    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);
    +    for (let i = 0; i < n; i++) {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 线性对数阶 */
    +function linearLogRecur(n: number): number {
    +    if (n <= 1) return 1;
    +    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);
    +    for (let i = 0; i < n; i++) {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 线性对数阶 */
    +int linearLogRecur(float n) {
    +    if (n <= 1) return 1;
    +    int count = linearLogRecur(n / 2) +
    +                linearLogRecur(n / 2);
    +    for (int i = 0; i < n; i++) {
    +        count ++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 线性对数阶 */
    +int linearLogRecur(float n)
    +{
    +    if (n <= 1) return 1;
    +    int count = linearLogRecur(n / 2) +
    +                linearLogRecur(n / 2);
    +    for (int i = 0; i < n; i++)
    +    {
    +        count++;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 线性对数阶 */
    +func linearLogRecur(n: Double) -> Int {
    +    if n <= 1 {
    +        return 1
    +    }
    +    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)
    +    for _ in 0 ..< Int(n) {
    +        count += 1
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 线性对数阶
    +fn linearLogRecur(n: f32) i32
    +{
    +    if (n <= 1) return 1;
    +    var count: i32 = linearLogRecur(n / 2) +
    +                linearLogRecur(n / 2);
    +    var i: f32 = 0;
    +    while (i < n) : (i += 1) {
    +        count += 1;
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    +

    time_complexity_logarithmic_linear

    +

    Fig. 线性对数阶的时间复杂度

    + +

    阶乘阶 \(O(n!)\)

    +

    阶乘阶对应数学上的「全排列」。即给定 \(n\) 个互不重复的元素,求其所有可能的排列方案,则方案数量为

    +
    \[ +n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1 +\]
    +

    阶乘常使用递归实现。例如以下代码,第一层分裂出 \(n\) 个,第二层分裂出 \(n - 1\) 个,…… ,直至到第 \(n\) 层时终止分裂。

    +
    +
    +
    +
    time_complexity.java
    /* 阶乘阶(递归实现) */
    +int factorialRecur(int n) {
    +    if (n == 0) return 1;
    +    int count = 0;
    +    // 从 1 个分裂出 n 个
    +    for (int i = 0; i < n; i++) {
    +        count += factorialRecur(n - 1);
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cpp
    /* 阶乘阶(递归实现) */
    +int factorialRecur(int n) {
    +    if (n == 0) return 1;
    +    int count = 0;
    +    // 从 1 个分裂出 n 个
    +    for (int i = 0; i < n; i++) {
    +        count += factorialRecur(n - 1);
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.py
    """ 阶乘阶(递归实现)"""
    +def factorial_recur(n):
    +    if n == 0: return 1
    +    count = 0
    +    # 从 1 个分裂出 n 个
    +    for _ in range(n):
    +        count += factorial_recur(n - 1)
    +    return count
    +
    +
    +
    +
    time_complexity.go
    /* 阶乘阶(递归实现) */
    +func factorialRecur(n int) int {
    +    if n == 0 {
    +        return 1
    +    }
    +    count := 0
    +    // 从 1 个分裂出 n 个
    +    for i := 0; i < n; i++ {
    +        count += factorialRecur(n - 1)
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.js
    /* 阶乘阶(递归实现) */
    +function factorialRecur(n) {
    +    if (n == 0) return 1;
    +    let count = 0;
    +    // 从 1 个分裂出 n 个
    +    for (let i = 0; i < n; i++) {
    +        count += factorialRecur(n - 1);
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.ts
    /* 阶乘阶(递归实现) */
    +function factorialRecur(n: number): number {
    +    if (n == 0) return 1;
    +    let count = 0;
    +    // 从 1 个分裂出 n 个
    +    for (let i = 0; i < n; i++) {
    +        count += factorialRecur(n - 1);
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.c
    /* 阶乘阶(递归实现) */
    +int factorialRecur(int n) {
    +    if (n == 0) return 1;
    +    int count = 0;
    +    for (int i = 0; i < n; i++) {
    +        count += factorialRecur(n - 1);
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.cs
    /* 阶乘阶(递归实现) */
    +int factorialRecur(int n)
    +{
    +    if (n == 0) return 1;
    +    int count = 0;
    +    // 从 1 个分裂出 n 个
    +    for (int i = 0; i < n; i++)
    +    {
    +        count += factorialRecur(n - 1);
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    time_complexity.swift
    /* 阶乘阶(递归实现) */
    +func factorialRecur(n: Int) -> Int {
    +    if n == 0 {
    +        return 1
    +    }
    +    var count = 0
    +    // 从 1 个分裂出 n 个
    +    for _ in 0 ..< n {
    +        count += factorialRecur(n: n - 1)
    +    }
    +    return count
    +}
    +
    +
    +
    +
    time_complexity.zig
    // 阶乘阶(递归实现)
    +fn factorialRecur(n: i32) i32 {
    +    if (n == 0) return 1;
    +    var count: i32 = 0;
    +    var i: i32 = 0;
    +    // 从 1 个分裂出 n 个
    +    while (i < n) : (i += 1) {
    +        count += factorialRecur(n - 1);
    +    }
    +    return count;
    +}
    +
    +
    +
    +
    +

    time_complexity_factorial

    +

    Fig. 阶乘阶的时间复杂度

    + +

    2.2.6. 最差、最佳、平均时间复杂度

    +

    某些算法的时间复杂度不是恒定的,而是与输入数据的分布有关。举一个例子,输入一个长度为 \(n\) 数组 nums ,其中 nums 由从 \(1\)\(n\) 的数字组成,但元素顺序是随机打乱的;算法的任务是返回元素 \(1\) 的索引。我们可以得出以下结论:

    +
      +
    • nums = [?, ?, ..., 1],即当末尾元素是 \(1\) 时,则需完整遍历数组,此时达到 最差时间复杂度 \(O(n)\)
    • +
    • nums = [1, ?, ?, ...] ,即当首个数字为 \(1\) 时,无论数组多长都不需要继续遍历,此时达到 最佳时间复杂度 \(\Omega(1)\)
    • +
    +

    「函数渐近上界」使用大 \(O\) 记号表示,代表「最差时间复杂度」。与之对应,「函数渐近下界」用 \(\Omega\) 记号(Omega Notation)来表示,代表「最佳时间复杂度」。

    +
    +
    +
    +
    worst_best_time_complexity.java
    /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */
    +int[] randomNumbers(int n) {
    +    Integer[] nums = new Integer[n];
    +    // 生成数组 nums = { 1, 2, 3, ..., n }
    +    for (int i = 0; i < n; i++) {
    +        nums[i] = i + 1;
    +    }
    +    // 随机打乱数组元素
    +    Collections.shuffle(Arrays.asList(nums));
    +    // Integer[] -> int[]
    +    int[] res = new int[n];
    +    for (int i = 0; i < n; i++) {
    +        res[i] = nums[i];
    +    }
    +    return res;
    +}
    +
    +/* 查找数组 nums 中数字 1 所在索引 */
    +int findOne(int[] nums) {
    +    for (int i = 0; i < nums.length; i++) {
    +        // 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        // 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if (nums[i] == 1)
    +            return i;
    +    }
    +    return -1;
    +}
    +
    +
    +
    +
    worst_best_time_complexity.cpp
    /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */
    +vector<int> randomNumbers(int n) {
    +    vector<int> nums(n);
    +    // 生成数组 nums = { 1, 2, 3, ..., n }
    +    for (int i = 0; i < n; i++) {
    +        nums[i] = i + 1;
    +    }
    +    // 使用系统时间生成随机种子
    +    unsigned seed = chrono::system_clock::now().time_since_epoch().count();
    +    // 随机打乱数组元素
    +    shuffle(nums.begin(), nums.end(), default_random_engine(seed));
    +    return nums;
    +}
    +
    +/* 查找数组 nums 中数字 1 所在索引 */
    +int findOne(vector<int>& nums) {
    +    for (int i = 0; i < nums.size(); i++) {
    +        // 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        // 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if (nums[i] == 1)
    +            return i;
    +    }
    +    return -1;
    +}
    +
    +
    +
    +
    worst_best_time_complexity.py
    """ 生成一个数组,元素为: 1, 2, ..., n ,顺序被打乱 """
    +def random_numbers(n):
    +    # 生成数组 nums =: 1, 2, 3, ..., n
    +    nums = [i for i in range(1, n + 1)]
    +    # 随机打乱数组元素
    +    random.shuffle(nums)
    +    return nums
    +
    +""" 查找数组 nums 中数字 1 所在索引 """
    +def find_one(nums):
    +    for i in range(len(nums)):
    +        # 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        # 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if nums[i] == 1:
    +            return i
    +    return -1
    +
    +
    +
    +
    worst_best_time_complexity.go
    /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */
    +func randomNumbers(n int) []int {
    +    nums := make([]int, n)
    +    // 生成数组 nums = { 1, 2, 3, ..., n }
    +    for i := 0; i < n; i++ {
    +        nums[i] = i + 1
    +    }
    +    // 随机打乱数组元素
    +    rand.Shuffle(len(nums), func(i, j int) {
    +        nums[i], nums[j] = nums[j], nums[i]
    +    })
    +    return nums
    +}
    +
    +/* 查找数组 nums 中数字 1 所在索引 */
    +func findOne(nums []int) int {
    +    for i := 0; i < len(nums); i++ {
    +        // 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        // 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if nums[i] == 1 {
    +            return i
    +        }
    +    }
    +    return -1
    +}
    +
    +
    +
    +
    worst_best_time_complexity.js
    /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */
    +function randomNumbers(n) {
    +    const nums = Array(n);
    +    // 生成数组 nums = { 1, 2, 3, ..., n }
    +    for (let i = 0; i < n; i++) {
    +        nums[i] = i + 1;
    +    }
    +    // 随机打乱数组元素
    +    for (let i = 0; i < n; i++) {
    +        const r = Math.floor(Math.random() * (i + 1));
    +        const temp = nums[i];
    +        nums[i] = nums[r];
    +        nums[r] = temp;
    +    }
    +    return nums;
    +}
    +
    +/* 查找数组 nums 中数字 1 所在索引 */
    +function findOne(nums) {
    +    for (let i = 0; i < nums.length; i++) {
    +        // 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        // 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if (nums[i] === 1) {
    +            return i;
    +        }
    +    }
    +    return -1;
    +}
    +
    +
    +
    +
    worst_best_time_complexity.ts
    /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */
    +function randomNumbers(n: number): number[] {
    +    const nums = Array(n);
    +    // 生成数组 nums = { 1, 2, 3, ..., n }
    +    for (let i = 0; i < n; i++) {
    +        nums[i] = i + 1;
    +    }
    +    // 随机打乱数组元素
    +    for (let i = 0; i < n; i++) {
    +        const r = Math.floor(Math.random() * (i + 1));
    +        const temp = nums[i];
    +        nums[i] = nums[r];
    +        nums[r] = temp;
    +    }
    +    return nums;
    +}
    +
    +/* 查找数组 nums 中数字 1 所在索引 */
    +function findOne(nums: number[]): number {
    +    for (let i = 0; i < nums.length; i++) {
    +        // 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        // 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if (nums[i] === 1) {
    +            return i;
    +        }
    +    }
    +    return -1;
    +}
    +
    +
    +
    +
    worst_best_time_complexity.c
    /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */
    +int *randomNumbers(int n) {
    +    // 分配堆区内存(创建一维可变长数组:数组中元素数量为n,元素类型为int)
    +    int *nums = (int *)malloc(n * sizeof(int));
    +    // 生成数组 nums = { 1, 2, 3, ..., n }
    +    for (int i = 0; i < n; i++) {
    +        nums[i] = i + 1;
    +    }
    +    // 随机打乱数组元素 
    +    for (int i = n - 1; i > 0; i--) {
    +        int j = rand() % (i + 1);
    +        int temp = nums[i];
    +        nums[i] = nums[j];
    +        nums[j] = temp; 
    +    }
    +    return nums;
    +}
    +
    +/* 查找数组 nums 中数字 1 所在索引 */
    +int findOne(int *nums, int n) {
    +    for (int i = 0; i < n; i++) {
    +        // 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        // 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if (nums[i] == 1) return i;
    +    }
    +    return -1;
    +}
    +
    +
    +
    +
    worst_best_time_complexity.cs
    /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */
    +int[] randomNumbers(int n)
    +{
    +    int[] nums = new int[n];
    +    // 生成数组 nums = { 1, 2, 3, ..., n }
    +    for (int i = 0; i < n; i++)
    +    {
    +        nums[i] = i + 1;
    +    }
    +
    +    // 随机打乱数组元素
    +    for (int i = 0; i < nums.Length; i++)
    +    {
    +        var index = new Random().Next(i, nums.Length);
    +        var tmp = nums[i];
    +        var ran = nums[index];
    +        nums[i] = ran;
    +        nums[index] = tmp;
    +    }
    +    return nums;
    +}
    +
    +/* 查找数组 nums 中数字 1 所在索引 */
    +int findOne(int[] nums)
    +{
    +    for (int i = 0; i < nums.Length; i++)
    +    {
    +        // 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        // 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if (nums[i] == 1)
    +            return i;
    +    }
    +    return -1;
    +}
    +
    +
    +
    +
    worst_best_time_complexity.swift
    /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */
    +func randomNumbers(n: Int) -> [Int] {
    +    // 生成数组 nums = { 1, 2, 3, ..., n }
    +    var nums = Array(1 ... n)
    +    // 随机打乱数组元素
    +    nums.shuffle()
    +    return nums
    +}
    +
    +/* 查找数组 nums 中数字 1 所在索引 */
    +func findOne(nums: [Int]) -> Int {
    +    for i in nums.indices {
    +        // 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        // 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if nums[i] == 1 {
    +            return i
    +        }
    +    }
    +    return -1
    +}
    +
    +
    +
    +
    worst_best_time_complexity.zig
    // 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱
    +pub fn randomNumbers(comptime n: usize) [n]i32 {
    +    var nums: [n]i32 = undefined;
    +    // 生成数组 nums = { 1, 2, 3, ..., n }
    +    for (nums) |*num, i| {
    +        num.* = @intCast(i32, i) + 1;
    +    }
    +    // 随机打乱数组元素
    +    const rand = std.crypto.random;
    +    rand.shuffle(i32, &nums);
    +    return nums;
    +}
    +
    +// 查找数组 nums 中数字 1 所在索引
    +pub fn findOne(nums: []i32) i32 {
    +    for (nums) |num, i| {
    +        // 当元素 1 在数组头部时,达到最佳时间复杂度 O(1)
    +        // 当元素 1 在数组尾部时,达到最差时间复杂度 O(n)
    +        if (num == 1) return @intCast(i32, i);
    +    }
    +    return -1;
    +}
    +
    +
    +
    +
    +
    +

    Tip

    +

    我们在实际应用中很少使用「最佳时间复杂度」,因为往往只有很小概率下才能达到,会带来一定的误导性。反之,「最差时间复杂度」最为实用,因为它给出了一个“效率安全值”,让我们可以放心地使用算法。

    +
    +

    从上述示例可以看出,最差或最佳时间复杂度只出现在“特殊分布的数据”中,这些情况的出现概率往往很小,因此并不能最真实地反映算法运行效率。相对地,「平均时间复杂度」可以体现算法在随机输入数据下的运行效率,用 \(\Theta\) 记号(Theta Notation)来表示

    +

    对于部分算法,我们可以简单地推算出随机数据分布下的平均情况。比如上述示例,由于输入数组是被打乱的,因此元素 \(1\) 出现在任意索引的概率都是相等的,那么算法的平均循环次数则是数组长度的一半 \(\frac{n}{2}\) ,平均时间复杂度为 \(\Theta(\frac{n}{2}) = \Theta(n)\)

    +

    但在实际应用中,尤其是较为复杂的算法,计算平均时间复杂度比较困难,因为很难简便地分析出在数据分布下的整体数学期望。这种情况下,我们一般使用最差时间复杂度来作为算法效率的评判标准。

    +
    +

    为什么很少看到 \(\Theta\) 符号?

