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+ +!!! 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。 + +=== "Java" + + ```java title="array.java" + /* 初始化数组 */ + int[] arr = new int[5]; // { 0, 0, 0, 0, 0 } + int[] nums = { 1, 3, 2, 5, 4 }; + ``` + +=== "C++" + + ```cpp title="array.cpp" + + ``` + +=== "Python" + + ```python title="array.py" + + ``` + +## 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](array.assets/array_memory_location_calculation.png) + +

Fig. 数组元素的内存地址计算

+ +```java title="" +// 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。 + +=== "Java" + + ```java title="array.java" + /* 随机返回一个数组元素 */ + int randomAccess(int[] nums) { + int randomIndex = ThreadLocalRandom.current(). + nextInt(0, nums.length); + int randomNum = nums[randomIndex]; + return randomNum; + } + ``` + +=== "C++" + + ```cpp title="array.cpp" + + ``` + +=== "Python" + + ```python title="array.py" + + ``` + +## 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。 + +=== "Java" + + ```java title="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; + } + ``` + +=== "C++" + + ```cpp title="array.cpp" + + ``` + +=== "Python" + + ```python title="array.py" + + ``` + +**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。 +- **丢失元素或:** 由于数组的长度不可变,因此在插入元素后,数组原来的末尾元素会丢失。 +- **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](array.assets/array_insert_remove_element.png) + +

Fig. 在数组中插入与删除元素

+ +=== "Java" + + ```java title="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]; + } + } + ``` + +=== "C++" + + ```cpp title="array.cpp" + + ``` + +=== "Python" + + ```python title="array.py" + + ``` + +## 数组常用操作 + +**数组遍历。** 以下介绍两种常用的遍历方法。 + +=== "Java" + + ```java title="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++; + } + } + ``` + +=== "C++" + + ```cpp title="array.cpp" + + ``` + +=== "Python" + + ```python title="array.py" + + ``` + +**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。 + +=== "Java" + + ```java title="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; + } + ``` + +=== "C++" + + ```cpp title="array.cpp" + + ``` + +=== "Python" + + ```python title="array.py" + + ``` + +## 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。 diff --git a/docs/chapter_array_and_linkedlist/linked_list.assets/linkedlist_common_types.png b/docs/chapter_array_and_linkedlist/linked_list.assets/linkedlist_common_types.png new file mode 100644 index 0000000000000000000000000000000000000000..3ced93c657ec77cd6f46487f56e7d0a94e4ba047 GIT binary patch literal 64512 zcmeGEi9b~D|38iokyMrvLP$ag*+YW}*|HAVmE9Q1nz5Iq?0eZ|H}-wsEBkH+GsaT3 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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。 + +「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。 + +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」)。 + +![linkedlist_definition](linked_list.assets/linkedlist_definition.png) + +

Fig. 链表定义与存储方式

+ +=== "Java" + + ```java title="" + /* 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 + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + """ 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) + ``` + +**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。 + +=== "Java" + + ```java title="" + /* 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; + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +## 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](linked_list.assets/linkedlist_insert_remove_node.png) + +

Fig. 在链表中插入与删除结点

+ +=== "Java" + + ```java title="" + /* 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; + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +## 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。 + +=== "Java" + + ```java title="" + /* 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++) { + head = head.next; + if (head == null) + return null; + } + return head; + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +**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。 + +## 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。 + +=== "Java" + + ```java title="" + /* 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; + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +## 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。 + +=== "Java" + + ```java title="" + /* 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 + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +![linkedlist_common_types](linked_list.assets/linkedlist_common_types.png) + +

Fig. 常见链表类型

diff --git a/docs/chapter_array_and_linkedlist/list.md b/docs/chapter_array_and_linkedlist/list.md new file mode 100644 index 000000000..4f584dba2 --- /dev/null +++ b/docs/chapter_array_and_linkedlist/list.md @@ -0,0 +1,273 @@ +--- +comments: true +--- + +# 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。 + +## 列表常用操作 + +**初始化列表。** 我们通常使用 `Integer[]` 包装类和 `Arrays.asList()` 作为中转,来初始化一个带有初始值的列表。 + +=== "Java" + + ```java title="list.java" + /* 初始化列表 */ + // 注意数组的元素类型是 int[] 的包装类 Integer[] + Integer[] numbers = new Integer[] { 1, 3, 2, 5, 4 }; + List list = new ArrayList<>(Arrays.asList(numbers)); + ``` + +=== "C++" + + ```cpp title="list.cpp" + + ``` + +=== "Python" + + ```python title="list.py" + + ``` + +**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。 + +=== "Java" + + ```java title="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 + ``` + +=== "C++" + + ```cpp title="list.cpp" + + ``` + +=== "Python" + + ```python title="list.py" + + ``` + +**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 $O(N)$ 。 + +=== "Java" + + ```java title="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 + ``` + +=== "C++" + + ```cpp title="list.cpp" + + ``` + +=== "Python" + + ```python title="list.py" + + ``` + +**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。 + +=== "Java" + + ```java title="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++; + } + ``` + +=== "C++" + + ```cpp title="list.cpp" + + ``` + +=== "Python" + + ```python title="list.py" + + ``` + +**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。 + +=== "Java" + + ```java title="list.java" + /* 拼接两个列表 */ + List 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 + ``` + +=== "C++" + + ```cpp title="list.cpp" + + ``` + +=== "Python" + + ```python title="list.py" + + ``` + +**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。 + +=== "Java" + + ```java title="list.java" + /* 排序列表 */ + Collections.sort(list); // 排序后,列表元素从小到大排列 + ``` + +=== "C++" + + ```cpp title="list.cpp" + + ``` + +=== "Python" + + ```python title="list.py" + + ``` + +## 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。 + +=== "Java" + + ```java title="my_list.java" + /* 列表类简易实现 */ + class MyList { + int[] nums; // 数组(存储列表元素) + int initialCapacity = 10; // 列表初始容量 + int size = 0; // 列表长度(即当前元素数量) + int extendRatio = 2; // 每次列表扩容的倍数 + + /* 构造函数 */ + public MyList() { + nums = new int[initialCapacity]; + } + + /* 获取列表容量 */ + public int size() { + return size; + } + + /* 获取列表长度(即当前元素数量) */ + public int capacity() { + return nums.length; + } + + /* 访问元素 */ + public int get(int index) { + // 索引如果越界则抛出异常,下同 + if (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 >= 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 == nums.length) + extendCapacity(); + nums[size] = num; + // 更新元素数量 + size++; + } + + /* 中间插入元素 */ + public void add(int index, int num) { + if (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 == nums.length) + 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 + size++; + } + + /* 删除元素 */ + public void remove(int index) { + if (index >= size) + throw new IndexOutOfBoundsException("索引越界"); + // 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 + size--; + } + + /* 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, nums.length * extendRatio); + } + } + ``` + +=== "C++" + + ```cpp title="my_list.cpp" + + ``` + +=== "Python" + + ```python title="my_list.py" + + ``` diff --git a/docs/chapter_array_and_linkedlist/summary.md b/docs/chapter_array_and_linkedlist/summary.md new file mode 100644 index 000000000..60c5b45a6 --- /dev/null +++ b/docs/chapter_array_and_linkedlist/summary.md @@ -0,0 +1,41 @@ +--- +comments: true +--- + +# 小结 + +- 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。 + +## 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 | $O(N)$ | $O(1)$ | + +
diff --git a/docs/chapter_computational_complexity/performance_evaluation.md b/docs/chapter_computational_complexity/performance_evaluation.md new file mode 100644 index 000000000..e4fadff69 --- /dev/null +++ b/docs/chapter_computational_complexity/performance_evaluation.md @@ -0,0 +1,43 @@ +--- +comments: true +--- + +# 算法效率评估 + +## 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. **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。 + +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 “运行地快、内存占用少” ,而如何去评价算法效率则是非常重要的问题。 + +## 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「渐进复杂度分析 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。 + +## 复杂度分析的重要性 + +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**。 diff --git a/docs/chapter_computational_complexity/space_complexity.assets/space_complexity_common_types.png b/docs/chapter_computational_complexity/space_complexity.assets/space_complexity_common_types.png new file mode 100644 index 0000000000000000000000000000000000000000..59773c1bdcf9f7265f6c711986152c3447f07641 GIT binary patch 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zeJn13(Fgmb%=Wh_w?5rXKTgda<)Idl`t#u-`Y9Sun1|1QGQGgF#%N^0XV1b z=lq`c#`D`tyFbBAb#`{{%<{D2?Z(;PH-(KUoQ+B5i<+lyJSsZQb$Z42K!>Pl~ za~wJ~ZWk;g?575;N9V_3i=*%6Ho)Rn=(xgAm~++rpd^x7`W>C=uNldXMiV#ETmmio z21=IBYg+EyxwAN&yNK21qJgjvTAChaZTRK5=D2q{POqMkC5V2X&-=rAz$1t;bBsH8 zSV%ea9L20&zPA@$Ps0-4+uNgu`?a*zv4th6^XGc} Fig. 算法使用的相关空间

+ +=== "Java" + + ```java title="" + /* 类 */ + class Node { + int val; + Node next; + Node(int x) { val = x; } + } + + /* 函数(或称方法) */ + 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 + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +## 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 $O(n)$ ; + +=== "Java" + + ```java title="" + 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) + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +**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 $n$ 次 `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。 + +=== "Java" + + ```java title="" + int function() { + // do something + return 0; + } + /* 循环 */ + void loop(int n) { + for (int i = 0; i < n; i++) { + function(); + } + } + /* 递归 */ + void recur(int n) { + if (n == 1) return; + return recur(n - 1); + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +## 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](space_complexity.assets/space_complexity_common_types.png) + +

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时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 $O(1)$ 。 + +=== "Java" + + ```java title="space_complexity_types.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(); + } + } + ``` + +=== "C++" + + ```cpp title="space_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="space_complexity_types.py" + + ``` + +### 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。 + +=== "Java" + + ```java title="space_complexity_types.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 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 map = new HashMap<>(); + for (int i = 0; i < n; i++) { + map.put(i, String.valueOf(i)); + } + } + ``` + +=== "C++" + + ```cpp title="space_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="space_complexity_types.py" + + ``` + +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。 + +=== "Java" + + ```java title="space_complexity_types.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); + } + ``` + +=== "C++" + + ```cpp title="space_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="space_complexity_types.py" + + ``` + +![space_complexity_recursive_linear](space_complexity.assets/space_complexity_recursive_linear.png) + +

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。 + +=== "Java" + + ```java title="space_complexity_types.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> numList = new ArrayList<>(); + for (int i = 0; i < n; i++) { + List tmp = new ArrayList<>(); + for (int j = 0; j < n; j++) { + tmp.add(0); + } + numList.add(tmp); + } + } + ``` + +=== "C++" + + ```cpp title="space_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="space_complexity_types.py" + + ``` + +在以下递归函数中,同时存在 $n$ 个未返回的 `algorihtm()` ,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。 + +=== "Java" + + ```java title="space_complexity_types.java" + /* 平方阶(递归实现) */ + int quadraticRecur(int n) { + if (n <= 0) return 0; + int[] nums = new int[n]; + System.out.println("递归 n = " + n + " 中的 nums 长度 = " + nums.length); + return quadraticRecur(n - 1); + } + ``` + +=== "C++" + + ```cpp title="space_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="space_complexity_types.py" + + ``` + +![space_complexity_recursive_quadratic](space_complexity.assets/space_complexity_recursive_quadratic.png) + +

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。 + +=== "Java" + + ```java title="space_complexity_types.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; + } + ``` + +=== "C++" + + ```cpp title="space_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="space_complexity_types.py" + + ``` + +![space_complexity_exponential](space_complexity.assets/space_complexity_exponential.png) + +