    +

    实际中我们经常使用「大 \(O\) 符号」来表示「平均复杂度」,这样严格意义上来说是不规范的。这可能是因为 \(O\) 符号实在是太朗朗上口了。
    如果在本书和其他资料中看到类似 平均时间复杂度 \(O(n)\) 的表述,请你直接理解为 \(\Theta(n)\) 即可。

    +
    + + + + + +

    评论

    + + + + + + +
    +
    + + + + +
    + + + +
    + + + +
    +
    +
    +
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    3.2. 数据结构分类

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    数据结构主要可根据「逻辑结构」和「物理结构」两种角度进行分类。

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    3.2.1. 逻辑结构:线性与非线性

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    「逻辑结构」反映了数据之间的逻辑关系。数组和链表的数据按照顺序依次排列,反映了数据间的线性关系;树从顶至底按层级排列,反映了祖先与后代之间的派生关系;图由结点和边组成,反映了复杂网络关系。

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    我们一般将逻辑结构分为「线性」和「非线性」两种。“线性”这个概念很直观,即表明数据在逻辑关系上是排成一条线的;而如果数据之间的逻辑关系是非线性的(例如是网状或树状的),那么就是非线性数据结构。

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    • 线性数据结构:数组、链表、栈、队列、哈希表;
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    • 非线性数据结构:树、图、堆、哈希表;
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    classification_logic_structure

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    Fig. 线性与非线性数据结构

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    3.2.2. 物理结构:连续与离散

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    Note

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    若感到阅读困难,建议先看完下个章节「数组与链表」,再回过头来理解物理结构的含义。

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    「物理结构」反映了数据在计算机内存中的存储方式。从本质上看,分别是 数组的连续空间存储链表的离散空间存储。物理结构从底层上决定了数据的访问、更新、增删等操作方法,在时间效率和空间效率方面呈现出此消彼长的特性。

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    classification_phisical_structure

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    Fig. 连续空间存储与离散空间存储

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    所有数据结构都是基于数组、或链表、或两者组合实现的。例如栈和队列,既可以使用数组实现、也可以使用链表实现,而例如哈希表,其实现同时包含了数组和链表。

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    • 基于数组可实现:栈、队列、哈希表、树、堆、图、矩阵、张量(维度 \(\geq 3\) 的数组)等;
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    • 基于链表可实现:栈、队列、哈希表、树、堆、图等;
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    基于数组实现的数据结构也被称为「静态数据结构」,这意味着该数据结构在在被初始化后,长度不可变。相反地,基于链表实现的数据结构被称为「动态数据结构」,该数据结构在被初始化后,我们也可以在程序运行中修改其长度。

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    Tip

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    数组与链表是其他所有数据结构的“底层积木”,建议读者一定要多花些时间了解。

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zo2UBK_P6jDp>k>^(?rNF3q+wcOWEBEW#ebxLRi;;bvOQx>E@}`s%uiPczdJFI z_DEU@MR}6{-Mbv`KU(qu9!~ne7-)@RBxw5naK22_m;G2<7I)sN^2_vAUAm}kr7Ixe z9>rp}|4X9pb$;!NvOl{6e-065DFJ?#gN}oqttyJKHuw9iziL}dYH}Y7Ru$%>KGzh4 z(MbpX#!0L~~=Nve*pfLLw(RZ3xXl8dpSoCSv zaOoY3G`Ft=Gse{oUUD zkUHm{rr>B=|G+>=qN4A&QqUai>`BSV=|~_dc`Q;L8aA20ml3+fzrbOtC27{vgDfrOUKWcP*y*UC;%E z=6`}Bf_#Eh*m~*V+|UEtB$0FNSZMZSXa6dZb<5u7w4xrn8Z?Y9Mls_|4Q0E4s(`l& zo_G0N_M8oERrrkmlbiq + + + + + + + + + + + + + + + + + + + + + + + + + + 3.1. 数据与内存 - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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    3.1. 数据与内存

    +

    3.1.1. 基本数据类型

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    谈到计算机中的数据,我们能够想到文本、图片、视频、语音、3D 模型等等,这些数据虽然组织形式不同,但是有一个共同点,即都是由各种基本数据类型构成的。

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    「基本数据类型」是 CPU 可以直接进行运算的类型,在算法中直接被使用。

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    • 「整数」根据不同的长度分为 byte, short, int, long ,根据算法需求选用,即在满足取值范围的情况下尽量减小内存空间占用;
    • +
    • 「浮点数」代表小数,根据长度分为 float, double ,同样根据算法的实际需求选用;
    • +
    • 「字符」在计算机中是以字符集的形式保存的,char 的值实际上是数字,代表字符集中的编号,计算机通过字符集查表来完成编号到字符的转换。占用空间与具体编程语言有关,通常为 2 bytes 或 1 byte ;
    • +
    • 「布尔」代表逻辑中的 “是” 与 “否” ,其占用空间需要具体根据编程语言确定,通常为 1 byte 或 1 bit ;
    • +
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    字节与比特

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    1 字节 (byte) = 8 比特 (bit) , 1 比特即最基本的 1 个二进制位

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    Table. Java 的基本数据类型

    + +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    类别符号占用空间取值范围默认值
    整数byte1 byte\(-2^7\) ~ \(2^7 - 1\) ( \(-128\) ~ \(127\) )\(0\)
    short2 bytes\(-2^{15}\) ~ \(2^{15} - 1\)\(0\)
    int4 bytes\(-2^{31}\) ~ \(2^{31} - 1\)\(0\)
    long8 bytes\(-2^{63}\) ~ \(2^{63} - 1\)\(0\)
    浮点数float4 bytes\(-3.4 \times 10^{38}\) ~ \(3.4 \times 10^{38}\)\(0.0\) f
    double8 bytes\(-1.7 \times 10^{308}\) ~ \(1.7 \times 10^{308}\)\(0.0\)
    字符char2 bytes / 1 byte\(0\) ~ \(2^{16} - 1\)\(0\)
    布尔boolean(bool)1 byte / 1 bit\(\text{true}\)\(\text{false}\)\(\text{false}\)
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    Tip

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    以上表格中,加粗项在「算法题」中最为常用。此表格无需硬背,大致理解即可,需要时可以通过查表来回忆。

    +
    +

    「基本数据类型」与「数据结构」之间的联系与区别

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    我们知道,数据结构是在计算机中 组织与存储数据的方式,它的主语是“结构”,而不是“数据”。比如,我们想要表示“一排数字”,自然应该使用「数组」这个数据结构。数组的存储方式使之可以表示数字的相邻关系、先后关系等一系列我们需要的信息,但至于其中存储的是整数 int ,还是小数 float ,或是字符 char ,则与所谓的数据的结构无关了

    +
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    /* 使用多种「基本数据类型」来初始化「数组」 */
    +int[] numbers = new int[5];
    +float[] decimals = new float[5];
    +char[] characters = new char[5];
    +boolean[] booleans = new boolean[5];
    +
    +
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    +
    /* 使用多种「基本数据类型」来初始化「数组」 */
    +int numbers[5];
    +float decimals[5];
    +char characters[5];
    +bool booleans[5];
    +
    +
    +
    +
    """ Python 的 list 可以自由存储各种基本数据类型和对象 """
    +list = [0, 0.0, 'a', False]
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    +
    +
    // 使用多种「基本数据类型」来初始化「数组」
    +var numbers = [5]int{}
    +var decimals = [5]float64{}
    +var characters = [5]byte{}
    +var booleans = [5]bool{}
    +
    +
    +
    +
    /* JavaScript 的数组可以自由存储各种基本数据类型和对象 */
    +const array = [0, 0.0, 'a', false];
    +
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    +
    +
    /* 使用多种「基本数据类型」来初始化「数组」 */
    +const numbers: number[] = [];
    +const characters: string[] = [];
    +const booleans: boolean[] = [];
    +
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    +
    /* 使用多种「基本数据类型」来初始化「数组」 */
    +int numbers[10];
    +float decimals[10];
    +char characters[10];
    +bool booleans[10];
    +
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    +
    /* 使用多种「基本数据类型」来初始化「数组」 */
    +int[] numbers = new int[5];
    +float[] decimals = new float[5];
    +char[] characters = new char[5];
    +bool[] booleans = new bool[5];
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    +
    /* 使用多种「基本数据类型」来初始化「数组」 */
    +let numbers = Array(repeating: Int(), count: 5)
    +let decimals = Array(repeating: Double(), count: 5)
    +let characters = Array(repeating: Character("a"), count: 5)
    +let booleans = Array(repeating: Bool(), count: 5)
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    3.1.2. 计算机内存

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    在计算机中,内存和硬盘是两种主要的存储硬件设备。「硬盘」主要用于长期存储数据,容量较大(通常可达到 TB 级别)、速度较慢。「内存」用于运行程序时暂存数据,速度较快,但容量较小(通常为 GB 级别)。

    +

    算法运行中,相关数据都被存储在内存中。下图展示了一个计算机内存条,其中每个黑色方块都包含一块内存空间。我们可以将内存想象成一个巨大的 Excel 表格,其中每个单元格都可以存储 1 byte 的数据,在算法运行时,所有数据都被存储在这些单元格中。

    +

    系统通过「内存地址 Memory Location」来访问目标内存位置的数据。计算机根据特定规则给表格中每个单元格编号,保证每块内存空间都有独立的内存地址。自此,程序便通过这些地址,访问内存中的数据。

    +

    computer_memory_location

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    Fig. 内存条、内存空间、内存地址

    + +

    内存资源是设计数据结构与算法的重要考虑因素。内存是所有程序的公共资源,当内存被某程序占用时,不能被其它程序同时使用。我们需要根据剩余内存资源的情况来设计算法。例如,若剩余内存空间有限,则要求算法占用的峰值内存不能超过系统剩余内存;若运行的程序很多、缺少大块连续的内存空间,则要求选取的数据结构必须能够存储在离散的内存空间内。

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    3.3. 小结

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    • 整数 byte, short, int, long 、浮点数 float, double 、字符 char 、布尔 boolean 是计算机中的基本数据类型,占用空间的大小决定了它们的取值范围。
    • +
    • 在程序运行时,数据存储在计算机的内存中。内存中每块空间都有独立的内存地址,程序是通过内存地址来访问数据的。
    • +
    • 数据结构主要可以从逻辑结构和物理结构两个角度进行分类。逻辑结构反映了数据中元素之间的逻辑关系,物理结构反映了数据在计算机内存中的存储形式。
    • +
    • 常见的逻辑结构有线性、树状、网状等。我们一般根据逻辑结构将数据结构分为线性(数组、链表、栈、队列)和非线性(树、图、堆)两种。根据实现方式的不同,哈希表可能是线性或非线性。
    • +
    • 物理结构主要有两种,分别是连续空间存储(数组)和离散空间存储(链表),所有的数据结构都是由数组、或链表、或两者组合实现的。
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zjc;p4jZf)luCs~6xcx110gHP3>+;AvJqTfk3c85cj*~J3LQ3lj)pR32V{ObgbcZQzwT^*d>I-Ebl^VbRe5x+ zyj)!K5{;hWreuQd=bWc*linX!|84^Mv6d-eGm6ieNbOs|@tgp`e8pSl=ED~;)0aAJ z-;vftrhbU7BHQ*$uO2k*r|lAtewR0SrO%ib0EhOck7vf_#B5p%@+O+tzV8^~PhD5H zg>#!ef1o+MG?mpq;%f4tVj@mAdWT_5^XgEr>aQ2)br}TGu8b;Xi|wbiI%NJ_PSotI z_$3UpU=4T7iLoi@CX_kA&&#j>t($I>@1x{@AN)U~PzU$f{*m3J9?Et+Pvw)(Rfi~D p$V37a + + + + + + + + + + + + + + + + + + + + + + + + + + 9.1. 图(Graph) - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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    9.1. 图

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    「图 Graph」是一种非线性数据结构,由「顶点 Vertex」和「边 Edge」组成。我们可将图 \(G\) 抽象地表示为一组顶点 \(V\) 和一组边 \(E\) 的集合。例如,以下表示一个包含 5 个顶点和 7 条边的图

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    \[ +\begin{aligned} +V & = \{ 1, 2, 3, 4, 5 \} \newline +E & = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline +G & = \{ V, E \} \newline +\end{aligned} +\]
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    linkedlist_tree_graph

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    那么,图与其他数据结构的关系是什么?如果我们把「顶点」看作结点,把「边」看作连接各个结点的指针,则可将「图」看成一种从「链表」拓展而来的数据结构。相比线性关系(链表)和分治关系(树),网络关系(图)的自由度更高,也从而更为复杂

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    9.1.1. 图常见类型

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    根据边是否有方向,分为「无向图 Undirected Graph」和「有向图 Directed Graph」。

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    • 在无向图中,边表示两结点之间“双向”的连接关系,例如微信或 QQ 中的“好友关系”;
    • +
    • 在有向图中,边是有方向的,即 \(A \rightarrow B\)\(A \leftarrow B\) 两个方向的边是相互独立的,例如微博或抖音上的“关注”与“被关注”关系;
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    directed_graph

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    根据所有顶点是否连通,分为「连通图 Connected Graph」和「非连通图 Disconnected Graph」。

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    • 对于连通图,从某个结点出发,可以到达其余任意结点;
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    • 对于非连通图,从某个结点出发,至少有一个结点无法到达;
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    connected_graph

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    我们可以给边添加“权重”变量,得到「有权图 Weighted Graph」。例如,在王者荣耀等游戏中,系统会根据共同游戏时间来计算玩家之间的“亲密度”,这种亲密度网络就可以使用有权图来表示。

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    weighted_graph

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    9.1.2. 图常用术语

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    • 「邻接 Adjacency」:当两顶点之间有边相连时,称此两顶点“邻接”。
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    • 「路径 Path」:从顶点 A 到顶点 B 走过的边构成的序列,被称为从 A 到 B 的“路径”。
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    • 「度 Degree」表示一个顶点具有多少条边。对于有向图,「入度 In-Degree」表示有多少条边指向该顶点,「出度 Out-Degree」表示有多少条边从该顶点指出。
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    9.1.3. 图的表示

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    图的常用表示方法有「邻接矩阵」和「邻接表」。以下使用「无向图」来举例。

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    邻接矩阵

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    设图的顶点数量为 \(n\) ,「邻接矩阵 Adjacency Matrix」使用一个 \(n \times n\) 大小的矩阵来表示图,每一行(列)代表一个顶点,矩阵元素代表边,使用 \(1\)\(0\) 来表示两个顶点之间有边或无边。

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    adjacency_matrix

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    邻接矩阵具有以下性质:

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    • 顶点不能与自身相连,因而邻接矩阵主对角线元素没有意义。
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    • 「无向图」两个方向的边等价,此时邻接矩阵关于主对角线对称。
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    • 将邻接矩阵的元素从 \(1\) , \(0\) 替换为权重,则能够表示「有权图」。
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    使用邻接矩阵表示图时,我们可以直接通过访问矩阵元素来获取边,因此增删查操作的效率很高,时间复杂度均为 \(O(1)\) 。然而,矩阵的空间复杂度为 \(O(n^2)\) ,内存占用较大。

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    邻接表

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    「邻接表 Adjacency List」使用 \(n\) 个链表来表示图,链表结点表示顶点。第 \(i\) 条链表对应顶点 \(i\) ,其中存储了所有与该顶点相连的顶点。

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    adjacency_list

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    邻接表仅存储存在的边,而边的总数往往远小于 \(n^2\) ,因此更加节省空间。但是,因为在邻接表中需要通过遍历链表来查找边,所以其时间效率不如邻接矩阵。

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    观察上图发现,邻接表结构与哈希表「链地址法」非常相似,因此我们也可以用类似方法来优化效率。比如,当链表较长时,可以把链表转化为「AVL 树」,从而将时间效率从 \(O(n)\) 优化至 \(O(\log n)\) ,还可以通过中序遍历获取有序序列;还可以将链表转化为 HashSet(即哈希表),将时间复杂度降低至 \(O(1)\) ,。

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    9.1.4. 图常见应用

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    现实中的许多系统都可以使用图来建模,对应的待求解问题也可以被约化为图计算问题。

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    顶点图计算问题
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    评论

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ze*m2dPtN9r8!H!L5N4vPRH>%Il25DrUP&$Xh_9e`bhP~G$f0sxX$yH=Ia68JBXo?% zVy~_FgrGv456?apS@bi`uh*QtJHwwJAd&`M^nqiJO8O+D{W58-{rL3)FK8snu`@5N7Wt~>va-!qZv+_UuW`S$lEz~4J#zPPu3((gR~KYlNX zi0u`(R9f;Mw@l{Kf_u>H++dqn^*??u4-Nl4v)VlWpQrozP{GPjFxLNl^#6|f|05HI zPaYbmEveNzy_JtsVDSK0-lWMx)kZhx)#*%Ah!j$Wcoa*8s4o-5iWmaJAgQyVKw6PX z7T@ciL|012hQ4kwyCdr7alZtFi)3tKEuUZOXh!Y((&y)iEEd!J_X+GqWrZ8ox%W*@ ztbt14wDbDmUMj#FEq75Q>CX(ftuZcl1QpG8g;U9U06c0)7~%{%g%m!QleIxyc1j +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + + + + + +

    9.2. 图基础操作

    +

    图的基础操作分为对「边」的操作和对「顶点」的操作,在「邻接矩阵」和「邻接表」这两种表示下的实现方式不同。

    +

    9.2.1. 基于邻接矩阵的实现

    +

    设图的顶点总数为 \(n\) ,则有:

    +
      +
    • 添加或删除边:直接在邻接矩阵中修改指定边的对应元素即可,使用 \(O(1)\) 时间。而由于是无向图,因此需要同时更新两个方向的边。
    • +
    • 添加顶点:在邻接矩阵的尾部添加一行一列,并全部填 \(0\) 即可,使用 \(O(n)\) 时间。
    • +
    • 删除顶点:在邻接矩阵中删除一行一列。当删除首行首列时达到最差情况,需要将 \((n-1)^2\) 个元素“向左上移动”,从而使用 \(O(n^2)\) 时间。
    • +
    • 初始化:传入 \(n\) 个顶点,初始化长度为 \(n\) 的顶点列表 vertices ,使用 \(O(n)\) 时间;初始化 \(n \times n\) 大小的邻接矩阵 adjMat ,使用 \(O(n^2)\) 时间。
    • +
    +
    +
    +
    +

    adjacency_matrix_initialization

    +
    +
    +

    adjacency_matrix_add_edge

    +
    +
    +

    adjacency_matrix_remove_edge

    +
    +
    +

    adjacency_matrix_add_vertex

    +
    +
    +

    adjacency_matrix_remove_vertex

    +
    +
    +
    +

    以下是基于邻接矩阵表示图的实现代码。

    +
    +
    +
    +
    graph_adjacency_matrix.java
    /* 基于邻接矩阵实现的无向图类 */
    +class GraphAdjMat {
    +    List<Integer> vertices;     // 顶点列表,元素代表“顶点值”,索引代表“顶点索引”
    +    List<List<Integer>> adjMat; // 邻接矩阵,行列索引对应“顶点索引”
    +
    +    /* 构造函数 */
    +    public GraphAdjMat(int[] vertices, int[][] edges) {
    +        this.vertices = new ArrayList<>();
    +        this.adjMat = new ArrayList<>();
    +        // 添加顶点
    +        for (int val : vertices) {
    +            addVertex(val);
    +        }
    +        // 添加边
    +        // 请注意,edges 元素代表顶点索引,即对应 vertices 元素索引
    +        for (int[] e : edges) {
    +            addEdge(e[0], e[1]);
    +        }
    +    }
    +
    +    /* 获取顶点数量 */
    +    public int size() {
    +        return vertices.size();
    +    }
    +
    +    /* 添加顶点 */
    +    public void addVertex(int val) {
    +        int n = size();
    +        // 向顶点列表中添加新顶点的值
    +        vertices.add(val);
    +        // 在邻接矩阵中添加一行
    +        List<Integer> newRow = new ArrayList<>(n);
    +        for (int j = 0; j < n; j++) {
    +            newRow.add(0);
    +        }
    +        adjMat.add(newRow);
    +        // 在邻接矩阵中添加一列
    +        for (List<Integer> row : adjMat) {
    +            row.add(0);
    +        }
    +    }
    +
    +    /* 删除顶点 */
    +    public void removeVertex(int index) {
    +        if (index >= size())
    +            throw new IndexOutOfBoundsException();
    +        // 在顶点列表中移除索引 index 的顶点
    +        vertices.remove(index);
    +        // 在邻接矩阵中删除索引 index 的行
    +        adjMat.remove(index);
    +        // 在邻接矩阵中删除索引 index 的列
    +        for (List<Integer> row : adjMat) {
    +            row.remove(index);
    +        }
    +    }
    +
    +    /* 添加边 */
    +    // 参数 i, j 对应 vertices 元素索引
    +    public void addEdge(int i, int j) {
    +        // 索引越界与相等处理
    +        if (i < 0 || j < 0 || i >= size() || j >= size() || i == j)
    +            throw new IndexOutOfBoundsException();
    +        // 在无向图中,邻接矩阵沿主对角线对称,即满足 (i, j) == (j, i)
    +        adjMat.get(i).set(j, 1);
    +        adjMat.get(j).set(i, 1);
    +    }
    +
    +    /* 删除边 */
    +    // 参数 i, j 对应 vertices 元素索引
    +    public void removeEdge(int i, int j) {
    +        // 索引越界与相等处理
    +        if (i < 0 || j < 0 || i >= size() || j >= size() || i == j)
    +            throw new IndexOutOfBoundsException();
    +        adjMat.get(i).set(j, 0);
    +        adjMat.get(j).set(i, 0);
    +    }
    +
    +    /* 打印邻接矩阵 */
    +    public void print() {
    +        System.out.print("顶点列表 = ");
    +        System.out.println(vertices);
    +        System.out.println("邻接矩阵 =");
    +        PrintUtil.printMatrix(adjMat);
    +    }
    +}
    +
    +
    +
    +
    graph_adjacency_matrix.cpp
    
    +
    +
    +
    +
    graph_adjacency_matrix.py
    
    +
    +
    +
    +
    graph_adjacency_matrix.go
    /* 基于邻接矩阵实现的无向图类 */
    +type graphAdjMat struct {
    +    // 顶点列表,元素代表“顶点值”,索引代表“顶点索引”
    +    vertices []int
    +    // 邻接矩阵,行列索引对应“顶点索引”
    +    adjMat [][]int
    +}
    +
    +func newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {
    +    // 添加顶点
    +    n := len(vertices)
    +    adjMat := make([][]int, n)
    +    for i := range adjMat {
    +        adjMat[i] = make([]int, n)
    +    }
    +    // 初始化图
    +    g := &graphAdjMat{
    +        vertices: vertices,
    +        adjMat:   adjMat,
    +    }
    +    // 添加边
    +    // 请注意,edges 元素代表顶点索引,即对应 vertices 元素索引
    +    for i := range edges {
    +        g.addEdge(edges[i][0], edges[i][1])
    +    }
    +    return g
    +}
    +
    +/* 获取顶点数量 */
    +func (g *graphAdjMat) size() int {
    +    return len(g.vertices)
    +}
    +
    +/* 添加顶点 */
    +func (g *graphAdjMat) addVertex(val int) {
    +    n := g.size()
    +    // 向顶点列表中添加新顶点的值
    +    g.vertices = append(g.vertices, val)
    +    // 在邻接矩阵中添加一行
    +    newRow := make([]int, n)
    +    g.adjMat = append(g.adjMat, newRow)
    +    // 在邻接矩阵中添加一列
    +    for i := range g.adjMat {
    +        g.adjMat[i] = append(g.adjMat[i], 0)
    +    }
    +}
    +
    +/* 删除顶点 */
    +func (g *graphAdjMat) removeVertex(index int) {
    +    if index >= g.size() {
    +        return
    +    }
    +    // 在顶点列表中移除索引 index 的顶点
    +    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)
    +    // 在邻接矩阵中删除索引 index 的行
    +    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)
    +    // 在邻接矩阵中删除索引 index 的列
    +    for i := range g.adjMat {
    +        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)
    +    }
    +}
    +
    +/* 添加边 */
    +// 参数 i, j 对应 vertices 元素索引
    +func (g *graphAdjMat) addEdge(i, j int) {
    +    // 索引越界与相等处理
    +    if i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {
    +        fmt.Errorf("%s", "Index Out Of Bounds Exception")
    +    }
    +    // 在无向图中,邻接矩阵沿主对角线对称,即满足 (i, j) == (j, i)
    +    g.adjMat[i][j] = 1
    +    g.adjMat[j][i] = 1
    +}
    +
    +/* 删除边 */
    +// 参数 i, j 对应 vertices 元素索引
    +func (g *graphAdjMat) removeEdge(i, j int) {
    +    // 索引越界与相等处理
    +    if i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {
    +        fmt.Errorf("%s", "Index Out Of Bounds Exception")
    +    }
    +    g.adjMat[i][j] = 0
    +    g.adjMat[j][i] = 0
    +}
    +
    +
    +
    +
    graph_adjacency_matrix.js
    
    +
    +
    +
    +
    graph_adjacency_matrix.ts
    
    +
    +
    +
    +
    graph_adjacency_matrix.c
    
    +
    +
    +
    +
    graph_adjacency_matrix.cs
    
    +
    +
    +
    +
    graph_adjacency_matrix.swift
    /* 基于邻接矩阵实现的无向图类 */
    +class GraphAdjMat {
    +    private var vertices: [Int] // 顶点列表,元素代表“顶点值”,索引代表“顶点索引”
    +    private var adjMat: [[Int]] // 邻接矩阵,行列索引对应“顶点索引”
    +
    +    /* 构造函数 */
    +    init(vertices: [Int], edges: [[Int]]) {
    +        self.vertices = []
    +        adjMat = []
    +        // 添加顶点
    +        for val in vertices {
    +            addVertex(val: val)
    +        }
    +        // 添加边
    +        // 请注意,edges 元素代表顶点索引,即对应 vertices 元素索引
    +        for e in edges {
    +            addEdge(i: e[0], j: e[1])
    +        }
    +    }
    +
    +    /* 获取顶点数量 */
    +    func size() -> Int {
    +        vertices.count
    +    }
    +
    +    /* 添加顶点 */
    +    func addVertex(val: Int) {
    +        let n = size()
    +        // 向顶点列表中添加新顶点的值
    +        vertices.append(val)
    +        // 在邻接矩阵中添加一行
    +        let newRow = Array(repeating: 0, count: n)
    +        adjMat.append(newRow)
    +        // 在邻接矩阵中添加一列
    +        for i in adjMat.indices {
    +            adjMat[i].append(0)
    +        }
    +    }
    +
    +    /* 删除顶点 */
    +    func removeVertex(index: Int) {
    +        if index >= size() {
    +            fatalError("越界")
    +        }
    +        // 在顶点列表中移除索引 index 的顶点
    +        vertices.remove(at: index)
    +        // 在邻接矩阵中删除索引 index 的行
    +        adjMat.remove(at: index)
    +        // 在邻接矩阵中删除索引 index 的列
    +        for i in adjMat.indices {
    +            adjMat[i].remove(at: index)
    +        }
    +    }
    +
    +    /* 添加边 */
    +    // 参数 i, j 对应 vertices 元素索引
    +    func addEdge(i: Int, j: Int) {
    +        // 索引越界与相等处理
    +        if i < 0 || j < 0 || i >= size() || j >= size() || i == j {
    +            fatalError("越界")
    +        }
    +        // 在无向图中,邻接矩阵沿主对角线对称,即满足 (i, j) == (j, i)
    +        adjMat[i][j] = 1
    +        adjMat[j][i] = 1
    +    }
    +
    +    /* 删除边 */
    +    // 参数 i, j 对应 vertices 元素索引
    +    func removeEdge(i: Int, j: Int) {
    +        // 索引越界与相等处理
    +        if i < 0 || j < 0 || i >= size() || j >= size() || i == j {
    +            fatalError("越界")
    +        }
    +        adjMat[i][j] = 0
    +        adjMat[j][i] = 0
    +    }
    +}
    +
    +
    +
    +
    graph_adjacency_matrix.zig
    
    +
    +
    +
    +
    +

    9.2.2. 基于邻接表的实现

    +

    设图的顶点总数为 \(n\) 、边总数为 \(m\) ,则有:

    +
      +
    • 添加边:在顶点对应链表的尾部添加边即可,使用 \(O(1)\) 时间。因为是无向图,所以需要同时添加两个方向的边。
    • +
    • 删除边:在顶点对应链表中查询与删除指定边,使用 \(O(m)\) 时间。与添加边一样,需要同时删除两个方向的边。
    • +
    • 添加顶点:在邻接表中添加一个链表即可,并以新增顶点为链表头结点,使用 \(O(1)\) 时间。
    • +
    • 删除顶点:需要遍历整个邻接表,删除包含指定顶点的所有边,使用 \(O(n + m)\) 时间。
    • +
    • 初始化:需要在邻接表中建立 \(n\) 个结点和 \(2m\) 条边,使用 \(O(n + m)\) 时间。
    • +
    +
    +
    +
    +

    adjacency_list_initialization

    +
    +
    +

    adjacency_list_add_edge

    +
    +
    +

    adjacency_list_remove_edge

    +
    +
    +

    adjacency_list_add_vertex

    +
    +
    +

    adjacency_list_remove_vertex

    +
    +
    +
    +

    基于邻接表实现图的代码如下所示。

    +
    +
    +
    +
    graph_adjacency_list.java
    /* 顶点类 */
    +class Vertex {
    +    int val;
    +    public Vertex(int val) {
    +        this.val = val;
    +    }
    +}
    +
    +/* 基于邻接表实现的无向图类 */
    +class GraphAdjList {
    +    // 请注意,vertices 和 adjList 中存储的都是 Vertex 对象
    +    Map<Vertex, Set<Vertex>> adjList; // 邻接表(使用哈希表实现)
    +
    +    /* 构造函数 */
    +    public GraphAdjList(Vertex[][] edges) {
    +        this.adjList = new HashMap<>();
    +        // 添加所有顶点和边
    +        for (Vertex[] edge : edges) {
    +            addVertex(edge[0]);
    +            addVertex(edge[1]);
    +            addEdge(edge[0], edge[1]);
    +        }
    +    }
    +
    +    /* 获取顶点数量 */
    +    public int size() {
    +        return adjList.size();
    +    }
    +
    +    /* 添加边 */
    +    public void addEdge(Vertex vet1, Vertex vet2) {
    +        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)
    +            throw new IllegalArgumentException();
    +        // 添加边 vet1 - vet2
    +        adjList.get(vet1).add(vet2);
    +        adjList.get(vet2).add(vet1);
    +    }
    +
    +    /* 删除边 */
    +    public void removeEdge(Vertex vet1, Vertex vet2) {
    +        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)
    +            throw new IllegalArgumentException();
    +        // 删除边 vet1 - vet2
    +        adjList.get(vet1).remove(vet2);
    +        adjList.get(vet2).remove(vet1);
    +    }
    +
    +    /* 添加顶点 */
    +    public void addVertex(Vertex vet) {
    +        if (adjList.containsKey(vet))
    +            return;
    +        // 在邻接表中添加一个新链表(即 HashSet)
    +        adjList.put(vet, new HashSet<>());
    +    }
    +
    +    /* 删除顶点 */
    +    public void removeVertex(Vertex vet) {
    +        if (!adjList.containsKey(vet))
    +            throw new IllegalArgumentException();
    +        // 在邻接表中删除顶点 vet 对应的链表(即 HashSet)
    +        adjList.remove(vet);
    +        // 遍历其它顶点的链表(即 HashSet),删除所有包含 vet 的边
    +        for (Set<Vertex> set : adjList.values()) {
    +            set.remove(vet);
    +        }
    +    }
    +
    +    /* 打印邻接表 */
    +    public void print() {
    +        System.out.println("邻接表 =");
    +        for (Map.Entry<Vertex, Set<Vertex>> entry : adjList.entrySet()) {
    +            List<Integer> tmp = new ArrayList<>();
    +            for (Vertex vertex : entry.getValue())
    +                tmp.add(vertex.val);
    +            System.out.println(entry.getKey().val + ": " + tmp + ",");
    +        }
    +    }
    +}
    +
    +
    +
    +
    graph_adjacency_list.cpp
    