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 $O(\log_{10} n) = O(\log n)$ 。 diff --git a/docs/chapter_computational_complexity/space_time_tradeoff.md b/docs/chapter_computational_complexity/space_time_tradeoff.md new file mode 100644 index 000000000..3753f050c --- /dev/null +++ b/docs/chapter_computational_complexity/space_time_tradeoff.md @@ -0,0 +1,79 @@ +# 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,内存空间不会成为算法瓶颈,因此以空间换时间更加常用。 + +## 示例题目 * + +以 LeetCode 全站第一题 [两数之和](https://leetcode.cn/problems/two-sum/) 为例,「暴力枚举」和「辅助哈希表」分别为 **空间最优** 和 **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。 + +=== "Java" + + ```java title="" title="leetcode_two_sum.java" + public int[] twoSum(int[] nums, int target) { + int size = nums.length; + // 外层 * 内层循环,时间复杂度为 O(n) + 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]; + } + ``` + +=== "C++" + + ```cpp title="leetcode_two_sum.cpp" + + ``` + +=== "Python" + + ```python title="leetcode_two_sum.py" + + ``` + +### 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。 + +=== "Java" + + ```java title="" title="leetcode_two_sum.java" + public int[] twoSum(int[] nums, int target) { + int size = nums.length; + // 辅助哈希表,空间复杂度 O(n) + Map 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]; + } + ``` + +=== "C++" + + ```cpp title="leetcode_two_sum.cpp" + + ``` + +=== "Python" + + ```python title="leetcode_two_sum.py" + + ``` diff --git a/docs/chapter_computational_complexity/summary.md b/docs/chapter_computational_complexity/summary.md new file mode 100644 index 000000000..6550e776c --- /dev/null +++ b/docs/chapter_computational_complexity/summary.md @@ -0,0 +1,28 @@ +--- +comments: true +--- + +# 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,即函数渐进上界,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,再判断渐进上界。 +- 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,空间复杂度不计入输入空间。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 $O(1)$ , $O(\log n)$ , $O(n)$ , $O(n^2)$ , $O(2^n)$ 。 diff --git a/docs/chapter_computational_complexity/time_complexity.assets/asymptotic_upper_bound.png b/docs/chapter_computational_complexity/time_complexity.assets/asymptotic_upper_bound.png new file mode 100644 index 0000000000000000000000000000000000000000..fabba53647a6596e2a619a3201004dfa81bd5fdc GIT binary patch literal 65804 zcmeFZcRbbYA3uJaD5D}oWfmbqWhUb^?2L$$T}k%d>u_3%L?K1?N@n&x3Mm-}*_-Tf z$T%Dh-|Ky(?)(1y{`!6Y{njH+b-l0m>w3-SYh0dbsw*C%K0ytG!44@aUDJZWD8NTJ zmJ$xWAvjdXz&A2yEyXLa>}J+!@Rvtsx0LUzsloWcXG$0a89i(tv;_Qyk)0qt3_YSk z#`5R078&2~HDJ#GOBgw6Jq)G~zC*umgFnzSyWjW4ll|Q>9=^Al0v5k-?=u_=R>SafNq%bwaYg>$mRy9-reko!TXhch_Wj~+tHOBiD)w7 z3_gFtfhvf_k*Y`e0>Anz-f#CrpNZt`d~KAL5+Sra-{yVxc1rr2^jq#%G2eJ@aM@AX 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zdRfGV;l76d+g=dIjoMK`ih7Q;PKrPQ1<_JZ*u8xD5@4%%wNO^6k^$C?vjx 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。 + +**时间复杂度分析将统计「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。因此,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” 。 + +**时间复杂度也存在一定的局限性。** 比如,虽然算法 `A` 和 `C` 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 `B` 比 `C` 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。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、最常用方法。 + +## 函数渐进上界 + +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 +$$ +T(n) = 3 + 2n +$$ +=== "Java" + + ```java title="" + 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 + } + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +$T(n)$ 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「渐进上界 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 $T(n)$ 」的渐进上界。下面我们先来看看函数渐进上界的数学定义。 + +!!! abstract "函数渐进上界" + + 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 $T(n)$ 的一个渐进上界,记为 + $$ + T(n) = O(f(n)) + $$ + +![asymptotic_upper_bound](time_complexity.assets/asymptotic_upper_bound.png) + +

    Fig. 函数的渐进上界

    + +本质上看,计算渐进上界就是在找一个函数 $f(n)$ ,**使得在 $n$ 趋向于无穷大时,$T(n)$ 和 $f(n)$ 处于相同的增长级别(仅相差一个常数项 $c$ 的倍数)**。 + +!!! tip + + 渐进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。 + +## 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,如何来确定渐进上界 $f(n)$ 呢?总体分为两步,首先「统计操作数量」,然后「判断渐进上界」。 + +### 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. **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。 +3. **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.` 和 `2.` 技巧。 + +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 $O(n^2)$ 。 + +=== "Java" + + ```java title="" + 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); + } + } + } + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +### 2. 判断渐进上界 + +**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 $O(f(n))$ | +| ---------------------- | -------------------- | +| $100000$ | $O(1)$ | +| $3n + 2$ | $O(n)$ | +| $2n^2 + 3n + 2$ | $O(n^2)$ | +| $n^3 + 10000n^2$ | $O(n^3)$ | +| $2^n + 10000n^{10000}$ | $O(2^n)$ | + +
    + +## 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](time_complexity.assets/time_complexity_common_types.png) + +

    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 $O(1)$ 。 + +=== "Java" + + ```java title="" title="time_complexity_types.java" + /* 常数阶 */ + int constant(int n) { + int count = 0; + int size = 100000; + for (int i = 0; i < size; i++) + count++; + return count; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +### 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。 + +=== "Java" + + ```java title="" title="time_complexity_types.java" + /* 线性阶 */ + int linear(int n) { + int count = 0; + for (int i = 0; i < n; i++) + count++; + return count; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +「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。 + +=== "Java" + + ```java title="" title="time_complexity_types.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; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +### 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 $O(n^2)$ 。 + +=== "Java" + + ```java title="" title="time_complexity_types.java" + /* 平方阶 */ + int quadratic(int n) { + int count = 0; + for (int i = 0; i < n; i++) { + for (int j = 0; j < n; j++) { + count++; + } + } + return count; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +![time_complexity_constant_linear_quadratic](time_complexity.assets/time_complexity_constant_linear_quadratic.png) + +

    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 $O(n^2)$ 。 + +$$ +O((n - 1) \frac{n}{2}) = O(n^2) +$$ + +=== "Java" + + ```java title="" title="time_complexity_types.java" + /* 平方阶(冒泡排序) */ + void bubbleSort(int[] nums) { + int n = nums.length; + for (int i = 0; i < n - 1; i++) { + for (int j = 0; j < n - 1 - i; j++) { + if (nums[j] > nums[j + 1]) { + // 交换 nums[j] 和 nums[j + 1] + int tmp = nums[j]; + nums[j] = nums[j + 1]; + nums[j + 1] = tmp; + } + } + } + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +### 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。 + +指数阶增长地非常快,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。 + +=== "Java" + + ```java title="" title="time_complexity_types.java" + /* 指数阶(遍历实现) */ + 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; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +![time_complexity_exponential](time_complexity.assets/time_complexity_exponential.png) + +

    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。 + +=== "Java" + + ```java title="" title="time_complexity_types.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; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +### 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,时间增长的很慢,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。 + +=== "Java" + + ```java title="" title="time_complexity_types.java" + /* 对数阶(循环实现) */ + int logarithmic(float n) { + int count = 0; + while (n > 1) { + n = n / 2; + count++; + } + return count; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +![time_complexity_logarithmic](time_complexity.assets/time_complexity_logarithmic.png) + +

    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。 + +=== "Java" + + ```java title="" title="time_complexity_types.java" + /* 对数阶(递归实现) */ + int logRecur(float n) { + if (n <= 1) return 0; + return logRecur(n / 2) + 1; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +### 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 $O(\log n)$ 和 $O(n)$ 。 + +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。 + +=== "Java" + + ```java title="" title="time_complexity_types.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; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +![time_complexity_logarithmic_linear](time_complexity.assets/time_complexity_logarithmic_linear.png) + +

    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。 + +=== "Java" + + ```java title="" title="time_complexity_types.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; + } + ``` + +=== "C++" + + ```cpp title="time_complexity_types.cpp" + + ``` + +=== "Python" + + ```python title="time_complexity_types.py" + + ``` + +![time_complexity_factorial](time_complexity.assets/time_complexity_factorial.png) + +

    Fig. 阶乘阶的时间复杂度

    + +## 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 $1$ 至 $n$ 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: + +- 当 `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 $O(n)$** ; +- 当 `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 $\Omega(1)$** ; + +「函数渐进上界」使用大 $O$ 记号表示,代表「最差时间复杂度」。与之对应,「函数渐进下界」用 $\Omega$ 记号(Omega Notation)来表示,代表「最佳时间复杂度」。 + +=== "Java" + + ```java title="" title="worst_best_time_complexity.java" + public class worst_best_time_complexity { + /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */ + static 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 */ + static int findOne(int[] nums) { + for (int i = 0; i < nums.length; i++) { + if (nums[i] == 1) + return i; + } + return -1; + } + + /* Driver Code */ + public static void main(String[] args) { + for (int i = 0; i < 10; i++) { + int n = 100; + int[] nums = randomNumbers(n); + int index = findOne(nums); + System.out.println("打乱后的数组为 " + Arrays.toString(nums)); + System.out.println("数字 1 的索引为 " + index); + } + } + } + ``` + +=== "C++" + + ```cpp title="worst_best_time_complexity.cpp" + + ``` + +=== "Python" + + ```python title="worst_best_time_complexity.py" + + ``` + +!!! 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 $\Theta(\frac{n}{2}) = \Theta(n)$ 。 + 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树、图、堆、哈希表; + +![classification_logic_structure](classification_of_data_strcuture.assets/classification_logic_structure.png) + +

    Fig. 线性与非线性数据结构

    + +## 物理结构:连续与离散 + +!!! note + + 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。 + +**「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** 和 **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。 + +![classification_phisical_structure](classification_of_data_strcuture.assets/classification_phisical_structure.png) + +

    Fig. 连续空间存储与离散空间存储

    + +**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。 + +- **基于数组可实现:** 栈、队列、堆、哈希表、矩阵、张量(维度 $\geq 3$ 的数组)等; +- **基于链表可实现:** 栈、队列、堆、哈希表、树、图等; + +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。 + +!!! tip + + 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。 diff --git a/docs/chapter_data_structure/data_and_memory.assets/computer_memory_location.png b/docs/chapter_data_structure/data_and_memory.assets/computer_memory_location.png new file mode 100644 index 0000000000000000000000000000000000000000..2111f4903a6c5bab1ea777c0e98de70da7af9953 GIT binary patch literal 216550 zcmeEugQxVyW%4laY+;J)PU-u=Gs zz1ja@nP;9pLwDEduBxu8uKt}9`b9zV{ad`ZFfcIhrKQA_VPIarT*8gNfqN-F6U)H9 z6tK?9k|HprqXc^|KYo~MN}I{c!_dB5zkzuTiw^VZPm`Aq3@qNiI{)b+1B?HEua#k` z|4*A2n!%PZ@c(KL1Ecj){`oU_`Tptizspydu>VCd6Yk%wU&Ca+`u8>5_)BXTe=%X7 zmjcN_3g8R_gVO!y3mZ>|g8wpZ7-=zKRS($1mfKjuIoF*e@79_txVOmCUy(nD=7=#= 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xDAu3>Yu42Y7&K*0?VACwHP!z}H`#+wL_6sk*!izMPhg;b@-J0hlt>x>^*`!|3f}+# literal 0 HcmV?d00001 diff --git a/docs/chapter_data_structure/data_and_memory.md b/docs/chapter_data_structure/data_and_memory.md new file mode 100644 index 000000000..1282e899b --- /dev/null +++ b/docs/chapter_data_structure/data_and_memory.md @@ -0,0 +1,81 @@ +--- +comments: true +--- + +# 数据与内存 + +## 基本数据类型 + +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。 + +**「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。** + +- 「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。 +- 「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 。 + +!!! note "字节与比特" + + 1 字节 (byte) = 8 比特 (bit) , 1 比特即最基本的 1 个二进制位 + +

    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 | byte | 1 byte | $-2^7$ ~ $2^7 - 1$ ( $-128$ ~ $127$ ) | $0$ | +| | short | 2 bytes | $-2^{15}$ ~ $2^{15} - 1$ | $0$ | +| | **int** | 4 bytes | $-2^{31}$ ~ $2^{31} - 1$ | $0$ | +| | long | 8 bytes | $-2^{63}$ ~ $2^{63} - 1$ | $0$ | +| 浮点数 | **float** | 4 bytes | $-3.4 \times 10^{38}$ ~ $3.4 \times 10^{38}$ | $0.0$ f | +| | double | 8 bytes | $-1.7 \times 10^{308}$ ~ $1.7 \times 10^{308}$ | $0.0$ | +| 字符 | **char** | 2 bytes | $0$ ~ $2^{16} - 1$ | $0$ | +| 布尔 | **boolean(bool)** | 1 byte / 1 bit | $\text{true}$ 或 $\text{false}$ | $\text{false}$ | + +
    + +!!! tip + + 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** + +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**。 + +=== "Java" + + ```java + /* 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]; + ``` + +=== "C++" + + ```cpp title="" + + ``` + +=== "Python" + + ```python title="" + + ``` + +## 计算机内存 + +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,速度更快,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](data_and_memory.assets/computer_memory_location.png) + +

    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。 diff --git a/docs/chapter_data_structure/summary.md b/docs/chapter_data_structure/summary.md new file mode 100644 index 000000000..e51e4dc5b --- /dev/null +++ b/docs/chapter_data_structure/summary.md @@ -0,0 +1,11 @@ +--- +comments: true +--- + +# 小结 + +- 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。 diff --git a/docs/chapter_dsa_introduction/index.assets/look_up_dictionary.png b/docs/chapter_dsa_introduction/index.assets/look_up_dictionary.png new file mode 100644 index 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      Hello,算法!

      + +## 读者对象 + +- 数据结构与算法零基础的同学,但需具备 Java, C++, Python 任一语言的编程基础 +- 具有一定算法刷题基础,想要系统地学习或复习数据结构与算法的同学 +- 本书以代码实践为导向,尤其适合在职或即将成为工程师(程序员)的同学 + +!!! quote "" + +

      追风赶月莫停留,平芜尽处是春山

      +

      一起加油!

      + +## 内容结构 + +本书主要内容分为复杂度分析、数据结构、算法三个部分。 + +![mindmap](index.assets/mindmap.png) + +

      Fig. 知识点思维导图

      + +### 复杂度分析 + +首先介绍数据结构与算法的评价维度,以及算法效率评估方法,引出了计算复杂度概念。 + +接下来,介绍了函数渐进上界的含义,并分别介绍了时间复杂度和空间复杂度的定义、推算方法、常见类型等,以及最差、最佳、平均时间复杂度的区别。 + +### 数据结构 + +首先介绍了物理结构和逻辑结构两种数据结构的分类方法,随后介绍了各个数据结构,包括数组、链表、栈、队列、树、堆、图、散列表等,内容包括: + +- 基本定义:数据结构的设计来源、存在意义; +- 主要特点:在各项数据操作中的优势、劣势; +- 常用操作:例如访问、更新、插入、删除、遍历、搜索等; +- 常见类型:在算法题或工程实际中,经常碰到的数据结构类型; +- 典型应用:此数据结构经常搭配哪些算法使用; +- 实现方法:对于重要的数据结构,将给出完整的实现示例; + +### 算法 + +介绍了常见的算法类型,包括查找算法、排序算法、搜索与回溯算法、动态规划、分治算法等,主要关心以下内容: + +- 基本定义:算法的设计思想; +- 主要特点:使用前置要求、优势和劣势; +- 算法效率:最差和平均时间复杂度、空间复杂度; +- 应用场景:结合例题讲述算法应用; + +## 配套代码 + +完整可运行的代码在 GitHub 仓库 [![github-stars](https://img.shields.io/github/stars/krahets/hello-algo?style=social)](https://github.com/krahets/hello-algo) ,你可以使用 Git 将仓库 Clone 至本地,也可以下载仓库压缩包使用。 + +编程环境部署和代码使用方法请见下章 “编程环境安装” 和 “算法学习建议” 。 + +## 风格约定 + +- 文章中的重要名词会用「」符号标注,例如「数组 Array」。这类名词应该被牢记,英文翻译也建议记住,以便后续查阅资料时使用。 +- 重点内容、总起句、总结句会被 **加粗** ,此类文字值得更多关注。 +- 专有名词和有特指含义的词句会使用 “ ” 标注,以避免歧义。 +- 标题后标注 * 符号的是选读章节,如果你的时间有限,可以先跳过这些章节。 + +## 本书特点 + +**以实践为主。** 我们知道,学习英语期间光啃书本是远远不够的,需要多听、多说、多写,在实践中培养语感、积累经验。编程语言也是一门语言,因此学习方法也应是类似的,需要多看优秀代码、多敲键盘、多思考代码逻辑。 + +本书的理论部分占少量篇幅,主要分为两类:一是基础且必要的概念知识,以培养读者对于算法的感性认识;二是重要的分类、对比或总结,这是为了帮助你站在更高视角俯瞰各个知识点,形成连点成面的效果。 + +实践部分主要由示例和代码组成。代码配有简要注释,复杂示例会尽可能地使用视觉化的形式呈现。我强烈建议读者对照着代码自己敲一遍,如果时间有限,也至少逐行读、复制并运行一遍,配合着讲解将代码吃透。 + +!!! quote + + “Talk is cheap. Show me the code.” ― Linus Torvalds + + “少吹牛,看代码” + +**视觉化学习。** 信息时代以来,视觉化的脚步从未停止。媒体形式经历了文字短信、图文 Email 、动图、短(长)视频、交互式 Web 、3D 游戏等演变过程,信息的视觉化程度越来越高、愈加符合人类感官、信息传播效率大大提升。科技界也在向视觉化迈进,iPhone 就是一个典型例子,其相对于传统手机是高度视觉化的,包含精心设计的字体、主题配色、交互动画等。 + +近两年,短视频成为最受欢迎的信息媒介,可以在短时间内将高密度的信息 “灌” 给我们,有着极其舒适的观看体验。阅读则不然,读者与书本之间天然存在一种 “疏离感”,我们看书会累、会走神、会停下来想其他事、会划下喜欢的句子、会思考某一片段的含义,这种疏离感给了读者与书本之间对话的可能,拓宽了想象空间。 + +本书作为一本入门教材,希望可以保有书本的 “慢节奏” ,但也会避免与读者产生过多 “疏离感” ,而是努力将知识完整清晰地推送到你聪明的小脑袋瓜中。我将采用视觉化的方式(例如配图、动画),尽我可能清晰易懂地讲解复杂概念和抽象示例。 + +!!! quote + + “A picture is worth a thousand words.” + + “一图胜千言” + +**内容精简化。** 大多数的经典教科书,会把每个主题都讲的很透彻。虽然透彻性正是其获得读者青睐的原因,但对于想要快速入门的初学者来说,这些教材的实用性不足。本书会避免引入非必要的概念、名词、定义等,也避免展开不必要的理论分析,毕竟这不是一本真正意义上的教材,主要任务是尽快地带领读者入门。 + +引入一些生活案例或趣味内容,非常适合作为知识点的引子或者解释的补充,但当融入过多额外元素时,内容会稍显冗长,也许反而使读者容易迷失、抓不住重点,这也是本书需要避免的。 + +敲代码如同写字,“美” 是统一的追求。本书力求美观的代码,保证规范的变量命名、统一的空格与换行、对齐的缩进、整齐的注释等。 + +## 致谢 + +(TODO) diff --git a/docs/chapter_introduction/installation.assets/image-20221117201957848.png 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z$fs-5UwHe!^F5!C`e4bOQ@ccRCY^p07QA&cS`Kbtn#{fBrO`xVVSCode 官网](https://code.visualstudio.com/) 下载对应系统安装包,本地安装即可。 + +## Python 环境 + +1. 前往 [Miniconda3](https://docs.conda.io/en/latest/miniconda.html) ,选择对应系统安装包,下载并安装。 +2. 在 VSCode 中的插件市场中搜索 Python ,安装 Python Extension Pack 。 + +## Java 环境 + +1. 前往 [OpenJDK](https://jdk.java.net/18/) ,选择对应系统安装包,下载并安装。 +2. 在 VSCode 中的插件市场中搜索 Java ,安装 Java Extension Pack 。 + +## C++ 环境 + +1. 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+![learning_route](suggestions.assets/learning_route.png) + +## 图文搭配学 + +对比大部分教材,本书更倾向于以结构化的方式介绍知识。视频和图片相比于文字的信息密度和结构化程度更高,更容易让人理解。在本书中,重点和难点知识会主要以动画、图解的形式呈现,而文字的作用则是作为动画和图的解释与补充。 + +在阅读本书的过程中,若发现某段内容提供了动画或图解,**建议以图为主线,将文字内容(一般在图的上方)对齐到图中内容,综合来理解**。 + +![algorithm_animation](suggestions.assets/algorithm_animation.gif) + +## 运行源代码 + +编程是门实践技术,知识内容最终都会落地到一行一行的代码上。本书提供配套 Java, C++, Python 代码仓。**若学习时间紧张,请至少将所有代码通读并运行一遍;在时间允许下,强力建议你对照着代码自己敲一遍,逐渐锻炼肌肉记忆,相比于读代码,写的过程也会带来新的收获。** + +代码栏上若标有 `*.java` , `*.cpp` , `*.py` ,则可在仓库 codes 文件夹中找到对应的 **代码源文件** 。 + +![md_code](suggestions.assets/md_code.png) + +这些源文件中包含测试样例,可以直接运行,帮助你省去不必要的调试时间,可以将精力集中在学习内容上。 + +![running_code](suggestions.assets/running_code.gif) + +!!! tip + + 若你的 PC 仍没有编程环境,可以参照下节「编程环境安装」的内容进行安装。 + +## 参与讨论区 + +阅读本书时,请不要 “惯着” 那些弄不明白的知识点。如果有任何疑惑,可以在评论区留下你的问题,小伙伴们和我都会给予解答(我一般 2 ~ 3 天看一次评论区)。 + +同时,也希望你可以多花时间逛逛评论区。一方面,可以看看大家遇到了什么问题,反过来查漏补缺,这往往可以引起更加深度的思考。另一方面,也希望你可以慷慨地解答小伙伴们的问题、分享自己的见解,大家一起加油与进步! + +![comment](suggestions.assets/comment.gif) diff --git a/docs/chapter_reference/index.md b/docs/chapter_reference/index.md new file mode 100644 index 000000000..fcc5da56f --- /dev/null +++ b/docs/chapter_reference/index.md @@ -0,0 +1,17 @@ +# 参考文献 + +[1] Thomas H. Cormen, et al. Introduction to Algorithms (3rd Edition). + +[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1st Edition). + +[3] 程杰. 大话数据结构. + +[4] 王争. 数据结构与算法之美. + +[5] 严蔚敏. 数据结构( C 语言版). + +[6] 邓俊辉. 数据结构( C++ 语言版,第三版). + +[7] 马克·艾伦·维斯著,陈越译. 数据结构与算法分析:Java语言描述(第三版). + +[8] Gayle Laakmann McDowell. 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+- **二分查找仅适用于数组** ,而在链表中使用效率很低,因为其在循环中需要跳跃式(非连续地)访问元素。 + +## 算法实现 + +给定一个长度为 $n$ 的排序数组 `nums` ,元素从小到大排列。数组的索引取值范围为 + +$$ +0, 1, 2, \cdots, n-1 +$$ + +使用「区间」来表示这个取值范围的方法主要有两种: + +1. **双闭区间 $[0, n-1]$** ,即两个边界都包含自身;此方法下,区间 $[0, 0]$ 仍包含一个元素; +2. **左闭右开 $[0, n)$** ,即左边界包含自身、右边界不包含自身;此方法下,区间 $[0, 0)$ 为空; + +### “双闭区间” 实现 + +首先,我们先采用 “双闭区间” 的表示,在数组 `nums` 中查找目标元素 `target` 的对应索引。 + +=== "Step 1" + + ![binary_search_step1](binary_search.assets/binary_search_step1.png) + +=== "Step 2" + + ![binary_search_step2](binary_search.assets/binary_search_step2.png) + +=== "Step 3" + + ![binary_search_step3](binary_search.assets/binary_search_step3.png) + +=== "Step 4" + + ![binary_search_step4](binary_search.assets/binary_search_step4.png) + +=== "Step 5" + + ![binary_search_step5](binary_search.assets/binary_search_step5.png) + +=== "Step 6" + + ![binary_search_step6](binary_search.assets/binary_search_step6.png) + +=== "Step 7" + + ![binary_search_step7](binary_search.assets/binary_search_step7.png) + +二分查找 “双闭区间” 表示下的代码如下所示。 + +=== "Java" + + ```java title="binary_search.java" + /* 二分查找(双闭区间) */ + int binarySearch(int[] nums, int target) { + // 初始化双闭区间 [0, n-1] ,即 i, j 分别指向数组首元素、尾元素 + int i = 0, j = nums.length - 1; + // 循环,当搜索区间为空时跳出(当 i > j 时为空) + while (i <= j) { + int m = (i + j) / 2; // 计算中点索引 m + if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 + i = m + 1; + else if (nums[m] > target) // 此情况说明 target 在区间 [i, m-1] 中 + j = m - 1; + else // 找到目标元素,返回其索引 + return m; + } + // 未找到目标元素,返回 -1 + return -1; + } + ``` + +### “左闭右开” 实现 + +当然,我们也可以使用 “左闭右开” 的表示方法,写出相同功能的二分查找代码。 + +=== "Java" + + ```java title="binary_search.java" + /* 二分查找(左闭右开) */ + int binarySearch1(int[] nums, int target) { + // 初始化左闭右开 [0, n) ,即 i, j 分别指向数组首元素、尾元素+1 + int i = 0, j = nums.length; + // 循环,当搜索区间为空时跳出(当 i = j 时为空) + while (i < j) { + int m = (i + j) / 2; // 计算中点索引 m + if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 + i = m + 1; + else if (nums[m] > target) // 此情况说明 target 在区间 [i, m) 中 + j = m; + else // 找到目标元素,返回其索引 + return m; + } + // 未找到目标元素,返回 -1 + return -1; + } + ``` + +### 两种表示对比 + +对比下来,两种表示的代码写法有以下不同点: + +