    +
    +
    +
    +
    graph_adjacency_list.py
    
    +
    +
    +
    +
    graph_adjacency_list.go
    /* 顶点类 */
    +type vertex struct {
    +    val int
    +}
    +
    +func newVertex(val int) vertex {
    +    return vertex{
    +        val: val,
    +    }
    +}
    +
    +/* 基于邻接表实现的无向图类 */
    +type graphAdjList struct {
    +    // 请注意,vertices 和 adjList 中存储的都是 Vertex 对象
    +    // 邻接表(使用哈希表实现), 使用哈希表模拟集合
    +    adjList map[vertex]map[vertex]struct{}
    +}
    +
    +/* 构造函数 */
    +func newGraphAdjList(edges [][]vertex) *graphAdjList {
    +    g := &graphAdjList{
    +        adjList: make(map[vertex]map[vertex]struct{}),
    +    }
    +    // 添加所有顶点和边
    +    for _, edge := range edges {
    +        g.addVertex(edge[0])
    +        g.addVertex(edge[1])
    +        g.addEdge(edge[0], edge[1])
    +    }
    +    return g
    +}
    +
    +/* 获取顶点数量 */
    +func (g *graphAdjList) size() int {
    +    return len(g.adjList)
    +}
    +
    +/* 添加边 */
    +func (g *graphAdjList) addEdge(vet1 vertex, vet2 vertex) {
    +    _, ok1 := g.adjList[vet1]
    +    _, ok2 := g.adjList[vet2]
    +    if !ok1 || !ok2 || vet1 == vet2 {
    +        panic("error")
    +    }
    +    // 添加边 vet1 - vet2, 添加匿名 struct{},
    +    g.adjList[vet1][vet2] = struct{}{}
    +    g.adjList[vet2][vet1] = struct{}{}
    +}
    +
    +/* 删除边 */
    +func (g *graphAdjList) removeEdge(vet1 vertex, vet2 vertex) {
    +    _, ok1 := g.adjList[vet1]
    +    _, ok2 := g.adjList[vet2]
    +    if !ok1 || !ok2 || vet1 == vet2 {
    +        panic("error")
    +    }
    +    // 删除边 vet1 - vet2, 借助 delete 来删除 map 中的键
    +    delete(g.adjList[vet1], vet2)
    +    delete(g.adjList[vet2], vet1)
    +}
    +
    +/* 添加顶点 */
    +func (g *graphAdjList) addVertex(vet vertex) {
    +    _, ok := g.adjList[vet]
    +    if ok {
    +        return
    +    }
    +    // 在邻接表中添加一个新链表(即 set)
    +    g.adjList[vet] = make(map[vertex]struct{})
    +}
    +
    +/* 删除顶点 */
    +func (g *graphAdjList) removeVertex(vet vertex) {
    +    _, ok := g.adjList[vet]
    +    if !ok {
    +        panic("error")
    +    }
    +    // 在邻接表中删除顶点 vet 对应的链表
    +    delete(g.adjList, vet)
    +    // 遍历其它顶点的链表(即 Set),删除所有包含 vet 的边
    +    for _, set := range g.adjList {
    +        // 操作
    +        delete(set, vet)
    +    }
    +}
    +
    +
    +
    +
    graph_adjacency_list.js
    
    +
    +
    +
    +
    graph_adjacency_list.ts
    
    +
    +
    +
    +
    graph_adjacency_list.c
    
    +
    +
    +
    +
    graph_adjacency_list.cs
    
    +
    +
    +
    +
    graph_adjacency_list.swift
    /* 顶点类 */
    +class Vertex: Hashable {
    +    var val: Int
    +
    +    init(val: Int) {
    +        self.val = val
    +    }
    +
    +    static func == (lhs: Vertex, rhs: Vertex) -> Bool {
    +        lhs.val == rhs.val
    +    }
    +
    +    func hash(into hasher: inout Hasher) {
    +        hasher.combine(val)
    +    }
    +}
    +
    +/* 基于邻接表实现的无向图类 */
    +class GraphAdjList {
    +    // 请注意,vertices 和 adjList 中存储的都是 Vertex 对象
    +    private var adjList: [Vertex: Set<Vertex>] // 邻接表(使用哈希表实现)
    +
    +    init(edges: [[Vertex]]) {
    +        adjList = [:]
    +        // 添加所有顶点和边
    +        for edge in edges {
    +            addVertex(vet: edge[0])
    +            addVertex(vet: edge[1])
    +            addEdge(vet1: edge[0], vet2: edge[1])
    +        }
    +    }
    +
    +    /* 获取顶点数量 */
    +    func size() -> Int {
    +        adjList.count
    +    }
    +
    +    /* 添加边 */
    +    func addEdge(vet1: Vertex, vet2: Vertex) {
    +        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {
    +            fatalError("参数错误")
    +        }
    +        // 添加边 vet1 - vet2
    +        adjList[vet1]?.insert(vet2)
    +        adjList[vet2]?.insert(vet1)
    +    }
    +
    +    /* 删除边 */
    +    func removeEdge(vet1: Vertex, vet2: Vertex) {
    +        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {
    +            fatalError("参数错误")
    +        }
    +        // 删除边 vet1 - vet2
    +        adjList[vet1]?.remove(vet2)
    +        adjList[vet2]?.remove(vet1)
    +    }
    +
    +    /* 添加顶点 */
    +    func addVertex(vet: Vertex) {
    +        if adjList[vet] != nil {
    +            return
    +        }
    +        // 在邻接表中添加一个新链表(即 HashSet)
    +        adjList[vet] = []
    +    }
    +
    +    /* 删除顶点 */
    +    func removeVertex(vet: Vertex) {
    +        if adjList[vet] == nil {
    +            fatalError("参数错误")
    +        }
    +        // 在邻接表中删除顶点 vet 对应的链表(即 HashSet)
    +        adjList.removeValue(forKey: vet)
    +        // 遍历其它顶点的链表(即 HashSet),删除所有包含 vet 的边
    +        for key in adjList.keys {
    +            adjList[key]?.remove(vet)
    +        }
    +    }
    +}
    +
    +
    +
    +
    graph_adjacency_list.zig
    
    +
    +
    +
    +
    +

    9.2.3. 效率对比

    +

    设图中共有 \(n\) 个顶点和 \(m\) 条边,下表为邻接矩阵和邻接表的时间和空间效率对比。

    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    邻接矩阵邻接表(链表)邻接表(哈希表)
    判断是否邻接\(O(1)\)\(O(m)\)\(O(1)\)
    添加边\(O(1)\)\(O(1)\)\(O(1)\)
    删除边\(O(1)\)\(O(m)\)\(O(1)\)
    添加顶点\(O(n)\)\(O(1)\)\(O(1)\)
    删除顶点\(O(n^2)\)\(O(n + m)\)\(O(n)\)
    内存空间占用\(O(n^2)\)\(O(n + m)\)\(O(n + m)\)
    +
    +

    观察上表,貌似邻接表(哈希表)的时间与空间效率最优。但实际上,在邻接矩阵中操作边的效率更高,只需要一次数组访问或赋值操作即可。总结以上,邻接矩阵体现“以空间换时间”,邻接表体现“以时间换空间”

    + + + + + +

    评论

    + + + + + + +
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    + + + + +
    + + + +
    + + + +
    +
    +
    +
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h3o}NAwBFkDfIruQd!^EfqJV$mPo+h&g>~NiKLFcJ)D!>! literal 0 HcmV?d00001 diff --git a/chapter_hashing/hash_collision/index.html b/chapter_hashing/hash_collision/index.html new file mode 100644 index 000000000..c6efaf915 --- /dev/null +++ b/chapter_hashing/hash_collision/index.html @@ -0,0 +1,1846 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + 6.2. 哈希冲突处理 - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    6.2. 哈希冲突

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    理想情况下,哈希函数应该为每个输入产生唯一的输出,使得 key 和 value 一一对应。而实际上,往往存在向哈希函数输入不同的 key 而产生相同输出的情况,这种情况被称为「哈希冲突 Hash Collision」。哈希冲突会导致查询结果错误,从而严重影响哈希表的可用性。

    +

    那么,为什么会出现哈希冲突呢?本质上看,由于哈希函数的输入空间往往远大于输出空间,因此不可避免地会出现多个输入产生相同输出的情况,即为哈希冲突。比如,输入空间是全体整数,输出空间是一个固定大小的桶(数组)的索引范围,那么必定会有多个整数同时映射到一个桶索引。

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    为了缓解哈希冲突,一方面,我们可以通过「哈希表扩容」来减小冲突概率。极端情况下,当输入空间和输出空间大小相等时,哈希表就等价于数组了,可谓“大力出奇迹”。

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    另一方面,考虑通过优化数据结构以缓解哈希冲突,常见的方法有「链式地址」和「开放寻址」。

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    6.2.1. 哈希表扩容

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    「负载因子 Load Factor」定义为 哈希表中元素数量除以桶槽数量(即数组大小),代表哈希冲突的严重程度。

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    负载因子常用作哈希表扩容的触发条件。比如在 Java 中,当负载因子 \(> 0.75\) 时则触发扩容,将 HashMap 大小扩充至原先的 \(2\) 倍。

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    与数组扩容类似,哈希表扩容操作的开销很大,因为需要将所有键值对从原哈希表依次移动至新哈希表。

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    6.2.2. 链式地址

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    在原始哈希表中,桶内的每个地址只能存储一个元素(即键值对)。考虑将单个元素转化成一个链表,将所有冲突元素都存储在一个链表中

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    hash_collision_chaining

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    链式地址下,哈希表操作方法为:

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    • 查询元素:先将 key 输入到哈希函数得到桶内索引,即可访问链表头结点,再通过遍历链表查找对应 value 。
    • +
    • 添加元素:先通过哈希函数访问链表头部,再将结点(即键值对)添加到链表头部即可。
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    • 删除元素:同样先根据哈希函数结果访问链表头部,再遍历链表查找对应结点,删除之即可。
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    链式地址虽然解决了哈希冲突问题,但仍存在局限性,包括:

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    • 占用空间变大,因为链表或二叉树包含结点指针,相比于数组更加耗费内存空间;
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    • 查询效率降低,因为需要线性遍历链表来查找对应元素;
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    为了缓解时间效率问题,可以把「链表」转化为「AVL 树」或「红黑树」,将查询操作的时间复杂度优化至 \(O(\log n)\)

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    6.2.3. 开放寻址

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    「开放寻址」不引入额外数据结构,而是通过“多次探测”来解决哈希冲突。根据探测方法的不同,主要分为 线性探测、平方探测、多次哈希

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    线性探测

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    「线性探测」使用固定步长的线性查找来解决哈希冲突。

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    插入元素:如果出现哈希冲突,则从冲突位置向后线性遍历(步长一般取 1 ),直到找到一个空位,则将元素插入到该空位中。

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    查找元素:若出现哈希冲突,则使用相同步长执行线性查找,会遇到两种情况:

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    1. 找到对应元素,返回 value 即可;
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    3. 若遇到空位,则说明查找键值对不在哈希表中;
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    hash_collision_linear_probing

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    线性探测存在以下缺陷:

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    • 不能直接删除元素。删除元素会导致桶内出现一个空位,在查找其他元素时,该空位有可能导致程序认为元素不存在(即上述第 2. 种情况)。因此需要借助一个标志位来标记删除元素。
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    • 容易产生聚集。桶内被占用的连续位置越长,这些连续位置发生哈希冲突的可能性越大,从而进一步促进这一位置的“聚堆生长”,最终导致增删查改操作效率的劣化。
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    多次哈希

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    顾名思义,「多次哈希」的思路是使用多个哈希函数 \(f_1(x)\) , \(f_2(x)\) , \(f_3(x)\) , \(\cdots\) 进行探测。

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    插入元素:若哈希函数 \(f_1(x)\) 出现冲突,则尝试 \(f_2(x)\) ,以此类推……直到找到空位后插入元素。

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    查找元素:以相同的哈希函数顺序查找,存在两种情况:

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    1. 找到目标元素,则返回之;
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    3. 到空位或已尝试所有哈希函数,说明哈希表中无此元素;
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    相比于「线性探测」,「多次哈希」方法更不容易产生聚集,代价是多个哈希函数增加了额外计算量。

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    工业界方案

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    Java 采用「链式地址」。在 JDK 1.8 之后,HashMap 内数组长度大于 64 时,长度大于 8 的链表会被转化为「红黑树」,以提升查找性能。

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    Python 采用「开放寻址」。字典 dict 使用伪随机数进行探测。

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    评论

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    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + + + + + +

    6.1. 哈希表

    +

    哈希表通过建立「键 key」和「值 value」之间的映射,实现高效的元素查找。具体地,输入一个 key ,在哈希表中查询并获取 value ,时间复杂度为 \(O(1)\)

    +

    例如,给定一个包含 \(n\) 个学生的数据库,每个学生有“姓名 name ”和“学号 id ”两项数据,希望实现一个查询功能:输入一个学号,返回对应的姓名,则可以使用哈希表实现。

    +

    hash_map

    +

    Fig. 哈希表抽象表示

    + +

    6.1.1. 哈希表效率

    +

    除了哈希表之外,还可以使用以下数据结构来实现上述查询功能:

    +
      +
    1. 无序数组:每个元素为 [学号, 姓名]
    2. +
    3. 有序数组:将 1. 中的数组按照学号从小到大排序;
    4. +
    5. 链表:每个结点的值为 [学号, 姓名]
    6. +
    7. 二叉搜索树:每个结点的值为 [学号, 姓名] ,根据学号大小来构建树;
    8. +
    +

    使用上述方法,各项操作的时间复杂度如下表所示(在此不做赘述,详解可见 二叉搜索树章节)。无论是查找元素、还是增删元素,哈希表的时间复杂度都是 \(O(1)\) ,全面胜出!