      + +观察发现,在 “双闭区间” 表示中,由于对左右两边界的定义是相同的,因此缩小区间的 $i$ , $j$ 处理方法也是对称的,这样更不容易出错。综上所述,**建议你采用 “双闭区间” 的写法。** + +### 大数越界处理 + +当数组长度很大时,加法 $i + j$ 的结果有可能会超出 `int` 类型的取值范围。在此情况下,我们需要换一种计算中点的写法。 + +```java +// (i + j) 有可能超出 int 的取值范围 +int m = (i + j) / 2; +// 更换为此写法则不会越界 +int m = i + (j - i) / 2; +``` + +## 复杂度分析 + +**时间复杂度 $O(\log n)$ :** 其中 $n$ 为数组或链表长度;每轮排除一半的区间,因此循环轮数为 $\log_2 n$ ,使用 $O(\log n)$ 时间。 + +**空间复杂度 $O(1)$ :** 指针 `i` , `j` 使用常数大小空间。 + +## 优缺点 + +二分查找效率很高,体现在: + +- **二分查找时间复杂度低。** 对数阶在数据量很大时具有巨大优势,例如,当数据大小 $n = 2^{20}$ 时,线性查找需要 $2^{20} = 1048576$ 轮循环,而二分查找仅需要 $\log_2 2^{20} = 20$ 轮循环。 +- **二分查找不需要额外空间。** 相对于借助额外数据结构来实现查找的算法来说,其更加节约空间使用。 + +但并不意味着所有情况下都应使用二分查找,这是因为: + +- **二分查找仅适用于有序数据。** 如果输入数据是乱序的,为了使用二分查找而专门执行数据排序,那么是得不偿失的,因为排序算法的时间复杂度一般为 $O(n \log n)$ ,比线性查找和二分查找都更差。再例如,对于频繁插入元素的场景,为了保持数组的有序性,需要将元素插入到特定位置,时间复杂度为 $O(n)$ ,也是非常昂贵的。 + +- **二分查找仅适用于数组。** 由于在二分查找中,访问索引是 ”非连续“ 的,因此链表或者基于链表实现的数据结构都无法使用。 + +- **在小数据量下,线性查找的性能更好。** 在线性查找中,每轮只需要 1 次判断操作;而在二分查找中,需要 1 次加法、1 次除法、1 ~ 3 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b/docs/chapter_searching/hashing_search.md new file mode 100644 index 000000000..c8371fbc7 --- /dev/null +++ b/docs/chapter_searching/hashing_search.md @@ -0,0 +1,60 @@ +--- +comments: true +--- + +# 哈希查找 + +!!! question + + 在数据量很大时,「线性查找」太慢;而「二分查找」要求数据必须是有序的,并且只能在数组中应用。那么是否有方法可以同时避免上述缺点呢?答案是肯定的,此方法被称为「哈希查找」。 + +「哈希查找 Hash Searching」借助一个哈希表来存储需要的「键值对 Key Value Pair」,我们可以在 $O(1)$ 时间下实现 “键 $\rightarrow$ 值” 映射查找,体现着 “以空间换时间” 的算法思想。 + +## 算法实现 + +如果我们想要给定数组中的一个目标元素 `target` ,获取该元素的索引,那么可以借助一个哈希表实现查找。 + +![hash_search_index](hashing_search.assets/hash_search_index.png) + +=== "Java" + + ```java title="hashing_search.java" + /* 哈希查找(数组) */ + int hashingSearch(Map map, int target) { + // 哈希表的 key: 目标元素,value: 索引 + // 若哈希表中无此 key ,返回 -1 + return map.getOrDefault(target, -1); + } + ``` + +再比如,如果我们想要给定一个目标结点值 `target` ,获取对应的链表结点对象,那么也可以使用哈希查找实现。 + +![hash_search_listnode](hashing_search.assets/hash_search_listnode.png) + +=== "Java" + + ```java title="hashing_search.java" + /* 哈希查找(链表) */ + ListNode hashingSearch1(Map map, int target) { + // 哈希表的 key: 目标结点值,value: 结点对象 + // 若哈希表中无此 key ,返回 -1 + return map.getOrDefault(target, null); + } + ``` + +## 复杂度分析 + +**时间复杂度:** $O(1)$ ,哈希表的查找操作使用 $O(1)$ 时间。 + +**空间复杂度:** $O(n)$ ,其中 $n$ 为数组或链表长度。 + +## 优缺点 + +在哈希表中,**查找、插入、删除操作的平均时间复杂度都为 $O(1)$** ,这意味着无论是高频增删还是高频查找场景,哈希查找的性能表现都非常好。当然,一切的前提是保证哈希表未退化。 + +即使如此,哈希查找仍存在一些问题,在实际应用中,需要根据情况灵活选择方法。 + +- 辅助哈希表 **需要使用 $O(n)$ 的额外空间**,意味着需要预留更多的计算机内存; +- 建立和维护哈希表需要时间,因此哈希查找 **不适合高频增删、低频查找的使用场景**; +- 当哈希冲突严重时,哈希表会退化为链表,**时间复杂度劣化至 $O(n)$** ; +- **当数据量很小时,线性查找比哈希查找更快**。这是因为计算哈希映射函数可能比遍历一个小型数组更慢; diff --git a/docs/chapter_searching/linear_search.assets/linear_search.png b/docs/chapter_searching/linear_search.assets/linear_search.png new file mode 100644 index 0000000000000000000000000000000000000000..722904e3c30970accfc8c6f635724f759e46bb90 GIT binary patch literal 77042 zcmeFZ1y>zQw*`tz2<|SylHl$x!6A5Xhv4q+!6CRyfZ!6`-911E?(PnI!)x;CJ?H*_ 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a/docs/chapter_searching/linear_search.md b/docs/chapter_searching/linear_search.md new file mode 100644 index 000000000..b967de9d2 --- /dev/null +++ b/docs/chapter_searching/linear_search.md @@ -0,0 +1,60 @@ +--- +comments: true +--- + +# 线性查找 + +「线性查找 Linear Search」是一种最基础的查找方法,其从数据结构的一端开始,依次访问每个元素,直到另一端后停止。 + +## 算法实现 + +线性查找实质上就是遍历数据结构 + 判断条件。比如,我们想要在数组 `nums` 中查找目标元素 `target` 的对应索引,那么可以在数组中进行线性查找。 + +![linear_search](linear_search.assets/linear_search.png) + +=== "Java" + + ```java title="linear_search.java" + /* 线性查找(数组) */ + int linearSearch(int[] nums, int target) { + // 遍历数组 + for (int i = 0; i < nums.length; i++) { + // 找到目标元素,返回其索引 + if (nums[i] == target) + return i; + } + // 未找到目标元素,返回 -1 + return -1; + } + ``` + +再比如,我们想要在给定一个目标结点值 `target` ,返回此结点对象,也可以在链表中进行线性查找。 + +=== "Java" + + ```java title="linear_search.java" + /* 线性查找(链表) */ + ListNode linearSearch(ListNode head, int target) { + // 遍历链表 + while (head != null) { + // 找到目标结点,返回之 + if (head.val == target) + return head; + head = head.next; + } + // 未找到目标结点,返回 null + return null; + } + ``` + +## 复杂度分析 + +**时间复杂度 $O(n)$ :** 其中 $n$ 为数组或链表长度。 + +**空间复杂度 $O(1)$ :** 无需使用额外空间。 + +## 优缺点 + +**线性查找的通用性极佳。** 由于线性查找是依次访问元素的,即没有跳跃访问元素,因此数组或链表皆适用。 + +**线性查找的时间复杂度太高。** 在数据量 $n$ 很大时,查找效率很低。 diff --git a/docs/chapter_searching/summary.md b/docs/chapter_searching/summary.md new file mode 100644 index 000000000..97e378c1a --- /dev/null +++ b/docs/chapter_searching/summary.md @@ -0,0 +1,19 @@ +# 小结 + +- 线性查找是一种最基础的查找方法,通过遍历数据结构 + 判断条件实现查找。 +- 二分查找利用数据的有序性,通过循环不断缩小一半搜索区间来实现查找,其要求输入数据是有序的,并且仅适用于数组或基于数组实现的数据结构。 +- 哈希查找借助哈希表来实现常数阶时间复杂度的查找操作,体现以空间换时间的算法思想。 + +

      Table. 三种查找方法对比

      + +
      + +| | 线性查找 | 二分查找 | 哈希查找 | +| ------------------------------------- | ------------------------ | ----------------------------- | ------------------------ | +| 适用数据结构 | 数组、链表 | 数组 | 数组、链表 | +| 输入数据要求 | 无 | 有序 | 无 | +| 平均时间复杂度
      查找 / 插入 / 删除 | $O(n)$ / $O(1)$ / $O(n)$ | $O(\log n)$ / $O(n)$ / $O(n)$ | $O(1)$ / $O(1)$ / $O(1)$ | +| 最差时间复杂度
      查找 / 插入 / 删除 | $O(n)$ / $O(1)$ / $O(n)$ | $O(\log n)$ / $O(n)$ / $O(n)$ | $O(n)$ / $O(n)$ / $O(n)$ | +| 空间复杂度 | $O(1)$ | $O(1)$ | $O(n)$ | + +
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zj$P2JV|e5oy0M&!i#cOEmcYFtG^=!P!b$a^TVmSi)_*UJCpj{jMOF9m&L`oEtb}B0 zp%oHnMFjXnLN2;PN6j1{q^fl6oMjgxwUCs2bF?ttr$tK}*3UE}Isb7S{&3zb5IBfSs~Tl(-k=E_@esv}K%qlu_cBCaUuFu_}(yU)I+NhU-~^Xb2@ zD><(1GPekmbOB>*X7a4IkXWB#A%D=>^zDS>RL00+SRKfEalTMOHb}Tk-rc%qDJLDZ z8~&|F_t~#oaLBDT*DHm4kPW89XtFj{IdYlAeO6*<_7% zfywZ*1w}iPPf3?YZhl&1d;R>$|M*IGJhGwmi@WSEm3Er?cBb?EkDs6#0&1H@t1YJL zBrs{E!5m658jFe$eNkN;(A5+r)-2&awfe6(0e+0_)0v4a*O>!4Nyaw8)cEO2rhO5* zhb5qd zY2{#7VcoNV-%rL>&F{$6yK!T{ Oe@gP|mnvjUg8mN 右元素** 则将它俩交换,最终可将最大元素移动至数组最右端。 + +完成此次冒泡操作后,**数组最大元素已在正确位置,接下来只需排序剩余 $n - 1$ 个元素**。 + +=== "Step 1" + + ![bubble_operation_step1](bubble_sort.assets/bubble_operation_step1.png) + +=== "Step 2" + + ![bubble_operation_step2](bubble_sort.assets/bubble_operation_step2.png) + +=== "Step 3" + + ![bubble_operation_step3](bubble_sort.assets/bubble_operation_step3.png) + +=== "Step 4" + + ![bubble_operation_step4](bubble_sort.assets/bubble_operation_step4.png) + +=== "Step 5" + + ![bubble_operation_step5](bubble_sort.assets/bubble_operation_step5.png) + +=== "Step 6" + + ![bubble_operation_step6](bubble_sort.assets/bubble_operation_step6.png) + +=== "Step 7" + + ![bubble_operation_step7](bubble_sort.assets/bubble_operation_step7.png) + +## 算法流程 + +设数组长度为 $n$ ,完成第一轮「冒泡」后,数组最大元素已在正确位置,接下来只需排序剩余 $n - 1$ 个元素。 + +同理,对剩余 $n - 1$ 个元素执行「冒泡」,可将第二大元素交换至正确位置,因而待排序元素只剩 $n - 2$ 个。 + +以此类推…… **循环 $n - 1$ 轮「冒泡」,即可完成整个数组的排序**。 + +![bubble_sort](bubble_sort.assets/bubble_sort.png) + +=== "Java" + + ```java + /* 冒泡排序 */ + void bubbleSort(int[] nums) { + // 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; + } + } + } + } + ``` + +## 算法分析 + +**时间复杂度 $O(n^2)$ :** 各轮「冒泡」遍历的数组长度为 $n - 1$ , $n - 2$ , $\cdots$ , $2$ , $1$ 次,求和为 $\frac{(n - 1) n}{2}$ ,因此使用 $O(n^2)$ 时间。 + +**空间复杂度 $O(1)$ :** 指针 $i$ , $j$ 使用常数大小的额外空间。 + +**原地性:** 指针变量仅使用常数大小额外空间,因此是 **原地排序** 。 + +**稳定性:** 不交换相等元素,因此是 **稳定排序** 。 + +**自适应:** 引入 `flag` 优化后(见下文),可在输入数组已排序下达到最优时间复杂度 $O(N)$ ,因此是 **自适应排序** 。 + +## 效率优化 + +我们发现,若在某轮「冒泡」中未执行任何交换操作,则说明数组已经完成排序,可直接返回结果。考虑可以增加一个标志位 `flag` 来监听该情况,若出现则直接返回。 + +优化后,冒泡排序的最差和平均时间复杂度仍为 $O(n^2)$ ;而在输入数组 **已排序** 时,达到 **最佳时间复杂度** $O(n)$ 。 + +=== "Java" + + ```java + /* 冒泡排序(标志优化)*/ + void bubbleSortWithFlag(int[] nums) { + // 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--) { + boolean flag = false; // 初始化标志位 + // 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; + flag = true; // 记录交换元素 + } + } + if (!flag) break; // 此轮冒泡未交换任何元素,直接跳出 + } + } + ``` diff --git a/docs/chapter_sorting/insertion_sort.assets/insertion_operation.png b/docs/chapter_sorting/insertion_sort.assets/insertion_operation.png new file mode 100644 index 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z{=gqmnKR2(BU^^geeXlA1(L(9Z5sGyKafs%kWR8JRYuDV5HG;9#h=RJ5B9i^Qpx)7 zhF4%Bu6c>K)X9n=MjN~Ty+5}Nu+Stv<_NSrg>_6_h!+zKxdV=w1DUtf0vY@hKHwFf zWCZ@T@uku6Q1y#yCcF3G= 0 && nums[j] > base) { + nums[j + 1] = nums[j]; // 1. 将 nums[j] 向右移动一位 + j--; + } + nums[j + 1] = base; // 2. 将 base 赋值到正确位置 + } + } + ``` + +## 算法分析 + +**时间复杂度 $O(n^2)$ :** 各轮插入操作最多循环 $n - 1$ , $n-2$ , $\cdots$ , $2$ , $1$ 次,求和为 $\frac{(n - 1) n}{2}$ ,使用 $O(n^2)$ 时间。 + +**空间复杂度 $O(1)$ :** 指针 $i$ , $j$ 使用常数大小的额外空间。 + +**原地性:** 指针变量仅使用常数大小额外空间,因此是 **原地排序** 。 + +**稳定性:** 不交换相等元素,因此是 **稳定排序** 。 + +**自适应:** 当输入数组完全有序时,每次插入操作(即内循环)仅循环一次,此时时间复杂度为 $O(n)$ 。 + +## 插入排序 vs 冒泡排序 + +!!! question + + 虽然「插入排序」和「冒泡排序」的时间复杂度皆为 $O(n^2)$ ,但实际运行速度却有很大差别,这是为什么呢? + +回顾复杂度分析,两个方法的循环次数都是 $\frac{(n - 1) n}{2}$ 。但不同的是,「冒泡操作」是在做 **元素交换** ,需要借助一个临时变量实现,共 3 个单元操作;而「插入操作」是在做 **赋值** ,只需 1 个单元操作;因此,可以粗略估计出冒泡排序的计算开销约为插入排序的 3 倍。 + +插入排序运行速度快,并且具有原地、稳定、自适应的优点,因此很受欢迎。实际上,包括 Java 在内的许多编程语言的排序库函数的实现都用到了插入排序。库函数的大致思路: + +- 对于 **长数组**,采用基于分治的排序算法,例如「快速排序」,时间复杂度为 $O(n \log n)$ ; +- 对于 **短数组**,直接使用「插入排序」,时间复杂度为 $O(n^2)$ ; + +在数组较短时,复杂度中的常数项(即每轮中的单元操作数量)占主导作用,此时插入排序运行地更快。这个现象与「线性查找」和「二分查找」的情况类似。 + + + diff --git a/docs/chapter_sorting/merge_sort.md b/docs/chapter_sorting/merge_sort.md new file mode 100644 index 000000000..67f6f1e3a --- /dev/null +++ b/docs/chapter_sorting/merge_sort.md @@ -0,0 +1,6 @@ +--- +comments: true +--- + +# 归并排序 + diff --git a/docs/chapter_sorting/quick_sort.md b/docs/chapter_sorting/quick_sort.md new file mode 100644 index 000000000..3b1a107af --- /dev/null +++ b/docs/chapter_sorting/quick_sort.md @@ -0,0 +1,6 @@ +--- +comments: true +--- + +# 快速排序 + diff --git a/docs/chapter_stack_and_queue/deque.assets/deque_operations.png b/docs/chapter_stack_and_queue/deque.assets/deque_operations.png new file mode 100644 index 0000000000000000000000000000000000000000..94ede7b2bac408000e38fd67d3346ea0d7e93ae8 GIT binary patch literal 62181 zcmZr&1z1$w)}9$g1QkILQ9@8kS~?{}sS%Kdp#^EAyA@PKx*KKa7#e9*q(N%v?naOj 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b/docs/chapter_stack_and_queue/deque.md new file mode 100644 index 000000000..9b121f0fb --- /dev/null +++ b/docs/chapter_stack_and_queue/deque.md @@ -0,0 +1,78 @@ +# 双向队列 + +对于队列,我们只能在头部删除或在尾部添加元素,而「双向队列 Deque」更加灵活,在其头部和尾部都能执行元素添加或删除操作。 + +![deque_operations](deque.assets/deque_operations.png) + +

      Fig. 双向队列的操作

      + +## 双向队列常用操作 + +双向队列的常用操作见下表,方法名需根据编程语言设定来具体确定。 + +

      Table. 双向队列的常用操作

      + +
      + +| 方法 | 描述 | +| ------------ | ---------------- | +| offerFirst() | 将元素添加至队首 | +| offerLast() | 将元素添加至队尾 | +| pollFirst() | 删除队首元素 | +| pollLast() | 删除队尾元素 | +| peekFirst() | 访问队首元素 | +| peekLast() | 访问队尾元素 | +| size() | 获取队列的长度 | +| isEmpty() | 判断队列是否为空 | + +
      + +相同地,我们可以直接使用编程语言实现好的双向队列类。 + +=== "Java" + + ```java title="deque.java" + /* 初始化双向队列 */ + Deque deque = new LinkedList<>(); + + /* 元素入队 */ + deque.offerLast(2); + deque.offerLast(5); + deque.offerLast(4); + deque.offerFirst(3); + deque.offerFirst(1); + System.out.println("队列 deque = " + deque); + + /* 访问队首元素 */ + int peekFirst = deque.peekFirst(); + System.out.println("队首元素 peekFirst = " + peekFirst); + int peekLast = deque.peekLast(); + System.out.println("队尾元素 peekLast = " + peekLast); + + /* 元素出队 */ + int pollFirst = deque.pollFirst(); + System.out.println("队首出队元素 pollFirst = " + pollFirst + + ",队首出队后 deque = " + deque); + int pollLast = deque.pollLast(); + System.out.println("队尾出队元素 pollLast = " + pollLast + + ",队尾出队后 deque = " + deque); + + /* 获取队列的长度 */ + int size = deque.size(); + System.out.println("队列长度 size = " + size); + + /* 判断队列是否为空 */ + boolean isEmpty = deque.isEmpty(); + ``` + +=== "C++" + + ```cpp title="deque.cpp" + + ``` + +=== "Python" + + ```python title="deque.py" + + ``` diff --git a/docs/chapter_stack_and_queue/queue.assets/queue_operations.png b/docs/chapter_stack_and_queue/queue.assets/queue_operations.png new file mode 100644 index 0000000000000000000000000000000000000000..61e433c5c6027d833a3d5bf17a8d3bc6ab1cbae9 GIT binary patch literal 58934 zcmZ^L1yq#V_x8*%C}OA?lHxsVOfpOe1cEIqBdG#`po2?T4+adph*8Qx z!3)$$<(@dC`1AE8@JoQ1rmVSw0^}CB#(Fja1$y)E z_keo_TS4Hz-VcFjgV*yPL-7CnGk?9Jr9*$;F&*~LyU`)(X#ZTpdceCOev)^+!3)Ac zM#~8T!FhlF9~#SyLk`vrk(Ipr&>gy(f=DK68$AouGZY|+)G-twZlH(rD2s*j852Yg zlZM~p=3EvfBjgF?E-o&Es#Cdc=DsHby~g9!wt`!aA#f@P6hjOG!$bXb`94L=YM

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      + +## 队列常用操作 + +队列的常用操作见下表,方法命名需根据编程语言的设定来具体确定。 + +

      Table. 队列的常用操作

      + +
      + +| 方法 | 描述 | +| --------- | ---------------------------- | +| offer() | 元素入队,即将元素添加至队尾 | +| poll() | 队首元素出队 | +| front() | 访问队首元素 | +| size() | 获取队列的长度 | +| isEmpty() | 判断队列是否为空 | + +
      + +我们可以直接使用编程语言实现好的队列类。 + +=== "Java" + + ```java title="queue.java" + /* 初始化队列 */ + Queue queue = new LinkedList<>(); + + /* 元素入队 */ + queue.offer(1); + queue.offer(3); + queue.offer(2); + queue.offer(5); + queue.offer(4); + System.out.println("队列 queue = " + queue); + + /* 访问队首元素 */ + int peek = queue.peek(); + System.out.println("队首元素 peek = " + peek); + + /* 元素出队 */ + int poll = queue.poll(); + System.out.println("出队元素 poll = " + poll + ",出队后 queue = " + queue); + + /* 获取队列的长度 */ + int size = queue.size(); + System.out.println("队列长度 size = " + size); + + /* 判断队列是否为空 */ + boolean isEmpty = queue.isEmpty(); + ``` + +=== "C++" + + ```cpp title="queue.cpp" + + ``` + +=== "Python" + + ```python title="queue.py" + + ``` + +## 队列实现 + +队列需要一种可以在一端添加,并在另一端删除的数据结构,也可以使用链表或数组来实现。 + +### 基于链表的实现 + +我们将链表的「头结点」和「尾结点」分别看作是队首和队尾,并规定队尾只可添加结点,队首只可删除结点。 + +=== "Java" + + ```java title="linkedlist_queue.java" + /* 基于链表实现的队列 */ + class LinkedListQueue { + LinkedList list; + + public LinkedListQueue() { + // 初始化链表 + list = new LinkedList<>(); + } + /* 获取队列的长度 */ + public int size() { + return list.size(); + } + /* 判断队列是否为空 */ + public boolean isEmpty() { + return list.size() == 0; + } + /* 入队 */ + public void offer(int num) { + // 尾结点后添加 num + list.addLast(num); + } + /* 出队 */ + public int poll() { + // 删除头结点 + return list.removeFirst(); + } + /* 访问队首元素 */ + public int peek() { + return list.getFirst(); + } + } + ``` + +=== "C++" + + ```cpp title="linkedlist_queue.cpp" + + ``` + +=== "Python" + + ```python title="linkedlist_queue.py" + + ``` + +### 基于数组的实现 + +数组的删除首元素的时间复杂度为 $O(n)$ ,因此不适合直接用来实现队列。然而,我们可以借助两个指针 `front` , `rear` 来分别记录队首和队尾的索引位置,在入队 / 出队时分别将 `front` / `rear` 向后移动一位即可,这样每次仅需操作一个元素,时间复杂度降至 $O(1)$ 。 + +还有一个问题,在入队与出队的过程中,两个指针都在向后移动,而到达尾部后则无法继续移动了。为了解决此问题,我们可以采取一个取巧方案,即将数组看作是 “环形” 的。具体做法是规定指针越过数组尾部后,再次回到头部接续遍历,这样相当于使数组 “首尾相连” 了。 + +为了适应环形数组的设定,获取长度 `size()` 、入队 `offer()` 、出队 `poll()` 方法都需要做相应的取余操作处理,使得当尾指针绕回数组头部时,仍然可以正确处理操作。 + +基于数组实现的队列有一个缺点,即长度不可变。但这点我们可以通过动态数组来解决,有兴趣的同学可以自行实现。 + +=== "Java" + + ```java title="array_queue.java" + /* 基于环形数组实现的队列 */ + class ArrayQueue { + int[] nums; // 用于存储队列元素的数组 + int size = 0; // 队列长度(即元素个数) + int front = 0; // 头指针,指向队首 + int rear = 0; // 尾指针,指向队尾 + 1 + + public ArrayQueue(int capacity) { + // 初始化数组 + nums = new int[capacity]; + } + /* 获取队列的容量 */ + public int capacity() { + return nums.length; + } + /* 获取队列的长度 */ + public int size() { + int capacity = capacity(); + // 由于将数组看作为环形,可能 rear < front ,因此需要取余数 + return (capacity + rear - front) % capacity; + } + /* 判断队列是否为空 */ + public boolean isEmpty() { + return rear - front == 0; + } + /* 入队 */ + public void offer(int num) { + if (size() == capacity()) { + System.out.println("队列已满"); + return; + } + // 尾结点后添加 num + nums[rear] = num; + // 尾指针向后移动一位,越过尾部后返回到数组头部 + rear = (rear + 1) % capacity(); + } + /* 出队 */ + public int poll() { + // 删除头结点 + if (isEmpty()) + throw new EmptyStackException(); + int num = nums[front]; + // 队头指针向后移动,越过尾部后返回到数组头部 + front = (front + 1) % capacity(); + return num; + } + /* 访问队首元素 */ + public int peek() { + // 删除头结点 + if (isEmpty()) + throw new EmptyStackException(); + return nums[front]; + } + } + ``` + +=== "C++" + + ```cpp title="array_queue.cpp" + + ``` + +=== "Python" + + ```python title="array_queue.py" + + ``` + +## 队列典型应用 + +- **淘宝订单。** 购物者下单后,订单就被加入到队列之中,随后系统再根据顺序依次处理队列中的订单。在双十一时,在短时间内会产生海量的订单,如何处理「高并发」则是工程师们需要重点思考的问题。 + +- **各种待办事项。** 例如打印机的任务队列、餐厅的出餐队列等等。 diff --git a/docs/chapter_stack_and_queue/stack.assets/stack_operations.png b/docs/chapter_stack_and_queue/stack.assets/stack_operations.png new file mode 100644 index 0000000000000000000000000000000000000000..3c42ee229d4272246da959781e4eba3696681f2c GIT binary patch literal 57424 zcmeFZg&Vw1ncrAw|Ged|yR(zUxLCfF$x_C9UGC3( z*C<(R^tw8jb8GQ~^0_@{XlgJ~Ff~qqG8;&K;k5bs*|Nc5mz!rU$(=%?aUR@1#}O?y z2wN(8zTr=;8Yx4^k{FXs`=4_IJg^VruGOCt&y^0sWAkAx^M1O) zd%e`Dw@h2IjIcQ$O(~hE=eYw7h5XQV4UU^+<6)Pwd*=Ts-|jiho!q-N@%(kI{(sNc 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      + +## 栈常用操作 + +栈的常用操作见下表,方法名需根据编程语言设定来具体确定。 + +