    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    无序数组有序数组链表二叉搜索树哈希表
    查找元素\(O(n)\)\(O(\log n)\)\(O(n)\)\(O(\log n)\)\(O(1)\)
    插入元素\(O(1)\)\(O(n)\)\(O(1)\)\(O(\log n)\)\(O(1)\)
    删除元素\(O(n)\)\(O(n)\)\(O(n)\)\(O(\log n)\)\(O(1)\)
    +
    +

    6.1.2. 哈希表常用操作

    +

    哈希表的基本操作包括 初始化、查询操作、添加与删除键值对

    +
    +
    +
    +
    hash_map.java
    /* 初始化哈希表 */
    +Map<Integer, String> map = new HashMap<>();
    +
    +/* 添加操作 */
    +// 在哈希表中添加键值对 (key, value)
    +map.put(12836, "小哈");   
    +map.put(15937, "小啰");   
    +map.put(16750, "小算");   
    +map.put(13276, "小法");
    +map.put(10583, "小鸭");
    +
    +/* 查询操作 */
    +// 向哈希表输入键 key ,得到值 value
    +String name = map.get(15937);
    +
    +/* 删除操作 */
    +// 在哈希表中删除键值对 (key, value)
    +map.remove(10583);
    +
    +
    +
    +
    hash_map.cpp
    /* 初始化哈希表 */
    +unordered_map<int, string> map;
    +
    +/* 添加操作 */
    +// 在哈希表中添加键值对 (key, value)
    +map[12836] = "小哈";
    +map[15937] = "小啰";
    +map[16750] = "小算";
    +map[13276] = "小法";
    +map[10583] = "小鸭";
    +
    +/* 查询操作 */
    +// 向哈希表输入键 key ,得到值 value
    +string name = map[15937];
    +
    +/* 删除操作 */
    +// 在哈希表中删除键值对 (key, value)
    +map.erase(10583);
    +
    +
    +
    +
    hash_map.py
    """ 初始化哈希表 """
    +mapp = {}
    +
    +""" 添加操作 """
    +# 在哈希表中添加键值对 (key, value)
    +mapp[12836] = "小哈"
    +mapp[15937] = "小啰"
    +mapp[16750] = "小算"
    +mapp[13276] = "小法"
    +mapp[10583] = "小鸭"
    +
    +""" 查询操作 """
    +# 向哈希表输入键 key ,得到值 value
    +name = mapp[15937]
    +
    +""" 删除操作 """
    +# 在哈希表中删除键值对 (key, value)
    +mapp.pop(10583)
    +
    +
    +
    +
    hash_map.go
    /* 初始化哈希表 */
    +mapp := make(map[int]string)
    +
    +/* 添加操作 */
    +// 在哈希表中添加键值对 (key, value)
    +mapp[12836] = "小哈"
    +mapp[15937] = "小啰"
    +mapp[16750] = "小算"
    +mapp[13276] = "小法"
    +mapp[10583] = "小鸭"
    +
    +/* 查询操作 */
    +// 向哈希表输入键 key ,得到值 value
    +name := mapp[15937]
    +
    +/* 删除操作 */
    +// 在哈希表中删除键值对 (key, value)
    +delete(mapp, 10583)
    +
    +
    +
    +
    hash_map.js
    /* 初始化哈希表 */
    +const map = new ArrayHashMap();
    +/* 添加操作 */
    +// 在哈希表中添加键值对 (key, value)
    +map.set(12836, '小哈');
    +map.set(15937, '小啰');
    +map.set(16750, '小算');
    +map.set(13276, '小法');
    +map.set(10583, '小鸭');
    +
    +/* 查询操作 */
    +// 向哈希表输入键 key ,得到值 value
    +let name = map.get(15937);
    +
    +/* 删除操作 */
    +// 在哈希表中删除键值对 (key, value)
    +map.delete(10583);
    +
    +
    +
    +
    hash_map.ts
    /* 初始化哈希表 */
    +const map = new Map<number, string>();
    +/* 添加操作 */
    +// 在哈希表中添加键值对 (key, value)
    +map.set(12836, '小哈');
    +map.set(15937, '小啰');
    +map.set(16750, '小算');
    +map.set(13276, '小法');
    +map.set(10583, '小鸭');
    +console.info('\n添加完成后,哈希表为\nKey -> Value');
    +console.info(map);
    +
    +/* 查询操作 */
    +// 向哈希表输入键 key ,得到值 value
    +let name = map.get(15937);
    +console.info('\n输入学号 15937 ,查询到姓名 ' + name);
    +
    +/* 删除操作 */
    +// 在哈希表中删除键值对 (key, value)
    +map.delete(10583);
    +console.info('\n删除 10583 后,哈希表为\nKey -> Value');
    +console.info(map);
    +
    +
    +
    +
    hash_map.c
    
    +
    +
    +
    +
    hash_map.cs
    /* 初始化哈希表 */
    +Dictionary<int, String> map = new ();
    +
    +/* 添加操作 */
    +// 在哈希表中添加键值对 (key, value)
    +map.Add(12836, "小哈");
    +map.Add(15937, "小啰");
    +map.Add(16750, "小算");
    +map.Add(13276, "小法");
    +map.Add(10583, "小鸭");
    +
    +/* 查询操作 */
    +// 向哈希表输入键 key ,得到值 value
    +String name = map[15937];
    +
    +/* 删除操作 */
    +// 在哈希表中删除键值对 (key, value)
    +map.Remove(10583);
    +
    +
    +
    +
    hash_map.swift
    /* 初始化哈希表 */
    +var map: [Int: String] = [:]
    +
    +/* 添加操作 */
    +// 在哈希表中添加键值对 (key, value)
    +map[12836] = "小哈"
    +map[15937] = "小啰"
    +map[16750] = "小算"
    +map[13276] = "小法"
    +map[10583] = "小鸭"
    +
    +/* 查询操作 */
    +// 向哈希表输入键 key ,得到值 value
    +let name = map[15937]!
    +
    +/* 删除操作 */
    +// 在哈希表中删除键值对 (key, value)
    +map.removeValue(forKey: 10583)
    +
    +
    +
    +
    hash_map.zig
    
    +
    +
    +
    +
    +

    遍历哈希表有三种方式,即 遍历键值对、遍历键、遍历值

    +
    +
    +
    +
    hash_map.java
    /* 遍历哈希表 */
    +// 遍历键值对 key->value
    +for (Map.Entry <Integer, String> kv: map.entrySet()) {
    +    System.out.println(kv.getKey() + " -> " + kv.getValue());
    +}
    +// 单独遍历键 key
    +for (int key: map.keySet()) {
    +    System.out.println(key);
    +}
    +// 单独遍历值 value
    +for (String val: map.values()) {
    +    System.out.println(val);
    +}
    +
    +
    +
    +
    hash_map.cpp
    /* 遍历哈希表 */
    +// 遍历键值对 key->value
    +for (auto kv: map) {
    +    cout << kv.first << " -> " << kv.second << endl;
    +}
    +// 单独遍历键 key
    +for (auto key: map) {
    +    cout << key.first << endl;
    +}
    +// 单独遍历值 value
    +for (auto val: map) {
    +    cout << val.second << endl;
    +}
    +
    +
    +
    +
    hash_map.py
    """ 遍历哈希表 """
    +# 遍历键值对 key->value
    +for key, value in mapp.items():
    +    print(key, "->", value)
    +# 单独遍历键 key
    +for key in mapp.keys():
    +    print(key)
    +# 单独遍历值 value
    +for value in mapp.values():
    +    print(value)
    +
    +
    +
    +
    hash_map_test.go
    /* 遍历哈希表 */
    +// 遍历键值对 key->value
    +for key, value := range mapp {
    +    fmt.Println(key, "->", value)
    +}
    +// 单独遍历键 key
    +for key := range mapp {
    +    fmt.Println(key)
    +}
    +// 单独遍历值 value
    +for _, value := range mapp {
    +    fmt.Println(value)
    +}
    +
    +
    +
    +
    hash_map.js
    /* 遍历哈希表 */
    +// 遍历键值对 key->value
    +for (const entry of map.entries()) {
    +    if (!entry) continue;
    +    console.info(entry.key + ' -> ' + entry.val);
    +}
    +// 单独遍历键 key
    +for (const key of map.keys()) {
    +    console.info(key);
    +}
    +// 单独遍历值 value
    +for (const val of map.values()) {
    +    console.info(val);
    +}
    +
    +
    +
    +
    hash_map.ts
    /* 遍历哈希表 */
    +console.info('\n遍历键值对 Key->Value');
    +for (const [k, v] of map.entries()) {
    +    console.info(k + ' -> ' + v);
    +}
    +console.info('\n单独遍历键 Key');
    +for (const k of map.keys()) {
    +    console.info(k);
    +}
    +console.info('\n单独遍历值 Value');
    +for (const v of map.values()) {
    +    console.info(v);
    +}
    +
    +
    +
    +
    hash_map.c
    
    +
    +
    +
    +
    hash_map.cs
    /* 遍历哈希表 */
    +// 遍历键值对 Key->Value
    +foreach (var kv in map) {
    +    Console.WriteLine(kv.Key + " -> " + kv.Value);
    +}
    +// 单独遍历键 key
    +foreach (int key in map.Keys) {
    +    Console.WriteLine(key);
    +}
    +// 单独遍历值 value
    +foreach (String val in map.Values) {
    +    Console.WriteLine(val);
    +}
    +
    +
    +
    +
    hash_map.swift
    /* 遍历哈希表 */
    +// 遍历键值对 Key->Value
    +for (key, value) in map {
    +    print("\(key) -> \(value)")
    +}
    +// 单独遍历键 Key
    +for key in map.keys {
    +    print(key)
    +}
    +// 单独遍历值 Value
    +for value in map.values {
    +    print(value)
    +}
    +
    +
    +
    +
    hash_map.zig
    
    +
    +
    +
    +
    +

    6.1.3. 哈希函数

    +

    哈希表中存储元素的数据结构被称为「桶 Bucket」,底层实现可能是数组、链表、二叉树(红黑树),或是它们的组合。

    +

    最简单地,我们可以仅用一个「数组」来实现哈希表。首先,将所有 value 放入数组中,那么每个 value 在数组中都有唯一的「索引」。显然,访问 value 需要给定索引,而为了 建立 key 和索引之间的映射关系,我们需要使用「哈希函数 Hash Function」。

    +

    设数组为 bucket ,哈希函数为 f(x) ,输入键为 key 。那么获取 value 的步骤为:

    +
      +
    1. 通过哈希函数计算出索引,即 index = f(key)
    2. +
    3. 通过索引在数组中获取值,即 value = bucket[index]
    4. +
    +

    以上述学生数据 key 学号 -> value 姓名 为例,我们可以将「哈希函数」设计为

    +
    \[ +f(x) = x \% 100 +\]
    +

    hash_function

    +

    Fig. 哈希函数

    + +
    +
    +
    +
    array_hash_map.java
    /* 键值对 int->String */
    +class Entry {
    +    public int key;
    +    public String val;
    +    public Entry(int key, String val) {
    +        this.key = key;
    +        this.val = val;
    +    }
    +}
    +
    +/* 基于数组简易实现的哈希表 */
    +class ArrayHashMap {
    +    private List<Entry> bucket;
    +    public ArrayHashMap() {
    +        // 初始化一个长度为 100 的桶(数组)
    +        bucket = new ArrayList<>();
    +        for (int i = 0; i < 100; i++) {
    +            bucket.add(null);
    +        }
    +    }
    +
    +    /* 哈希函数 */
    +    private int hashFunc(int key) {
    +        int index = key % 100;
    +        return index;
    +    }
    +
    +    /* 查询操作 */
    +    public String get(int key) {
    +        int index = hashFunc(key);
    +        Entry pair = bucket.get(index);
    +        if (pair == null) return null;
    +        return pair.val;
    +    }
    +
    +    /* 添加操作 */
    +    public void put(int key, String val) {
    +        Entry pair = new Entry(key, val);
    +        int index = hashFunc(key);
    +        bucket.set(index, pair);
    +    }
    +
    +    /* 删除操作 */
    +    public void remove(int key) {
    +        int index = hashFunc(key);
    +        // 置为 null ,代表删除
    +        bucket.set(index, null);
    +    }
    +
    +    /* 获取所有键值对 */
    +    public List<Entry> entrySet() {
    +        List<Entry> entrySet = new ArrayList<>();
    +        for (Entry pair : bucket) {
    +            if (pair != null)
    +                entrySet.add(pair);
    +        }
    +        return entrySet;
    +    }
    +
    +    /* 获取所有键 */
    +    public List<Integer> keySet() {
    +        List<Integer> keySet = new ArrayList<>();
    +        for (Entry pair : bucket) {
    +            if (pair != null)
    +                keySet.add(pair.key);
    +        }
    +        return keySet;
    +    }
    +
    +    /* 获取所有值 */
    +    public List<String> valueSet() {
    +        List<String> valueSet = new ArrayList<>();
    +        for (Entry pair : bucket) {
    +            if (pair != null)
    +                valueSet.add(pair.val);
    +        }
    +        return valueSet;
    +    }
    +
    +    /* 打印哈希表 */
    +    public void print() {
    +        for (Entry kv: entrySet()) {
    +            System.out.println(kv.key + " -> " + kv.val);
    +        }
    +    }
    +}
    +
    +
    +
    +
    array_hash_map.cpp
    /* 键值对 int->String */
    +struct Entry {
    +public:
    +    int key;
    +    string val;
    +    Entry(int key, string val) {
    +        this->key = key;
    +        this->val = val;
    +    }
    +};
    +
    +/* 基于数组简易实现的哈希表 */
    +class ArrayHashMap {
    +private:
    +    vector<Entry*> bucket;
    +public:
    +    ArrayHashMap() {
    +        // 初始化一个长度为 100 的桶(数组)
    +        bucket= vector<Entry*>(100);
    +    }
    +
    +    /* 哈希函数 */
    +    int hashFunc(int key) {
    +        int index = key % 100;
    +        return index;
    +    }
    +
    +    /* 查询操作 */
    +    string get(int key) {
    +        int index = hashFunc(key);
    +        Entry* pair = bucket[index];
    +        if (pair == nullptr)
    +            return nullptr;
    +        return pair->val;
    +    }
    +
    +    /* 添加操作 */
    +    void put(int key, string val) {
    +        Entry* pair = new Entry(key, val);
    +        int index = hashFunc(key);
    +        bucket[index] = pair;
    +    }
    +
    +    /* 删除操作 */
    +    void remove(int key) {
    +        int index = hashFunc(key);
    +        // 置为 nullptr ,代表删除
    +        bucket[index] = nullptr;
    +    }
    +
    +    /* 获取所有键值对 */
    +    vector<Entry*> entrySet() {
    +        vector<Entry*> entrySet;
    +        for (Entry* pair: bucket) {
    +            if (pair != nullptr) {
    +                entrySet.push_back(pair);
    +            }
    +        }
    +        return entrySet;
    +    }
    +
    +    /* 获取所有键 */
    +    vector<int> keySet() {
    +        vector<int> keySet;
    +        for (Entry* pair: bucket) {
    +            if (pair != nullptr) {
    +                keySet.push_back(pair->key);
    +            }
    +        }
    +        return keySet;
    +    }
    +
    +    /* 获取所有值 */
    +    vector<string> valueSet() {
    +        vector<string> valueSet;
    +        for (Entry* pair: bucket) {
    +            if (pair != nullptr){
    +                valueSet.push_back(pair->val);
    +            }
    +        }
    +        return valueSet;
    +    }
    +
    +    /* 打印哈希表 */
    +    void print() {
    +        for (Entry* kv: entrySet()) {
    +            cout << kv->key << " -> " << kv->val << endl;
    +        }
    +    }
    +};
    +
    +
    +
    +
    array_hash_map.py
    """ 键值对 int->String """
    +class Entry:
    +    def __init__(self, key, val):
    +        self.key = key
    +        self.val = val
    +
    +""" 基于数组简易实现的哈希表 """
    +class ArrayHashMap:
    +    def __init__(self):
    +        # 初始化一个长度为 100 的桶(数组)
    +        self.bucket = [None] * 100
    +
    +    """ 哈希函数 """
    +    def hash_func(self, key):
    +        index = key % 100
    +        return index
    +
    +    """ 查询操作 """
    +    def get(self, key):
    +        index = self.hash_func(key)
    +        pair = self.bucket[index]
    +        if pair is None:
    +            return None
    +        return pair.val
    +
    +    """ 添加操作 """
    +    def put(self, key, val):
    +        pair = Entry(key, val)
    +        index = self.hash_func(key)
    +        self.bucket[index] = pair
    +
    +    """ 删除操作 """
    +    def remove(self, key):
    +        index = self.hash_func(key)
    +        # 置为 None ,代表删除
    +        self.bucket[index] = None
    +
    +    """ 获取所有键值对 """
    +    def entry_set(self):
    +        result = []
    +        for pair in self.bucket:
    +            if pair is not None:
    +                result.append(pair)
    +        return result
    +
    +    """ 获取所有键 """
    +    def key_set(self):
    +        result = []
    +        for pair in self.bucket:
    +            if pair is not None:
    +                result.append(pair.key)
    +        return result
    +
    +    """ 获取所有值 """
    +    def value_set(self):
    +        result = []
    +        for pair in self.bucket:
    +            if pair is not None:
    +                result.append(pair.val)
    +        return result
    +
    +    """ 打印哈希表 """
    +    def print(self):
    +        for pair in self.bucket:
    +            if pair is not None:
    +                print(pair.key, "->", pair.val)
    +
    +
    +
    +
    array_hash_map.go
    /* 键值对 int->String */
    +type entry struct {
    +    key int
    +    val string
    +}
    +
    +/* 基于数组简易实现的哈希表 */
    +type arrayHashMap struct {
    +    bucket []*entry
    +}
    +
    +func newArrayHashMap() *arrayHashMap {
    +    // 初始化一个长度为 100 的桶(数组)
    +    bucket := make([]*entry, 100)
    +    return &arrayHashMap{bucket: bucket}
    +}
    +
    +/* 哈希函数 */
    +func (a *arrayHashMap) hashFunc(key int) int {
    +    index := key % 100
    +    return index
    +}
    +
    +/* 查询操作 */
    +func (a *arrayHashMap) get(key int) string {
    +    index := a.hashFunc(key)
    +    pair := a.bucket[index]
    +    if pair == nil {
    +        return "Not Found"
    +    }
    +    return pair.val
    +}
    +
    +/* 添加操作 */
    +func (a *arrayHashMap) put(key int, val string) {
    +    pair := &entry{key: key, val: val}
    +    index := a.hashFunc(key)
    +    a.bucket[index] = pair
    +}
    +
    +/* 删除操作 */
    +func (a *arrayHashMap) remove(key int) {
    +    index := a.hashFunc(key)
    +    // 置为 nil ,代表删除
    +    a.bucket[index] = nil
    +}
    +
    +
    +
    +
    array_hash_map.js
    /* 键值对 Number -> String */
    +class Entry {
    +    constructor(key, val) {
    +        this.key = key;
    +        this.val = val;
    +    }
    +}
    +
    +/* 基于数组简易实现的哈希表 */
    +class ArrayHashMap {
    +    #bucket;
    +    constructor() {
    +        // 初始化一个长度为 100 的桶(数组)
    +        this.#bucket = new Array(100).fill(null);
    +    }
    +
    +    /* 哈希函数 */
    +    #hashFunc(key) {
    +        return key % 100;
    +    }
    +
    +    /* 查询操作 */
    +    get(key) {
    +        let index = this.#hashFunc(key);
    +        let entry = this.#bucket[index];
    +        if (entry === null) return null;
    +        return entry.val;
    +    }
    +
    +    /* 添加操作 */
    +    set(key, val) {
    +        let index = this.#hashFunc(key);
    +        this.#bucket[index] = new Entry(key, val);
    +    }
    +
    +    /* 删除操作 */
    +    delete(key) {
    +        let index = this.#hashFunc(key);
    +        // 置为 null ,代表删除
    +        this.#bucket[index] = null;
    +    }
    +}
    +
    +
    +
    +
    array_hash_map.ts
    /* 键值对 Number -> String */
    +class Entry {
    +public key: number;
    +public val: string;
    +
    +    constructor(key: number, val: string) {
    +        this.key = key;
    +        this.val = val;
    +    }
    +}
    +
    +/* 基于数组简易实现的哈希表 */
    +class ArrayHashMap {
    +
    +    private readonly bucket: (Entry | null)[];
    +
    +    constructor() {
    +        // 初始化一个长度为 100 的桶(数组)
    +        this.bucket = (new Array(100)).fill(null);
    +    }
    +
    +    /* 哈希函数 */
    +    private hashFunc(key: number): number {
    +        return key % 100;
    +    }
    +
    +    /* 查询操作 */
    +    public get(key: number): string | null {
    +        let index = this.hashFunc(key);
    +        let entry = this.bucket[index];
    +        if (entry === null) return null;
    +        return entry.val;
    +    }
    +
    +    /* 添加操作 */
    +    public set(key: number, val: string) {
    +        let index = this.hashFunc(key);
    +        this.bucket[index] = new Entry(key, val);
    +    }
    +
    +    /* 删除操作 */
    +    public delete(key: number) {
    +        let index = this.hashFunc(key);
    +        // 置为 null ,代表删除
    +        this.bucket[index] = null;
    +    }
    +}
    +
    +
    +
    +
    array_hash_map.c
    