      Table. 栈的常用操作

      + +
      + +| 方法 | 描述 | +| --------- | ---------------------- | +| push() | 元素入栈(添加至栈顶) | +| pop() | 栈顶元素出栈 | +| peek() | 访问栈顶元素 | +| size() | 获取栈的长度 | +| isEmpty() | 判断栈是否为空 | + +
      + +我们可以直接使用编程语言实现好的栈类。 + +=== "Java" + + ```java title="stack.java" + /* 初始化栈 */ + Stack stack = new Stack<>(); + + /* 元素入栈 */ + stack.push(1); + stack.push(3); + stack.push(2); + stack.push(5); + stack.push(4); + System.out.println("栈 stack = " + stack); + + /* 访问栈顶元素 */ + int peek = stack.peek(); + System.out.println("栈顶元素 peek = " + peek); + + /* 元素出栈 */ + int pop = stack.pop(); + System.out.println("出栈元素 pop = " + pop + ",出栈后 stack = " + stack); + + /* 获取栈的长度 */ + int size = stack.size(); + System.out.println("栈的长度 size = " + size); + + /* 判断是否为空 */ + boolean isEmpty = stack.isEmpty(); + ``` + +=== "C++" + + ```cpp title="stack.cpp" + + ``` + +=== "Python" + + ```python title="stack.py" + + ``` + +## 栈的实现 + +为了更加清晰地了解栈的运行机制,接下来我们来自己动手实现一个栈类。 + +栈规定元素是先入后出的,因此我们只能在栈顶添加或删除元素。然而,数组或链表都可以在任意位置添加删除元素,因此 **栈可被看作是一种受约束的数组或链表**。换言之,我们可以 “屏蔽” 数组或链表的部分无关操作,使之对外的表现逻辑符合栈的规定即可。 + +### 基于链表的实现 + +使用「链表」实现栈时,将链表的尾结点看作栈顶即可。 + +受益于链表的离散存储方式,栈的扩容更加灵活,删除元素的内存也会被系统自动回收;缺点是无法像数组一样高效地随机访问,并且由于链表结点需存储指针,导致单个元素占用空间更大。 + +=== "Java" + + ```java title="linkedlist_stack.java" + /* 基于链表实现的栈 */ + class LinkedListStack { + LinkedList list; + public LinkedListStack() { + // 初始化链表 + list = new LinkedList<>(); + } + /* 获取栈的长度 */ + public int size() { + return list.size(); + } + /* 判断栈是否为空 */ + public boolean isEmpty() { + return size() == 0; + } + /* 入栈 */ + public void push(int num) { + list.addLast(num); + } + /* 出栈 */ + public int pop() { + return list.removeLast(); + } + /* 访问栈顶元素 */ + public int peek() { + return list.getLast(); + } + } + ``` + +=== "C++" + + ```cpp title="linkedlist_stack.cpp" + + ``` + +=== "Python" + + ```python title="linkedlist_stack.py" + + ``` + +### 基于数组的实现 + +使用「数组」实现栈时,将数组的尾部当作栈顶。准确地说,我们需要使用「列表」,因为入栈的元素可能是源源不断的,因此使用动态数组可以方便扩容。 + +基于数组实现的栈,优点是支持随机访问,缺点是会造成一定的空间浪费,因为列表的容量始终 $\geq$ 元素数量。 + +=== "Java" + + ```java title="array_stack.java" + /* 基于数组实现的栈 */ + class ArrayStack { + List list; + public ArrayStack() { + // 初始化列表(动态数组) + list = new ArrayList<>(); + } + /* 获取栈的长度 */ + public int size() { + return list.size(); + } + /* 判断栈是否为空 */ + public boolean isEmpty() { + return size() == 0; + } + /* 入栈 */ + public void push(int num) { + list.add(num); + } + /* 出栈 */ + public int pop() { + return list.remove(size() - 1); + } + /* 访问栈顶元素 */ + public int peek() { + return list.get(size() - 1); + } + /* 访问索引 index 处元素 */ + public int get(int index) { + return list.get(index); + } + } + ``` + +=== "C++" + + ```cpp title="array_stack.cpp" + + ``` + +=== "Python" + + ```python title="array_stack.py" + + ``` + +!!! tip + + 实际编程中,我们一般直接将 `ArrayList` 或 `LinkedList` 当作「栈」来使用。我们仅需通过脑补来屏蔽无关操作,而不用专门去包装它。 + +## 栈典型应用 + +- **浏览器中的后退与前进、软件中的撤销与反撤销。** 每当我们打开新的网页,浏览器就讲上一个网页执行入栈,这样我们就可以通过「后退」操作来回到上一页面,后退操作实际上是在执行出栈。如果要同时支持后退和前进,那么则需要两个栈来配合实现。 + +- **程序内存管理。** 每当调用函数时,系统就会站栈顶添加一个栈帧,用来记录函数的上下文信息。在递归函数中,向下递推会不断执行入栈,向上回溯阶段时出栈。 diff --git a/docs/chapter_stack_and_queue/summary.md 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+二叉搜索树的查找操作和二分查找算法如出一辙,也是在每轮排除一半情况。循环次数最多为二叉树的高度,当二叉树平衡时,使用 $O(\log n)$ 时间。 + +=== "Java" + + ```java title="binary_search_tree.java" + /* 查找结点 */ + TreeNode search(int num) { + TreeNode cur = root; + // 循环查找,越过叶结点后跳出 + while (cur != null) { + // 目标结点在 root 的右子树中 + if (cur.val < num) cur = cur.right; + // 目标结点在 root 的左子树中 + else if (cur.val > num) cur = cur.left; + // 找到目标结点,跳出循环 + else break; + } + // 返回目标结点 + return cur; + } + ``` + +### 插入结点 + +给定一个待插入元素 `num` ,为了保持二叉搜索树 “左子树 < 根结点 < 右子树” 的性质,插入操作分为两步: + +1. **查找插入位置:** 与查找操作类似,我们从根结点出发,根据当前节点值和 `num` 的大小关系循环向下搜索,直到越过叶结点(遍历到 $\text{null}$ )时跳出循环; + +2. **在该位置插入结点:** 初始化结点 `num` ,将该结点放到 $\text{null}$ 的位置 ; + +二叉搜索树不允许存在重复节点,否则将会违背其定义。因此若待插入结点在树中已经存在,则不执行插入,直接返回即可。 + +![bst_insert](binary_search_tree.assets/bst_insert.png) + +=== "Java" + + ```java title="binary_search_tree.java" + /* 插入结点 */ + TreeNode insert(int num) { + // 若树为空,直接提前返回 + if (root == null) return null; + TreeNode cur = root, pre = null; + // 循环查找,越过叶结点后跳出 + while (cur != null) { + // 找到重复结点,直接返回 + if (cur.val == num) return null; + pre = cur; + // 插入位置在 root 的右子树中 + if (cur.val < num) cur = cur.right; + // 插入位置在 root 的左子树中 + else cur = cur.left; + } + // 插入结点 val + TreeNode node = new TreeNode(num); + if (pre.val < num) pre.right = node; + else pre.left = node; + return node; + } + ``` + +为了插入结点,需要借助 **辅助结点 `prev`** 保存上一轮循环的结点,这样在遍历到 $\text{null}$ 时,我们也可以获取到其父结点,从而完成结点插入操作。 + +与查找结点相同,插入结点使用 $O(\log n)$ 时间。 + +### 删除结点 + +与插入结点一样,我们需要在删除操作后维持二叉搜索树的 “左子树 < 根结点 < 右子树” 的性质。首先,我们需要在二叉树中执行查找操作,获取待删除结点。接下来,根据待删除结点的子结点数量,删除操作需要分为三种情况: + +**待删除结点的子结点数量 $= 0$ 。** 表明待删除结点是叶结点,直接删除即可。 + +![bst_remove_case1](binary_search_tree.assets/bst_remove_case1.png) + +**待删除结点的子结点数量 $= 1$ 。** 将待删除结点替换为其子结点。 + +![bst_remove_case2](binary_search_tree.assets/bst_remove_case2.png) + +**待删除结点的子结点数量 $= 2$ 。** 删除操作分为三步: + +1. 找到待删除结点在 **中序遍历序列** 中的下一个结点,记为 `nex` ; +2. 在树中递归删除结点 `nex` ; +3. 使用 `nex` 替换待删除结点; + +=== "Step 1" + + ![bst_remove_case3_1](binary_search_tree.assets/bst_remove_case3_1.png) + +=== "Step 2" + + ![bst_remove_case3_2](binary_search_tree.assets/bst_remove_case3_2.png) + +=== "Step 3" + + ![bst_remove_case3_3](binary_search_tree.assets/bst_remove_case3_3.png) + +=== "Step 4" + + ![bst_remove_case3_4](binary_search_tree.assets/bst_remove_case3_4.png) + +删除结点操作也使用 $O(\log n)$ 时间,其中查找待删除结点 $O(\log n)$ ,获取中序遍历后继结点 $O(\log n)$ 。 + +=== "Java" + + ```java title="binary_search_tree.java" + /* 删除结点 */ + TreeNode remove(int num) { + // 若树为空,直接提前返回 + if (root == null) return null; + TreeNode cur = root, pre = null; + // 循环查找,越过叶结点后跳出 + while (cur != null) { + // 找到待删除结点,跳出循环 + if (cur.val == num) break; + pre = cur; + // 待删除结点在 root 的右子树中 + if (cur.val < num) cur = cur.right; + // 待删除结点在 root 的左子树中 + else cur = cur.left; + } + // 若无待删除结点,则直接返回 + if (cur == null) return null; + // 子结点数量 = 0 or 1 + if (cur.left == null || cur.right == null) { + // 当子结点数量 = 0 / 1 时, child = null / 该子结点 + TreeNode child = cur.left != null ? cur.left : cur.right; + // 删除结点 cur + if (pre.left == cur) pre.left = child; + else pre.right = child; + } + // 子结点数量 = 2 + else { + // 获取中序遍历中 cur 的下一个结点 + TreeNode nex = min(cur.right); + int tmp = nex.val; + // 递归删除结点 nex + remove(nex.val); + // 将 nex 的值复制给 cur + cur.val = tmp; + } + return cur; + } + /* 获取最小结点 */ + TreeNode min(TreeNode root) { + if (root == null) return root; + // 循环访问左子结点,直到叶结点时为最小结点,跳出 + while (root.left != null) { + root = root.left; + } + return root; + } + ``` + +## 二叉搜索树的优势 + +假设给定 $n$ 个数字,最常用的存储方式是「数组」,那么对于这串乱序的数字,常见操作的效率为: + +- **查找元素:** 由于数组是乱序的,因此需要遍历数组来确定,使用 $O(n)$ 时间; +- **插入元素:** 只需将元素添加至数组尾部即可,使用 $O(1)$ 时间; +- **删除元素:** 先查找元素,使用 $O(\log n)$ 时间,再在数组中删除该元素,使用 $O(n)$ 时间; +- **获取最小 / 最大元素:** 需要遍历数组来确定,使用 $O(1)$ 时间; + +为了得到先验信息,我们也可以预先将数组元素进行排序,得到一个「排序数组」,此时操作效率为: + +- **查找元素:** 由于数组已排序,可以使用二分查找,使用 $O(\log n)$ 时间; +- **插入元素:** 为了保持数组是有序的,需插入到数组某位置,平均使用 $O(n)$ 时间; +- **删除元素:** 与乱序数组中的情况相同,使用 $O(n)$ 时间; +- **获取最小 / 最大元素:** 数组头部和尾部元素即是最小和最大元素,使用 $O(1)$ 时间; + +观察发现,乱序数组和排序数组中的各类操作的时间复杂度是 “偏科” 的,即有的快有的慢;**而二叉搜索树的各项操作的时间复杂度都是对数阶,在数据量 $n$ 很大时有巨大优势**。 + +
      + +| | 乱序数组 | 排序数组 | 二叉搜索树 | +| ------------------- | -------- | ----------- | ----------- | +| 查找指定元素 | $O(n)$ | $O(\log n)$ | $O(\log n)$ | +| 插入元素 | $O(1)$ | $O(n)$ | $O(\log n)$ | +| 删除元素 | $O(n)$ | $O(n)$ | $O(\log n)$ | +| 获取最小 / 最大元素 | $O(n)$ | $O(1)$ | $O(\log n)$ | + +
      + +## 二叉搜索树的退化 + +二叉搜索树的理想状态是「完美二叉树」,我们称这样的二叉树是 “平衡” 的,此时可以在 $\log n$ 轮循环内查找任意结点。 + +如果我们动态地在二叉搜索树中插入与删除结点,**则可能导致二叉树退化为链表**,此时各种操作的时间复杂度也退化之 $O(n)$ 。 + +!!! note + + 在实际应用中,如何保持二叉搜索树的平衡,也是一个需要重要考虑的问题。 + +![bst_degradation](binary_search_tree.assets/bst_degradation.png) + +## 二叉搜索树常见应用 + +- 系统中的多级索引,高效查找、插入、删除操作。 +- 各种搜索算法的底层数据结构。 +- 存储数据流,保持其已排序。 diff --git a/docs/chapter_tree/binary_tree.assets/binary_tree_add_remove.png b/docs/chapter_tree/binary_tree.assets/binary_tree_add_remove.png new file mode 100644 index 0000000000000000000000000000000000000000..f28476b961ddd358c1f20dcc12f1b0ce9166a48e GIT binary patch literal 77756 zcmeFZWmp{B)-?(QhY%bRT!XtyAP|BDcXxMpC%C&b9yGW+1b26LcXw{F-|w97p0oG; zegC{k^K|o6Ra3QAtuf~uW6TPcmK1&u{|O!p4D7wA$TwLqFeu<7#273D@WxLd1`fP| z+sO(GfR&Em?gPL08z_q!N=SfF0iR*PpukbV-o5Pt{09U7^w+^}hlqjW{QI*kIK}_j z2l&oF6EMiX_6Gw~1>WEOX#+oR$NbOxyG-!^`;D0p|JoZ0Ec4yJK0}NFdxLp>`|1w7 z!C8x_*nxo|b-n$7$5SKW0Ot)R`t7T{Gx$*p97w*qp^YA~lZ~dcwX+-+mWqOqn5tk_ 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m6$-s&OZ-o2eoJn?zLoI8phu^DVhI8KBQ7lS;-jF>+y4VK&MBM# literal 0 HcmV?d00001 diff --git a/docs/chapter_tree/binary_tree.md b/docs/chapter_tree/binary_tree.md new file mode 100644 index 000000000..8eb9eb249 --- /dev/null +++ b/docs/chapter_tree/binary_tree.md @@ -0,0 +1,196 @@ +--- +comments: true +--- + +# 二叉树 + +「二叉树 Binary Tree」是一种非线性数据结构,代表着祖先与后代之间的派生关系,体现着 “一分为二” 的分治逻辑。类似于链表,二叉树也是以结点为单位存储的,结点包含「值」和两个「指针」。 + +=== "Java" + + ```java + /* 链表结点类 */ + class TreeNode { + int val; // 结点值 + TreeNode left; // 左子结点指针 + TreeNode right; // 右子结点指针 + TreeNode(int x) { val = x; } + } + ``` + +结点的两个指针分别指向「左子结点 Left Child Node」和「右子结点 Right Child Node」,并且称该结点为两个子结点的「父结点 Parent Node」。给定二叉树某结点,将左子结点以下的树称为该结点的「左子树 Left Subtree」,右子树同理。 + +![binary_tree_definition](binary_tree.assets/binary_tree_definition.png) + +