    +
    +
    +
    +
    array_hash_map.cs
    /* 键值对 int->String */
    +class Entry
    +{
    +    public int key;
    +    public String val;
    +    public Entry(int key, String val)
    +    {
    +        this.key = key;
    +        this.val = val;
    +    }
    +}
    +
    +/* 基于数组简易实现的哈希表 */
    +class ArrayHashMap
    +{
    +    private List<Entry?> bucket;
    +    public ArrayHashMap()
    +    {
    +        // 初始化一个长度为 100 的桶(数组)
    +        bucket = new ();
    +        for (int i = 0; i < 100; i++)
    +        {
    +            bucket.Add(null);
    +        }
    +    }
    +    /* 哈希函数 */
    +    private int hashFunc(int key)
    +    {
    +        int index = key % 100;
    +        return index;
    +    }
    +    /* 查询操作 */
    +    public String? get(int key)
    +    {
    +        int index = hashFunc(key);
    +        Entry? pair = bucket[index];
    +        if (pair == null) return null;
    +        return pair.val;
    +    }
    +    /* 添加操作 */
    +    public void put(int key, String val)
    +    {
    +        Entry pair = new Entry(key, val);
    +        int index = hashFunc(key);
    +        bucket[index]=pair;
    +    }
    +    /* 删除操作 */
    +    public void remove(int key)
    +    {
    +        int index = hashFunc(key);
    +        // 置为 null ,代表删除
    +        bucket[index]=null;
    +    }
    +}
    +
    +
    +
    +
    array_hash_map.swift
    /* 键值对 int->String */
    +class Entry {
    +    var key: Int
    +    var val: String
    +
    +    init(key: Int, val: String) {
    +        self.key = key
    +        self.val = val
    +    }
    +}
    +
    +/* 基于数组简易实现的哈希表 */
    +class ArrayHashMap {
    +    private var bucket: [Entry?] = []
    +
    +    init() {
    +        // 初始化一个长度为 100 的桶(数组)
    +        for _ in 0 ..< 100 {
    +            bucket.append(nil)
    +        }
    +    }
    +
    +    /* 哈希函数 */
    +    private func hashFunc(key: Int) -> Int {
    +        let index = key % 100
    +        return index
    +    }
    +
    +    /* 查询操作 */
    +    func get(key: Int) -> String? {
    +        let index = hashFunc(key: key)
    +        let pair = bucket[index]
    +        return pair?.val
    +    }
    +
    +    /* 添加操作 */
    +    func put(key: Int, val: String) {
    +        let pair = Entry(key: key, val: val)
    +        let index = hashFunc(key: key)
    +        bucket[index] = pair
    +    }
    +
    +    /* 删除操作 */
    +    func remove(key: Int) {
    +        let index = hashFunc(key: key)
    +        // 置为 nil ,代表删除
    +        bucket[index] = nil
    +    }
    +}
    +
    +
    +
    +
    array_hash_map.zig
    
    +
    +
    +
    +
    +

    6.1.4. 哈希冲突

    +

    细心的同学可能会发现,哈希函数 \(f(x) = x \% 100\) 会在某些情况下失效。具体地,当输入的 key 后两位相同时,哈希函数的计算结果也相同,指向同一个 value 。例如,分别查询两个学号 \(12836\)\(20336\) ,则有

    +
    \[ +f(12836) = f(20336) = 36 +\]
    +

    两个学号指向了同一个姓名,这明显是不对的,我们将这种现象称为「哈希冲突 Hash Collision」。如何避免哈希冲突的问题将被留在下章讨论。

    +

    hash_collision

    +

    Fig. 哈希冲突

    + +

    综上所述,一个优秀的「哈希函数」应该具备以下特性:

    +
      +
    • 尽量少地发生哈希冲突;
    • +
    • 时间复杂度 \(O(1)\) ,计算尽可能高效;
    • +
    • 空间使用率高,即“键值对占用空间 / 哈希表总占用空间”尽可能大;
    • +
    + + + + + +

    评论

    + + + + + + +
    +
    + + + + +
    + + + +
    + + + +
    +
    +
    +
    + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/chapter_hashing/summary/index.html b/chapter_hashing/summary/index.html new file mode 100644 index 000000000..95af35d0a --- /dev/null +++ b/chapter_hashing/summary/index.html @@ -0,0 +1,1679 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + 6.3. 小结 - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + + + 跳转至 + + +
    +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + + + + + +

    6.3. 小结

    + + + + + +

    评论

    + + + + + + +
    +
    + + + + +
    + + + +
    + + + +
    +
    +
    +
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    8.1. 堆

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    「堆 Heap」是一棵限定条件下的「完全二叉树」。根据成立条件,堆主要分为两种类型:

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    • 「大顶堆 Max Heap」,任意结点的值 \(\geq\) 其子结点的值;
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    • 「小顶堆 Min Heap」,任意结点的值 \(\leq\) 其子结点的值;
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    min_heap_and_max_heap

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    8.1.1. 堆术语与性质

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    • 由于堆是完全二叉树,因此最底层结点靠左填充,其它层结点皆被填满。
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    • 二叉树中的根结点对应「堆顶」,底层最靠右结点对应「堆底」。
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    • 对于大顶堆 / 小顶堆,其堆顶元素(即根结点)的值最大 / 最小。
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    8.1.2. 堆常用操作

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    值得说明的是,多数编程语言提供的是「优先队列 Priority Queue」,其是一种抽象数据结构,定义为具有出队优先级的队列

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    而恰好,堆的定义与优先队列的操作逻辑完全吻合,大顶堆就是一个元素从大到小出队的优先队列。从使用角度看,我们可以将「优先队列」和「堆」理解为等价的数据结构。因此,本文与代码对两者不做特别区分,统一使用「堆」来命名。

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    堆的常用操作见下表(方法命名以 Java 为例)。

    +

    Table. 堆的常用操作

    + +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    方法描述时间复杂度
    add()元素入堆\(O(\log n)\)
    poll()堆顶元素出堆\(O(\log n)\)
    peek()访问堆顶元素(大 / 小顶堆分别为最大 / 小值)\(O(1)\)
    size()获取堆的元素数量\(O(1)\)
    isEmpty()判断堆是否为空\(O(1)\)
    +
    +

    我们可以直接使用编程语言提供的堆类(或优先队列类)。

    +
    +

    Tip

    +

    类似于排序中“从小到大排列”和“从大到小排列”,“大顶堆”和“小顶堆”可仅通过修改 Comparator 来互相转换。

    +
    +
    +
    +
    +
    heap.java
    /* 初始化堆 */
    +// 初始化小顶堆
    +Queue<Integer> minHeap = new PriorityQueue<>();
    +// 初始化大顶堆(使用 lambda 表达式修改 Comparator 即可)
    +Queue<Integer> maxHeap = new PriorityQueue<>((a, b) -> { return b - a; });
    +
    +/* 元素入堆 */
    +maxHeap.add(1);
    +maxHeap.add(3);
    +maxHeap.add(2);
    +maxHeap.add(5);
    +maxHeap.add(4);
    +
    +/* 获取堆顶元素 */
    +int peek = maxHeap.peek(); // 5
    +
    +/* 堆顶元素出堆 */
    +// 出堆元素会形成一个从大到小的序列
    +peek = heap.poll();  // 5
    +peek = heap.poll();  // 4
    +peek = heap.poll();  // 3
    +peek = heap.poll();  // 2
    +peek = heap.poll();  // 1
    +
    +/* 获取堆大小 */
    +int size = maxHeap.size();
    +
    +/* 判断堆是否为空 */
    +boolean isEmpty = maxHeap.isEmpty();
    +
    +/* 输入列表并建堆 */
    +minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4));
    +
    +
    +
    +
    heap.cpp
    /* 初始化堆 */
    +// 初始化小顶堆
    +priority_queue<int, vector<int>, greater<int>> minHeap;
    +// 初始化大顶堆
    +priority_queue<int, vector<int>, less<int>> maxHeap;
    +
    +/* 元素入堆 */
    +maxHeap.push(1);
    +maxHeap.push(3);
    +maxHeap.push(2);
    +maxHeap.push(5);
    +maxHeap.push(4);
    +
    +/* 获取堆顶元素 */
    +int peek = maxHeap.top(); // 5
    +
    +/* 堆顶元素出堆 */
    +// 出堆元素会形成一个从大到小的序列
    +maxHeap.pop(); // 5
    +maxHeap.pop(); // 4
    +maxHeap.pop(); // 3
    +maxHeap.pop(); // 2
    +maxHeap.pop(); // 1
    +
    +/* 获取堆大小 */
    +int size = maxHeap.size();
    +
    +/* 判断堆是否为空 */
    +bool isEmpty = maxHeap.empty();
    +
    +/* 输入列表并建堆 */
    +vector<int> input{1, 3, 2, 5, 4};
    +priority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());
    +
    +
    +
    +
    heap.py
    
    +
    +
    +
    +
    heap.go
    // Go 语言中可以通过实现 heap.Interface 来构建整数大顶堆
    +// 实现 heap.Interface 需要同时实现 sort.Interface
    +type intHeap []any
    +
    +// Push heap.Interface 的方法,实现推入元素到堆
    +func (h *intHeap) Push(x any) {
    +    // Push 和 Pop 使用 pointer receiver 作为参数
    +    // 因为它们不仅会对切片的内容进行调整,还会修改切片的长度。
    +    *h = append(*h, x.(int))
    +}
    +
    +// Pop heap.Interface 的方法,实现弹出堆顶元素
    +func (h *intHeap) Pop() any {
    +    // 待出堆元素存放在最后
    +    last := (*h)[len(*h)-1]
    +    *h = (*h)[:len(*h)-1]
    +    return last
    +}
    +
    +// Len sort.Interface 的方法
    +func (h *intHeap) Len() int {
    +    return len(*h)
    +}
    +
    +// Less sort.Interface 的方法
    +func (h *intHeap) Less(i, j int) bool {
    +    // 如果实现小顶堆,则需要调整为小于号
    +    return (*h)[i].(int) > (*h)[j].(int)
    +}
    +
    +// Swap sort.Interface 的方法
    +func (h *intHeap) Swap(i, j int) {
    +    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]
    +}
    +
    +// Top 获取堆顶元素
    +func (h *intHeap) Top() any {
    +    return (*h)[0]
    +}
    +
    +/* Driver Code */
    +func TestHeap(t *testing.T) {
    +    /* 初始化堆 */
    +    // 初始化大顶堆
    +    maxHeap := &intHeap{}
    +    heap.Init(maxHeap)
    +    /* 元素入堆 */
    +    // 调用 heap.Interface 的方法,来添加元素
    +    heap.Push(maxHeap, 1)
    +    heap.Push(maxHeap, 3)
    +    heap.Push(maxHeap, 2)
    +    heap.Push(maxHeap, 4)
    +    heap.Push(maxHeap, 5)
    +
    +    /* 获取堆顶元素 */
    +    top := maxHeap.Top()
    +    fmt.Printf("堆顶元素为 %d\n", top)
    +
    +    /* 堆顶元素出堆 */
    +    // 调用 heap.Interface 的方法,来移除元素
    +    heap.Pop(maxHeap)
    +    heap.Pop(maxHeap)
    +    heap.Pop(maxHeap)
    +    heap.Pop(maxHeap)
    +    heap.Pop(maxHeap)
    +
    +    /* 获取堆大小 */
    +    size := len(*maxHeap)
    +    fmt.Printf("堆元素数量为 %d\n", size)
    +
    +    /* 判断堆是否为空 */
    +    isEmpty := len(*maxHeap) == 0
    +    fmt.Printf("堆是否为空 %t\n", isEmpty)
    +}
    +
    +
    +
    +
    heap.js
    // JavaScript 未提供内置 heap 类
    +
    +
    +
    +
    heap.ts
    // TypeScript 未提供内置堆 Heap 类
    +
    +
    +
    +
    heap.c
    
    +
    +
    +
    +
    heap.cs
    
    +
    +
    +
    +
    heap.swift
    // Swift 未提供内置 heap 类
    +
    +
    +
    +
    heap.zig
    
    +
    +
    +
    +
    +

    8.1.3. 堆的实现

    +

    下文实现的是「大顶堆」,若想转换为「小顶堆」,将所有大小逻辑判断取逆(例如将 \(\geq\) 替换为 \(\leq\) )即可,有兴趣的同学可自行实现。

    +

    堆的存储与表示

    +

    在二叉树章节我们学过,「完全二叉树」非常适合使用「数组」来表示,而堆恰好是一棵完全二叉树,因而我们采用「数组」来存储「堆」

    +

    二叉树指针。使用数组表示二叉树时,元素代表结点值,索引代表结点在二叉树中的位置,而结点指针通过索引映射公式来实现

    +

    具体地,给定索引 \(i\) ,那么其左子结点索引为 \(2i + 1\) 、右子结点索引为 \(2i + 2\) 、父结点索引为 \((i - 1) / 2\) (向下整除)。当索引越界时,代表空结点或结点不存在。

    +

    representation_of_heap

    +

    我们将索引映射公式封装成函数,以便后续使用。

    +
    +
    +
    +
    my_heap.java
    /* 获取左子结点索引 */
    +int left(int i) {
    +    return 2 * i + 1;
    +}
    +
    +/* 获取右子结点索引 */
    +int right(int i) {
    +    return 2 * i + 2;
    +}
    +
    +/* 获取父结点索引 */
    +int parent(int i) {
    +    return (i - 1) / 2; // 向下整除
    +}
    +
    +
    +
    +
    my_heap.cpp
    // 使用动态数组,这样无需考虑扩容问题
    +vector<int> maxHeap;
    +
    +/* 获取左子结点索引 */
    +int left(int i) {
    +    return 2 * i + 1;
    +}
    +
    +/* 获取右子结点索引 */
    +int right(int i) {
    +    return 2 * i + 2;
    +} 
    +
    +/* 获取父结点索引 */
    +int parent(int i) {
    +    return (i - 1) / 2; // 向下取整
    +}
    +
    +
    +
    +
    my_heap.py
    
    +
    +
    +
    +
    my_heap.go
    type maxHeap struct {
    +    // 使用切片而非数组,这样无需考虑扩容问题
    +    data []any
    +}
    +
    +/* 构造函数,建立空堆 */
    +func newHeap() *maxHeap {
    +    return &maxHeap{
    +        data: make([]any, 0),
    +    }
    +}
    +
    +/* 获取左子结点索引 */
    +func (h *maxHeap) left(i int) int {
    +    return 2*i + 1
    +}
    +
    +/* 获取右子结点索引 */
    +func (h *maxHeap) right(i int) int {
    +    return 2*i + 2
    +}
    +
    +/* 获取父结点索引 */
    +func (h *maxHeap) parent(i int) int {
    +    // 向下整除
    +    return (i - 1) / 2
    +}
    +
    +
    +
    +
    my_heap.js
    #maxHeap;
    +
    +/* 构造函数,建立空堆或根据输入列表建堆 */
    +constructor(nums) {
    +    // 将列表元素原封不动添加进堆
    +    this.#maxHeap = nums === undefined ? [] : [...nums];
    +    // 堆化除叶结点以外的其他所有结点
    +    for (let i = this.#parent(this.size() - 1); i >= 0; i--) {
    +        this.#siftDown(i);
    +    }
    +}
    +
    +/* 获取左子结点索引 */
    +#left(i) {
    +    return 2 * i + 1;
    +}
    +
    +/* 获取右子结点索引 */
    +#right(i) {
    +    return 2 * i + 2;
    +}
    +
    +/* 获取父结点索引 */
    +#parent(i) {
    +    return Math.floor((i - 1) / 2); // 向下整除
    +}
    +
    +
    +
    +
    my_heap.ts
    private maxHeap: number[];
    +/* 构造函数,建立空堆或根据输入列表建堆 */
    +constructor(nums?: number[]) {
    +    // 将列表元素原封不动添加进堆
    +    this.maxHeap = nums === undefined ? [] : [...nums];
    +    // 堆化除叶结点以外的其他所有结点
    +    for (let i = this.parent(this.size() - 1); i >= 0; i--) {
    +        this.siftDown(i);
    +    }
    +}
    +
    +/* 获取左子结点索引 */
    +private left(i: number): number {
    +    return 2 * i + 1;
    +}
    +
    +/* 获取右子结点索引 */
    +private right(i: number): number {
    +    return 2 * i + 2;
    +}
    +
    +/* 获取父结点索引 */
    +private parent(i: number): number {
    +    return Math.floor((i - 1) / 2); // 向下整除
    +}
    +
    +
    +
    +
    my_heap.c
    
    +
    +
    +
    +
    my_heap.cs
    
    +
    +
    +
    +
    my_heap.swift
    var maxHeap: [Int]
    +
    +/* 构造函数,建立空堆 */
    +init() {
    +    maxHeap = []
    +}
    +
    +/* 获取左子结点索引 */
    +func left(i: Int) -> Int {
    +    2 * i + 1
    +}
    +
    +/* 获取右子结点索引 */
    +func right(i: Int) -> Int {
    +    2 * i + 2
    +}
    +
    +/* 获取父结点索引 */
    +func parent(i: Int) -> Int {
    +    (i - 1) / 2 // 向下整除
    +}
    +
    +
    +
    +
    my_heap.zig
    
    +
    +
    +
    +
    +

    访问堆顶元素

    +

    堆顶元素是二叉树的根结点,即列表首元素。

    +
    +
    +
    +
    my_heap.java
    /* 访问堆顶元素 */
    +int peek() {
    +    return maxHeap.get(0);
    +}
    +
    +
    +
    +
    my_heap.cpp
    /* 访问堆顶元素 */
    +int peek() {
    +    return maxHeap[0];
    +}
    +
    +
    +
    +
    my_heap.py
    
    +
    +
    +
    +
    my_heap.go
    /* 访问堆顶元素 */
    +func (h *maxHeap) peek() any {
    +    return h.data[0]
    +}
    +
    +
    +
    +
    my_heap.js
    /* 访问堆顶元素 */
    +peek() {
    +    return this.#maxHeap[0];
    +}
    +
    +
    +
    +
    my_heap.ts
    /* 访问堆顶元素 */
    +public peek(): number {
    +    return this.maxHeap[0];
    +}
    +
    +
    +
    +
    my_heap.c
    
    +
    +
    +
    +
    my_heap.cs
    
    +
    +
    +
    +
    my_heap.swift
    /* 访问堆顶元素 */
    +func peek() -> Int {
    +    maxHeap[0]
    +}
    +
    +
    +
    +
    my_heap.zig
    
    +
    +
    +
    +
    +

    元素入堆

    +

    给定元素 val ,我们先将其添加到堆底。添加后,由于 val 可能大于堆中其它元素,此时堆的成立条件可能已经被破坏,因此需要修复从插入结点到根结点这条路径上的各个结点,该操作被称为「堆化 Heapify」。

    +

    考虑从入堆结点开始,从底至顶执行堆化。具体地,比较插入结点与其父结点的值,若插入结点更大则将它们交换;并循环以上操作,从底至顶地修复堆中的各个结点;直至越过根结点时结束,或当遇到无需交换的结点时提前结束。

    +
    +
    +
    +

    heap_push_step1

    +
    +
    +

    heap_push_step2

    +
    +
    +

    heap_push_step3

    +
    +
    +

    heap_push_step4

    +
    +
    +

    heap_push_step5

    +
    +
    +

    heap_push_step6

    +
    +
    +
    +

    设结点总数为 \(n\) ,则树的高度为 \(O(\log n)\) ,易得堆化操作的循环轮数最多为 \(O(\log n)\)因而元素入堆操作的时间复杂度为 \(O(\log n)\)

    +
    +
    +
    +
    my_heap.java
    /* 元素入堆 */
    +void push(int val) {
    +    // 添加结点
    +    maxHeap.add(val);
    +    // 从底至顶堆化
    +    siftUp(size() - 1);
    +}
    +
    +/* 从结点 i 开始,从底至顶堆化 */
    +void siftUp(int i) {
    +    while (true) {
    +        // 获取结点 i 的父结点
    +        int p = parent(i);
    +        // 当“越过根结点”或“结点无需修复”时,结束堆化
    +        if (p < 0 || maxHeap.get(i) <= maxHeap.get(p))
    +            break;
    +        // 交换两结点
    +        swap(i, p);
    +        // 循环向上堆化
    +        i = p;
    +    }
    +}
    +
    +
    +
    +
    my_heap.cpp
    /* 元素入堆 */
    +void push(int val) {
    +    // 添加结点
    +    maxHeap.push_back(val);
    +    // 从底至顶堆化
    +    siftUp(size() - 1);
    +}
    +
    +/* 从结点 i 开始,从底至顶堆化 */
    +void siftUp(int i) {
    +    while (true) {
    +        // 获取结点 i 的父结点
    +        int p =  parent(i);
    +        // 当“越过根结点”或“结点无需修复”时,结束堆化
    +        if (p < 0 || maxHeap[i] <= maxHeap[p])
    +            break;
    +        // 交换两结点
    +        swap(maxHeap[i], maxHeap[p]);
    +        // 循环向上堆化
    +        i = p;
    +    }
    +}
    +
    +
    +
    +
    my_heap.py
    
    +
    +
    +
    +
    my_heap.go
    /* 元素入堆 */
    +func (h *maxHeap) push(val any) {
    +    // 添加结点
    +    h.data = append(h.data, val)
    +    // 从底至顶堆化
    +    h.siftUp(len(h.data) - 1)
    +}
    +
    +/* 从结点 i 开始,从底至顶堆化 */
    +func (h *maxHeap) siftUp(i int) {
    +    for true {
    +        // 获取结点 i 的父结点
    +        p := h.parent(i)
    +        // 当“越过根结点”或“结点无需修复”时,结束堆化
    +        if p < 0 || h.data[i].(int) <= h.data[p].(int) {
    +            break
    +        }
    +        // 交换两结点
    +        h.swap(i, p)
    +        // 循环向上堆化
    +        i = p
    +    }
    +}
    +
    +
    +
    +
    my_heap.js
    /* 元素入堆 */
    +push(val) {
    +    // 添加结点
    +    this.#maxHeap.push(val);
    +    // 从底至顶堆化
    +    this.#siftUp(this.size() - 1);
    +}
    +
    +/* 从结点 i 开始,从底至顶堆化 */
    +#siftUp(i) {
    +    while (true) {
    +        // 获取结点 i 的父结点
    +        const p = this.#parent(i);
    +        // 当“越过根结点”或“结点无需修复”时,结束堆化
    +        if (p < 0 || this.#maxHeap[i] <= this.#maxHeap[p]) break;
    +        // 交换两结点
    +        this.#swap(i, p);
    +        // 循环向上堆化
    +        i = p;
    +    }
    +}
    +
    +
    +
    +
    my_heap.ts
    /* 元素入堆 */
    +public push(val: number): void {
    +    // 添加结点
    +    this.maxHeap.push(val);
    +    // 从底至顶堆化
    +    this.siftUp(this.size() - 1);
    +}
    +
    +/* 从结点 i 开始,从底至顶堆化 */
    +private siftUp(i: number): void {
    +    while (true) {
    +        // 获取结点 i 的父结点
    +        const p = this.parent(i);
    +        // 当“越过根结点”或“结点无需修复”时,结束堆化
    +        if (p < 0 || this.maxHeap[i] <= this.maxHeap[p]) break;
    +        // 交换两结点
    +        this.swap(i, p);
    +        // 循环向上堆化
    +        i = p;
    +    }
    +}
    +
    +
    +
    +
    my_heap.c
    
    +
    +
    +
    +
    my_heap.cs
    
    +
    +
    +
    +
    my_heap.swift
    /* 元素入堆 */
    +func push(val: Int) {
    +    // 添加结点
    +    maxHeap.append(val)
    +    // 从底至顶堆化
    +    siftUp(i: size() - 1)
    +}
    +
    +/* 从结点 i 开始,从底至顶堆化 */
    +func siftUp(i: Int) {
    +    var i = i
    +    while true {
    +        // 获取结点 i 的父结点
    +        let p = parent(i: i)
    +        // 当“越过根结点”或“结点无需修复”时,结束堆化
    +        if p < 0 || maxHeap[i] <= maxHeap[p] {
    +            break
    +        }
    +        // 交换两结点
    +        swap(i: i, j: p)
    +        // 循环向上堆化
    +        i = p
    +    }
    +}
    +
    +
    +
    +
    my_heap.zig
    
    +
    +
    +
    +
    +

    堆顶元素出堆

    +

    堆顶元素是二叉树根结点,即列表首元素,如果我们直接将首元素从列表中删除,则二叉树中所有结点都会随之发生移位(索引发生变化),这样后续使用堆化修复就很麻烦了。为了尽量减少元素索引变动,采取以下操作步骤:

    +
      +
    1. 交换堆顶元素与堆底元素(即交换根结点与最右叶结点);
    2. +
    3. 交换完成后,将堆底从列表中删除(注意,因为已经交换,实际上删除的是原来的堆顶元素);
    4. +
    5. 从根结点开始,从顶至底执行堆化
    6. +
    +

    顾名思义,从顶至底堆化的操作方向与从底至顶堆化相反,我们比较根结点的值与其两个子结点的值,将最大的子结点与根结点执行交换,并循环以上操作,直到越过叶结点时结束,或当遇到无需交换的结点时提前结束。

    +
    +
    +
    +

    heap_poll_step1

    +
    +
    +

    heap_poll_step2

    +
    +
    +

    heap_poll_step3

    +
    +
    +

    heap_poll_step4

    +
    +
    +

    heap_poll_step5

    +
    +
    +

    heap_poll_step6

    +
    +
    +

    heap_poll_step7

    +
    +
    +

    heap_poll_step8

    +
    +
    +

    heap_poll_step9

    +
    +
    +

    heap_poll_step10

    +
    +
    +
    +

    与元素入堆操作类似,堆顶元素出堆操作的时间复杂度为 \(O(\log n)\)

    +
    +
    +
    +
    my_heap.java
    /* 元素出堆 */
    +int poll() {
    +    // 判空处理
    +    if (isEmpty())
    +        throw new EmptyStackException();
    +    // 交换根结点与最右叶结点(即交换首元素与尾元素)
    +    swap(0, size() - 1);
    +    // 删除结点
    +    int val = maxHeap.remove(size() - 1);
    +    // 从顶至底堆化
    +    siftDown(0);
    +    // 返回堆顶元素
    +    return val;
    +}
    +
    +/* 从结点 i 开始,从顶至底堆化 */
    +void siftDown(int i) {
    +    while (true) {
    +        // 判断结点 i, l, r 中值最大的结点,记为 ma
    +        int l = left(i), r = right(i), ma = i;
    +        if (l < size() && maxHeap.get(l) > maxHeap.get(ma))
    +            ma = l;
    +        if (r < size() && maxHeap.get(r) > maxHeap.get(ma))
    +            ma = r;
    +        // 若结点 i 最大或索引 l, r 越界,则无需继续堆化,跳出
    +        if (ma == i) break;
    +        // 交换两结点
    +        swap(i, ma);
    +        // 循环向下堆化
    +        i = ma;
    +    }
    +}
    +
    +
    +
    +
    my_heap.cpp
    /* 元素出堆 */
    +void poll() {
    +    // 判空处理
    +    if (empty()) {
    +        cout << "Error:堆为空" << endl;
    +        return;
    +    }
    +    // 交换根结点与最右叶结点(即交换首元素与尾元素)
    +    swap(maxHeap[0], maxHeap[size() - 1]);
    +    // 删除结点
    +    maxHeap.pop_back();
    +    // 从顶至底堆化
    +    siftDown(0);
    +}
    +
    +/* 从结点 i 开始,从顶至底堆化 */
    +void siftDown(int i) {
    +    while (true) {
    +        // 判断结点 i, l, r 中值最大的结点,记为 ma
    +        int l = left(i), r = right(i), ma = i;
    +        // 若结点 i 最大或索引 l, r 越界,则无需继续堆化,跳出
    +        if (l < size() && maxHeap[l] > maxHeap[ma]) 
    +            ma = l;
    +        if (r < size() && maxHeap[r] > maxHeap[ma])
    +            ma = r;
    +        // 若结点 i 最大或索引 l, r 越界,则无需继续堆化,跳出
    +        if (ma == i) 
    +            break;
    +        swap(maxHeap[i], maxHeap[ma]);
    +        // 循环向下堆化
    +        i = ma;
    +    }
    +}
    +
    +
    +
    +
    my_heap.py
    