      Fig. 子结点与子树

      + +需要注意,父结点、子结点、子树是可以向下递推的。例如,如果将上图的「结点 2」看作父结点,那么其左子节点和右子结点分别为「结点 4」和「结点 5」,左子树和右子树分别为「结点 4 以下的树」和「结点 5 以下的树」。 + +## 二叉树常见术语 + +「根结点 Root Node」:二叉树最顶层的结点,其没有父结点; + +「叶结点 Leaf Node」:没有子结点的结点,其两个指针都指向 $\text{null}$ ; + +结点「度 Degree」:结点的子结点数量,二叉树中度的范围是 0, 1, 2 ; + +结点「深度 Depth」 :根结点到该结点的层数; + +结点「高度 Height」:最远叶结点到该结点的层数; + +二叉树「高度」:二叉树中根结点到最远叶结点的层数; + +![binary_tree_terminology](binary_tree.assets/binary_tree_terminology.png) + +

      Fig. 二叉树的常见术语

      + +## 二叉树最佳和最差结构 + +当二叉树的每层的结点都被填满时,达到「完美二叉树」;而当所有结点都偏向一边时,二叉树退化为「链表」。 + +![binary_tree_corner_cases](binary_tree.assets/binary_tree_corner_cases.png) + +

      Fig. 二叉树的最佳和最差结构

      + +在最佳和最差结构下,二叉树的结点数量和高度等性质达到最大(最小)值。 + +
      + +| | 完美二叉树 | 链表 | +| ----------------------------- | ---------- | -------------- | +| 二叉树第 $i$ 层的结点数量 | $2^{i-1}$ | $1$ | +| 高度为 $h$ 的二叉树的结点总数 | $2^h - 1$ | $h$ | +| 结点总数为 $n$ 的二叉树的高度 | $n$ | $\log_2 n + 1$ | + +
      + +## 二叉树基本操作 + +**初始化二叉树。** 与链表类似,先初始化结点,再构建引用指向(即指针)。 + +=== "Java" + + ```java title="binary_tree.java" + // 初始化结点 + TreeNode n1 = new TreeNode(1); + TreeNode n2 = new TreeNode(2); + TreeNode n3 = new TreeNode(3); + TreeNode n4 = new TreeNode(4); + TreeNode n5 = new TreeNode(5); + // 构建引用指向(即指针) + n1.left = n2; + n1.right = n3; + n2.left = n4; + n2.right = n5; + ``` + +**插入与删除结点。** 与链表类似,插入与删除结点都可以通过修改指针实现。 + +![binary_tree_add_remove](binary_tree.assets/binary_tree_add_remove.png) + +

      Fig. 在二叉树中插入与删除结点

      + +```java title="binary_tree.java" +TreeNode P = new TreeNode(0); +// 在 n1 -> n2 中间插入结点 P +n1.left = P; +P.left = n2; +// 删除结点 P +n1.left = n2; +``` + +!!! note + + 插入结点会改变二叉树的原有逻辑结构,删除结点往往意味着删除了该结点的所有子树。因此,二叉树中的插入与删除一般都是由一套操作配合完成的,这样才能实现有意义的操作。 + +## 二叉树遍历 + +非线性数据结构的遍历操作比线性数据结构更加复杂,往往需要使用搜索算法来实现。常见的二叉树遍历方式有层序遍历、前序遍历、中序遍历、后序遍历。 + +### 层序遍历 + +「层序遍历 Hierarchical-Order Traversal」从顶至底、一层一层地遍历二叉树,并在每层中按照从左到右的顺序访问结点。 + +层序遍历本质上是「广度优先搜索 Breadth-First Traversal」,其体现着一种 “一圈一圈向外” 的层进遍历方式。 + +![binary_tree_bfs](binary_tree.assets/binary_tree_bfs.png) + +

      Fig. 二叉树的层序遍历

      + +广度优先遍历一般借助「队列」来实现。队列的规则是 “先进先出” ,广度优先遍历的规则是 ”一层层平推“ ,两者背后的思想是一致的。 + +=== "Java" + + ```java title="binary_tree_bfs.java" + /* 层序遍历 */ + List hierOrder(TreeNode root) { + // 初始化队列,加入根结点 + Queue queue = new LinkedList<>() {{ add(root); }}; + // 初始化一个列表,用于保存遍历序列 + List list = new ArrayList<>(); + while (!queue.isEmpty()) { + TreeNode node = queue.poll(); // 队列出队 + list.add(node.val); // 保存结点值 + if (node.left != null) + queue.offer(node.left); // 左子结点入队 + if (node.right != null) + queue.offer(node.right); // 右子结点入队 + } + return list; + } + ``` + +### 前序、中序、后序遍历 + +相对地,前、中、后序遍历皆属于「深度优先遍历 Depth-First Traversal」,其体现着一种 “先走到尽头,再回头继续” 的回溯遍历方式。 + +如下图所示,左侧是深度优先遍历的的示意图,右上方是对应的递归实现代码。深度优先遍历就像是绕着整个二叉树的外围 “走” 一圈,走的过程中,在每个结点都会遇到三个位置,分别对应前序遍历、中序遍历、后序遍历。 + +![binary_tree_dfs](binary_tree.assets/binary_tree_dfs.png) + +