    +
    +
    +
    +
    my_heap.go
    /* 元素出堆 */
    +func (h *maxHeap) poll() any {
    +    // 判空处理
    +    if h.isEmpty() {
    +        fmt.Println("error")
    +        return nil
    +    }
    +    // 交换根结点与最右叶结点(即交换首元素与尾元素)
    +    h.swap(0, h.size()-1)
    +    // 删除结点
    +    val := h.data[len(h.data)-1]
    +    h.data = h.data[:len(h.data)-1]
    +    // 从顶至底堆化
    +    h.siftDown(0)
    +
    +    // 返回堆顶元素
    +    return val
    +}
    +
    +/* 从结点 i 开始,从顶至底堆化 */
    +func (h *maxHeap) siftDown(i int) {
    +    for true {
    +        // 判断结点 i, l, r 中值最大的结点,记为 max
    +        l, r, max := h.left(i), h.right(i), i
    +        if l < h.size() && h.data[l].(int) > h.data[max].(int) {
    +            max = l
    +        }
    +        if r < h.size() && h.data[r].(int) > h.data[max].(int) {
    +            max = r
    +        }
    +        // 若结点 i 最大或索引 l, r 越界,则无需继续堆化,跳出
    +        if max == i {
    +            break
    +        }
    +        // 交换两结点
    +        h.swap(i, max)
    +        // 循环向下堆化
    +        i = max
    +    }
    +}
    +
    +
    +
    +
    my_heap.js
    /* 元素出堆 */
    +poll() {
    +    // 判空处理
    +    if (this.isEmpty()) throw new Error("堆为空");
    +    // 交换根结点与最右叶结点(即交换首元素与尾元素)
    +    this.#swap(0, this.size() - 1);
    +    // 删除结点
    +    const val = this.#maxHeap.pop();
    +    // 从顶至底堆化
    +    this.#siftDown(0);
    +    // 返回堆顶元素
    +    return val;
    +}
    +
    +/* 从结点 i 开始,从顶至底堆化 */
    +#siftDown(i) {
    +    while (true) {
    +        // 判断结点 i, l, r 中值最大的结点,记为 ma
    +        const l = this.#left(i),
    +            r = this.#right(i);
    +        let ma = i;
    +        if (l < this.size() && this.#maxHeap[l] > this.#maxHeap[ma]) ma = l;
    +        if (r < this.size() && this.#maxHeap[r] > this.#maxHeap[ma]) ma = r;
    +        // 若结点 i 最大或索引 l, r 越界,则无需继续堆化,跳出
    +        if (ma == i) break;
    +        // 交换两结点
    +        this.#swap(i, ma);
    +        // 循环向下堆化
    +        i = ma;
    +    }
    +}
    +
    +
    +
    +
    my_heap.ts
    /* 元素出堆 */
    +public poll(): number {
    +    // 判空处理
    +    if (this.isEmpty()) throw new RangeError("Heap is empty.");
    +    // 交换根结点与最右叶结点(即交换首元素与尾元素)
    +    this.swap(0, this.size() - 1);
    +    // 删除结点
    +    const val = this.maxHeap.pop();
    +    // 从顶至底堆化
    +    this.siftDown(0);
    +    // 返回堆顶元素
    +    return val;
    +}
    +
    +/* 从结点 i 开始,从顶至底堆化 */
    +private siftDown(i: number): void {
    +    while (true) {
    +        // 判断结点 i, l, r 中值最大的结点,记为 ma
    +        const l = this.left(i), r = this.right(i);
    +        let ma = i;
    +        if (l < this.size() && this.maxHeap[l] > this.maxHeap[ma]) ma = l;
    +        if (r < this.size() && this.maxHeap[r] > this.maxHeap[ma]) ma = r;
    +        // 若结点 i 最大或索引 l, r 越界,则无需继续堆化,跳出
    +        if (ma == i) break;
    +        // 交换两结点
    +        this.swap(i, ma);
    +        // 循环向下堆化
    +        i = ma;
    +    }
    +}
    +
    +
    +
    +
    my_heap.c
    
    +
    +
    +
    +
    my_heap.cs
    
    +
    +
    +
    +
    my_heap.swift
    /* 元素出堆 */
    +func poll() -> Int {
    +    // 判空处理
    +    if isEmpty() {
    +        fatalError("堆为空")
    +    }
    +    // 交换根结点与最右叶结点(即交换首元素与尾元素)
    +    swap(i: 0, j: size() - 1)
    +    // 删除结点
    +    let val = maxHeap.remove(at: size() - 1)
    +    // 从顶至底堆化
    +    siftDown(i: 0)
    +    // 返回堆顶元素
    +    return val
    +}
    +
    +/* 从结点 i 开始,从顶至底堆化 */
    +func siftDown(i: Int) {
    +    var i = i
    +    while true {
    +        // 判断结点 i, l, r 中值最大的结点,记为 ma
    +        let l = left(i: i)
    +        let r = right(i: i)
    +        var ma = i
    +        if l < size(), maxHeap[l] > maxHeap[ma] {
    +            ma = l
    +        }
    +        if r < size(), maxHeap[r] > maxHeap[ma] {
    +            ma = r
    +        }
    +        // 若结点 i 最大或索引 l, r 越界,则无需继续堆化,跳出
    +        if ma == i {
    +            break
    +        }
    +        // 交换两结点
    +        swap(i: i, j: ma)
    +        // 循环向下堆化
    +        i = ma
    +    }
    +}
    +
    +
    +
    +
    my_heap.zig
    
    +
    +
    +
    +
    +

    输入数据并建堆 *

    +

    如果我们想要直接输入一个列表并将其建堆,那么该怎么做呢?最直接地,考虑使用「元素入堆」方法,将列表元素依次入堆。元素入堆的时间复杂度为 \(O(n)\) ,而平均长度为 \(\frac{n}{2}\) ,因此该方法的总体时间复杂度为 \(O(n \log n)\)

    +

    然而,存在一种更加优雅的建堆方法。设结点数量为 \(n\) ,我们先将列表所有元素原封不动添加进堆,然后迭代地对各个结点执行「从顶至底堆化」。当然,无需对叶结点执行堆化,因为其没有子结点。

    +
    +
    +
    +
    my_heap.java
    /* 构造函数,根据输入列表建堆 */
    +MaxHeap(List<Integer> nums) {
    +    // 将列表元素原封不动添加进堆
    +    maxHeap = new ArrayList<>(nums);
    +    // 堆化除叶结点以外的其他所有结点
    +    for (int i = parent(size() - 1); i >= 0; i--) {
    +        siftDown(i);
    +    }
    +}
    +
    +
    +
    +
    my_heap.cpp
    /* 构造函数,根据输入列表建堆 */
    +MaxHeap(vector<int> nums) {
    +    // 将列表元素原封不动添加进堆
    +    maxHeap = nums;
    +    // 堆化除叶结点以外的其他所有结点
    +    for (int i = parent(size() - 1); i >= 0; i--) {
    +        siftDown(i);
    +    }
    +}
    +
    +
    +
    +
    my_heap.py
    
    +
    +
    +
    +
    my_heap.go
    /* 构造函数,根据切片建堆 */
    +func newMaxHeap(nums []any) *maxHeap {
    +    // 将列表元素原封不动添加进堆
    +    h := &maxHeap{data: nums}
    +    // 堆化除叶结点以外的其他所有结点
    +    for i := len(h.data) - 1; i >= 0; i-- {
    +        h.siftDown(i)
    +    }
    +    return h
    +}
    +
    +
    +
    +
    my_heap.js
    /* 构造函数,建立空堆或根据输入列表建堆 */
    +constructor(nums) {
    +    // 将列表元素原封不动添加进堆
    +    this.#maxHeap = nums === undefined ? [] : [...nums];
    +    // 堆化除叶结点以外的其他所有结点
    +    for (let i = this.#parent(this.size() - 1); i >= 0; i--) {
    +        this.#siftDown(i);
    +    }
    +}
    +
    +
    +
    +
    my_heap.ts
    /* 构造函数,建立空堆或根据输入列表建堆 */
    +constructor(nums?: number[]) {
    +    // 将列表元素原封不动添加进堆
    +    this.maxHeap = nums === undefined ? [] : [...nums];
    +    // 堆化除叶结点以外的其他所有结点
    +    for (let i = this.parent(this.size() - 1); i >= 0; i--) {
    +        this.siftDown(i);
    +    }
    +}
    +
    +
    +
    +
    my_heap.c
    
    +
    +
    +
    +
    my_heap.cs
    
    +
    +
    +
    +
    my_heap.swift
    /* 构造函数,根据输入列表建堆 */
    +init(nums: [Int]) {
    +    // 将列表元素原封不动添加进堆
    +    maxHeap = nums
    +    // 堆化除叶结点以外的其他所有结点
    +    for i in stride(from: parent(i: size() - 1), through: 0, by: -1) {
    +        siftDown(i: i)
    +    }
    +}
    +
    +
    +
    +
    my_heap.zig
    
    +
    +
    +
    +
    +

    那么,第二种建堆方法的时间复杂度时多少呢?我们来做一下简单推算。

    +
      +
    • 完全二叉树中,设结点总数为 \(n\) ,则叶结点数量为 \((n + 1) / 2\) ,其中 \(/\) 为向下整除。因此在排除叶结点后,需要堆化结点数量为 \((n - 1)/2\) ,即为 \(O(n)\)
    • +
    • 从顶至底堆化中,每个结点最多堆化至叶结点,因此最大迭代次数为二叉树高度 \(O(\log n)\)
    • +
    +

    将上述两者相乘,可得时间复杂度为 \(O(n \log n)\) 。然而,该估算结果仍不够准确,因为我们没有考虑到 二叉树底层结点远多于顶层结点 的性质。

    +

    下面我们来尝试展开计算。为了减小计算难度,我们假设树是一个「完美二叉树」,该假设不会影响计算结果的正确性。设二叉树(即堆)结点数量为 \(n\) ,树高度为 \(h\) 。上文提到,结点堆化最大迭代次数等于该结点到叶结点的距离,而这正是“结点高度”。因此,我们将各层的“结点数量 \(\times\) 结点高度”求和,即可得到所有结点的堆化的迭代次数总和。

    +
    \[ +T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{(h-1)}\times1 +\]
    +

    heapify_count

    +

    化简上式需要借助中学的数列知识,先对 \(T(h)\) 乘以 \(2\) ,易得

    +
    \[ +\begin{aligned} +T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{h-1}\times1 \newline +2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \cdots + 2^{h}\times1 \newline +\end{aligned} +\]
    +

    使用错位相减法,令下式 \(2 T(h)\) 减去上式 \(T(h)\) ,可得

    +
    \[ +2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \cdots + 2^{h-1} + 2^h +\]
    +

    观察上式,\(T(h)\) 是一个等比数列,可直接使用求和公式,得到时间复杂度为

    +
    \[ +\begin{aligned} +T(h) & = 2 \frac{1 - 2^h}{1 - 2} - h \newline +& = 2^{h+1} - h \newline +& = O(2^h) +\end{aligned} +\]
    +

    进一步地,高度为 \(h\) 的完美二叉树的结点数量为 \(n = 2^{h+1} - 1\) ,易得复杂度为 \(O(2^h) = O(n)\)。以上推算表明,输入列表并建堆的时间复杂度为 \(O(n)\) ,非常高效

    +

    8.1.4. 堆常见应用

    +
      +
    • 优先队列。堆常作为实现优先队列的首选数据结构,入队和出队操作时间复杂度为 \(O(\log n)\) ,建队操作为 \(O(n)\) ,皆非常高效。
    • +
    • 堆排序。给定一组数据,我们使用其建堆,并依次全部弹出,则可以得到有序的序列。当然,堆排序一般无需弹出元素,仅需每轮将堆顶元素交换至数组尾部并减小堆的长度即可。
    • +
    • 获取最大的 \(k\) 个元素。这既是一道经典算法题目,也是一种常见应用,例如选取热度前 10 的新闻作为微博热搜,选取前 10 销量的商品等。
    • +
    + + + + + +

    评论

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b/chapter_introduction/algorithms_are_everywhere/index.html new file mode 100644 index 000000000..4fab59b5d --- /dev/null +++ b/chapter_introduction/algorithms_are_everywhere/index.html @@ -0,0 +1,1714 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.1. 算法无处不在 - Hello 算法 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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    1.1. 算法无处不在

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    听到“算法”这个词,我们一般会联想到数学。但实际上,大多数算法并不包含复杂的数学,而更像是在考察基本逻辑,而这些逻辑在我们日常生活中处处可见。

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    在正式介绍算法之前,我想告诉你一件有趣的事:其实,你在过去已经学会了很多算法,并且已经习惯将它们应用到日常生活中。接下来,我将介绍两个具体例子来佐证。

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    例一:拼积木。一套积木,除了有许多部件之外,还会附送详细的拼装说明书。我们按照说明书上一步步操作,即可拼出复杂的积木模型。

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    如果从数据结构与算法的角度看,大大小小的「积木」就是数据结构,而「拼装说明书」上的一系列步骤就是算法。

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    例二:查字典。在字典中,每个汉字都有一个对应的拼音,而字典是按照拼音的英文字母表顺序排列的。假设需要在字典中查询任意一个拼音首字母为 \(r\) 的字,一般我们会这样做:

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    1. 打开字典大致一半页数的位置,查看此页的首字母是什么(假设为 \(m\) );
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    3. 由于在英文字母表中 \(r\)\(m\) 的后面,因此应排除字典前半部分,查找范围仅剩后半部分;
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    5. 循环执行步骤 1-2 ,直到找到拼音首字母为 \(r\) 的页码时终止。
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    look_up_dictionary_step_1

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    look_up_dictionary_step_2

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    look_up_dictionary_step_3

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    look_up_dictionary_step_4

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    look_up_dictionary_step_5

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    查字典这个小学生的标配技能,实际上就是大名鼎鼎的「二分查找」。从数据结构角度,我们可以将字典看作是一个已排序的「数组」;而从算法角度,我们可将上述查字典的一系列指令看作是「二分查找」算法。

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    小到烹饪一道菜、大到星际航行,几乎所有问题的解决都离不开算法。计算机的出现,使我们可以通过编程将数据结构存储在内存中,也可以编写代码来调用 CPU, GPU 执行算法,从而将生活中的问题搬运到计算机中,更加高效地解决各式各样的复杂问题。

    +
    +

    Tip

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    读到这里,如果你感到对数据结构、算法、数组、二分查找等此类概念一知半解,那么就太好了!因为这正是本书存在的价值,接下来,本书将会一步步地引导你进入数据结构与算法的知识殿堂。

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    评论

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    1.2. 算法是什么

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    1.2.1. 算法定义

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    「算法 Algorithm」是在有限时间内解决特定问题的一组指令或操作步骤。算法具有以下特性:

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    • 问题是明确的,需要拥有明确的输入和输出定义。
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    • 解具有确定性,即给定相同输入时,输出一定相同。
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    • 具有可行性,可在有限步骤、有限时间、有限内存空间下完成。
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    • 独立于编程语言,即可用多种语言实现。
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    1.2.2. 数据结构定义

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    「数据结构 Data Structure」是在计算机中组织与存储数据的方式。为了提高数据存储和操作性能,数据结构的设计原则有:

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    • 空间占用尽可能小,节省计算机内存。
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    • 数据操作尽量快,包括数据访问、添加、删除、更新等。
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    • 提供简洁的数据表示和逻辑信息,以便算法高效运行。
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    数据结构的设计是一个充满权衡的过程,这意味着如果获得某方面的优势,则往往需要在另一方面做出妥协。例如,链表相对于数组,数据添加删除操作更加方便,但牺牲了数据的访问速度;图相对于链表,提供了更多的逻辑信息,但需要占用更多的内存空间。

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    1.2.3. 数据结构与算法的关系

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    「数据结构」与「算法」是高度相关、紧密嵌合的,体现在:

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    • 数据结构是算法的底座。数据结构为算法提供结构化存储的数据,以及操作数据的对应方法。
    • +
    • 算法是发挥数据结构优势的舞台。数据结构仅存储数据信息,结合算法才可解决特定问题。
    • +
    • 算法有对应最优的数据结构。给定算法,一般可基于不同的数据结构实现,而最终执行效率往往相差很大。
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    relationship_between_data_structure_and_algorithm

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    Fig. 数据结构与算法的关系

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    如果将「LEGO 乐高」类比到「数据结构与算法」,那么可以得到下表所示的对应关系。

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    + + + + + + + + + + + + + + + + + + + + + + + + + +
    数据结构与算法LEGO 乐高
    输入数据未拼装的积木
    数据结构积木组织形式,包括形状、大小、连接方式等
    算法把积木拼成目标形态的一系列操作步骤
    输出数据积木模型
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    约定俗成的简称

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    在实际讨论中,我们通常会将「数据结构与算法」直接简称为「算法」。例如,我们熟称的 LeetCode 算法题目,实际上同时考察了数据结构和算法两部分知识。

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    评论

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