      Fig. 二叉树的前 / 中 / 后序遍历

      + +
      + +| 位置 | 含义 | 此处访问结点时对应 | +| ---------- | ------------------------------------ | ----------------------------- | +| 橙色圆圈处 | 刚进入此结点,即将访问该结点的左子树 | 前序遍历 Pre-Order Traversal | +| 蓝色圆圈处 | 已访问完左子树,即将访问右子树 | 中序遍历 In-Order Traversal | +| 紫色圆圈处 | 已访问完左子树和右子树,即将返回 | 后序遍历 Post-Order Traversal | + +
      + +=== "Java" + + ```java title="binary_tree_dfs.java" + /* 前序遍历 */ + void preOrder(TreeNode root) { + if (root == null) return; + // 访问优先级:根结点 -> 左子树 -> 右子树 + list.add(root.val); + preOrder(root.left); + preOrder(root.right); + } + + /* 中序遍历 */ + void inOrder(TreeNode root) { + if (root == null) return; + // 访问优先级:左子树 -> 根结点 -> 右子树 + inOrder(root.left); + list.add(root.val); + inOrder(root.right); + } + + /* 后序遍历 */ + void postOrder(TreeNode root) { + if (root == null) return; + // 访问优先级:左子树 -> 右子树 -> 根结点 + postOrder(root.left); + postOrder(root.right); + list.add(root.val); + } + ``` + +!!! note + + 使用循环一样可以实现前、中、后序遍历,但代码相对繁琐,有兴趣的同学可以自行实现。 diff --git a/docs/chapter_tree/binary_tree_types.assets/balanced_binary_tree.png b/docs/chapter_tree/binary_tree_types.assets/balanced_binary_tree.png new file mode 100644 index 0000000000000000000000000000000000000000..030cd7abf35dd6f0f4ef138ab56f6d321053e25a GIT binary patch literal 66095 zcmeFZg6C6zq#LAb2uLWQbeDjz!I17DwlI2hD%}X9 z2hTO|j-Su-{sr%iAGg8hy3Xr3kNh5SE=p5ffs~k*7z6^5Dk(nH0)YsCk9a+Vc)*(s zlQI~11AAyGJOGt<(JukNgjpFVS*xjmxPi}vAOi4J&;{Hjz&{X}_UvZdEy`fJe}C2j zbN;yoSTn*7gnxED2xJ7j<9;lF|G0b3-!J5X|FdE)-oLIU0Oelz*Jr#Q;A&9t!}|fi z8EV4HAMok~QGgD5;?%A3(pAE+3PMG^$CU951O%aq z#wmK5u)#SuJEy=#2jd*R(^`$AXH~UQHDmq7Wll+7YdpCU8}lovHOUF_qOY>vzAy8w z{Y^aPRP72QLLlMkz3-jl{tE*9W~yATa)ay=*qg z*zc3VVK8yGpP#-baICDX(9zP;($P^%aMT3oL@M#*G2o+TU~ee@ahnXYLEDRDp<$i& zw+RUe+FDv#+S&s5?)mL3zD`c23Gn+SxpoQGD5I;3esB4e3Y6hG(X)L|nW{A7Qrk<5uLs*Iofl=2@X@5q zhE&4ihyvBdVbxOH0}T*!HK)!Kq4FO$|C=W`7LAfu^cE^{7Ap;xT|MMHgn%m;2ha)$ z`A2&jh>_l#DpwdlLjwc!BCawV_z#g_aK%0RPi;35AJqR|NbFXwfqrpDnUY`xa)Y3) z>$-u$rOVgj)6#APG?0Tgw|;#kV-S05F!T_qoErDUl&!jkiMsfiYu@Vpe+o(_w}XIc ztF$xu+up#JuvSz-8!wZ<6}Y9Zudkt@A()zw{@UO$Lf}yMGCX%kJ)|}dp(N7bF5h0H z-c|6H;xDNan&-w`@4o@!d4kp_KoSS8aVaB@)|;ZAUicCseFqP|5VW^8R_u|e;ZfwY z<8U!3>m#zXn%9WPv^xNjhmfQS8v7RY9|4|yf>OK6r?Afo`X_#1t%fkk0o%Dg?1*ae zukqNN{?N4$B25lTNlIEWRbbyi>ro(Tqpg#A6o|HaF^=teU98p87G&mslNqThbKh|@ zihUg&l+dH+hLBR`+|g~vZyQ}ERSLaGTFKTdO@{0Be2F_HQ!3%jkOfdE8TI}9laDib%F+B&$Xu4>I7|K zxqDdq)S<8@_B@syU1RZM7|Ra2_JhgCuvT^_ve$ocQzmyCuPeU)ty&uK1C+iKN&yUi zV&Sx=A9Ims0M*qc^xCmfw5qgJ`I2hx%H>csBVSc#$~k*F@~auxESKwSe{1qIRfXF- zuk>nCMGSg%U@&%mG+s|3N$(vDIz;7TkH0_s-coG(I2B&>tf@v$oU$l4-hcJx)qj{2 zAd!d6xicmUhS3@rw4>UC8&sY4uq^hOS58TKZ5as(1)Zo{ZASabOc@`;nhx!+7MWD( zS_eK@ceB1TGR*Q{#)*?{j}n+5qsyoNt?as8*FtKAW2gz`b}20iyctEw;V6~xF#tWN zBPjBCFC%lF$M|mp$iGQWUYM?e|Fu*oBPIyN7~UdQ=u5l<_?nsy(OHLXH}=eHJwbsG z*-mTe|NCZci|3k@@c^?DMzPAnaYUZfyQNQXw6hw0O{@@AZC~vj3|LhUL|auuY_h~U zgZ~zJWD%6;ftpO|ib#RZHhNt!LN&;VdUsjV0|fWX$4rRg!Ltri;X-*1sU5ZsWHnEp zhW@44GOzHd;zy?kSREsmRXNMfNpMGjEA+yJE~#2MU195FG|nHf+{-9%{J#{q5K5}9 z){b9R3T<7sd~T-ARHg73(JFMQtqaZ`F_8`*9d#6{LdoL&)475|u9Kur4>)G!!K!H4 zQ_+kGWuhQ2YsgYOM{3q?fDL2c-_ri1#mp>1dSbt8>mXG+RstFx(DHZkbV|Qz?(pER z#oMy*gr3!WfG^wiv-i@2#HDvD7!+z1)^0XM;Q!MxgYev_TrZZgGu*o>1UYUKXzNnG zl({7FkoGTX!F5FValb}s(`7fLFTgJdJ*y>Z>Tmc;bN{a=gF^O+Q(e(Fsz<9OpI+V2 z2pRMi*Mg7oICV$4Sk{aE{qPyWYKGx#TCm>zu0`QY_wRoT2HHWO`tIn2u!1yjEuQ}) zl5gWdX~aK;N3PW-O1OI2J?Q|;)pN7^DaYT>I3Z^S?J7w35&u(bL%i?_p=02mcm8=j zR2uj*wl}W&uZ`t0fbpiipdzpS`SfWrFrG#5&aJY){DrLo6jaZ^eC5_Z?QCBb3L0j@ zD)RiL!lM+-Am>YwAODj1|49EI*O}2ws9frE#R%hah0iZa@=tnnZF7821Ai%|QnY9*uR--Qr6OXE#hOx`-5E3;>X~;^lz@?lU>RENkPJzy2RqP7~1qcj`5p;5s0kVO?j~Rg@ws^3l1Kf09}BY_1Jw62nap=7jTHkeu@fpAdeA!G zjd|3O*-__e_gnVjlc~}0IgTF~8<&GgI-AL-0vVrbv13#M;@(BY)(>y7>r9yW9wikO z$uHE1C3c~f{lD1Rh-j3QmUF{y3;4$l>WGrnyq?r){HypnK^GgZMxROQWawBmZ{8_8fhzQZHhps&Ur*)=`{@q50AJdv8>> zB}ZJ~%uM&#@l?xf6M^r?lZxMaR@B8$6g@a4A=nTNsdvw`wY6m}TTo2*S=dA&TwLEX z4czj7W?u`UxPNmu@zDi&tm{eK>d^G352Y76CQTTdr3H9@^+j!m?p6I^SPl_6=m;)r zcLnD=WMFuw0Z!lhgp)@<{<;*C^XzceMQV$*NrC8d+pdEeRw?63nt@64;s^coTQiZ; zLs^~^$4VqDQY{4>>tb4m!!M8oIZ2s`sv!E#cE&Rwb&6XapEI&;dk~o{#RriUe z^6}oqBSCod+Emr)wXW0ltnP6`ZF9@=z9t11Ho?md{OJR01HGxYvh`{~Wkd}+1yDmIqenTGKBZzdE zM6-Dw*2vTt+i$ZW0h@-u{9^9-pmjpQ%fahC>E=Ccfzw~NPV=64GfY}_rQZ#h*PjoZ z+kF4g+ z1v{QZoCIKG&BgtPN4l_ou<~X;2wZW^KSm7aq=|0k-uSjx-stVrVXuk|ToRTx+AQd- zUN4$UTPnUf55tqDJAw!aaSrM}g-IL#s0}Dz^uYR)`qVrNRGTPyZI<=>U>lix-0k8& zDdrr!5OeBt9Q$KgnG$P{r1^9>m@w6Rvbw@?InB084$>>x(sZ%Vn~4js$L$s6Av9#b z)MF6hJA&m_Pdvq-k!c2|4kjq^U$V8dE8b-u7u!4(x@#hgRGw+9oGgvx+*M#Q+^dlH zPEyxULO8v1TBBOq=4Zm$sy@AwLrw6^K z$1~crPo2#^|A6DI-PS&$_7LN^ARd>#GxE{K*4cIQWOMHo{hc3ah7|h@aTAXfO0Qkx zKbdXF>9hROF=gCfKQkIm8KJs7RFxyTQo^e-(tl))LFd+CM~zs-Z}|^K8WyVuujR0V z^j1xcC-gw$Ms}+TXW3DkF&=)~o8pd9c3h_wZLJq{O)Y8|K~i z8`SA=wRe9V-*NI!Cy&PG3pWE_*gkUD)_wHe^oB7-YgZph@@lx^d5SP?3gC&2T?Pnl z^T+&~Y1_`|P{f8#i)H%Z;-Lb@ET2oZILxuZZ0%?v82jCKu}UtdW}|{Z(sd_gKB$E4 z5Lss@sJ8mPh z9Zn#)VUJP6VPh2n={Jg|J%Z>z>Ny&(wV_@YiwK59_)eyhI6|npS6*8OqMH((K+P6? 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      {{ lang.t("meta.comments") }}

      + + + + +{% endif %} \ No newline at end of file diff --git a/mkdocs.yml b/mkdocs.yml index 8edf56df9..3fc9c919c 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -14,7 +14,7 @@ copyright: Copyright © 2020 - 2022 Krahets # Configuration theme: name: material - custom_dir: overrides + custom_dir: docs/overrides language: zh features: - announce.dismiss

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Q0DsP&vau*RaV7fy0B^i^asU7T literal 0 HcmV?d00001 diff --git a/docs/chapter_introduction/index.md b/docs/chapter_introduction/index.md new file mode 100644 index 000000000..6355dfe51 --- /dev/null +++ b/docs/chapter_introduction/index.md @@ -0,0 +1,109 @@ +--- +comments: true +--- + +# 关于本书 + +在 2018 年 10 月发生了一件事,成为了我职业生涯的重要转折点。当时的我在交大读研,对互联网求职一无所知,但仍然硬着头皮申请了 Microsoft 的软件工程师实习。在二面中,面试官让我在白板上写出 “快速排序” 代码,我摇了摇头,畏畏缩缩地写了一个 “冒泡排序” ,并且还写错了。从面试官的表情上,我看到了一个大大的 Game Over 。 + +从那次失利开始,找工作的压力就倒逼我开始刷算法题。我采用 “扫雷游戏” 式的学习方法,两眼一抹黑刷题,扫到不会的 “雷” 就通过查资料把它解决掉,配合周期性的总结,逐渐形成了数据结构与算法的知识图景。幸运地,我在 2020 年秋招斩获了多家大厂的 Offer 。 + +之后,分享题解成为了我的爱好。经常刷题的同学可能遇见过一个顶着路飞笑脸头像,名为「Krahets」的力扣 ID ,那便是我。截至目前,我已在力扣(LeetCode)上分享了近 100 道题目解析,累积了 1700 万阅读量,回复了数千条读者的评论和问题,并编写了 LeetBook《图解算法数据结构》,已免费售出 21 万多本。 + +回想自己当初在 “扫雷式” 刷题中被炸的满头包的痛苦,我意识到有一本 “刷题前必看” 的读物可以使算法小白少走许多弯路,而这正与我的擅长点契合。强烈的写作意愿席卷而来,那就来吧: + +

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000000000..d54963aa9 --- /dev/null +++ b/docs/chapter_introduction/contribution.md @@ -0,0 +1,46 @@ +--- +comments: true +--- + +# 一起参与创作 + +由于作者水平有限,书中内容难免疏漏谬误,请您谅解。此外,希望您可以一同参与到本书的内容创作中来。如果你发现笔误、无效链接、内容缺失、文字歧义、解释不清晰、行文结构不合理等问题,烦请您帮忙修正内容,以帮助其他读者获取更优质的学习内容。 + +!!! quote "" + + 纸质书籍的两次印刷的间隔时间往往需要数年,内容更新非常不方便。
    但在本开源 HTML 书中,内容更迭的时间被缩短至数日甚至几个小时。 + +## 修改文字 + +每个页面的右上角都有一个「编辑」按钮,你可以按照以下步骤修改文章: + +1. 点击编辑按钮,如果遇到提示 “需要 Fork 此仓库” ,请通过; +2. 修改 Markdown 源文件内容; +3. 在页面底部填写更改说明,然后单击 “Propose file change” 按钮; +4. 页面跳转后,点击 “Create pull request” 按钮发起拉取请求即可,我会第一时间查看处理并及时更新内容。 + +![edit_markdown](contribution.assets/edit_markdown.png) + +## 修改图片 + +书中的配图无法直接修改,需要通过以下途径提出修改意见: + +1. 新建一个 Issue ,将需要修改的图片复制或截图,粘贴在面板中; +2. 描述图片问题,应如何修改; +3. 提交 Issue 即可,我会第一时间重新画图并替换图片。 + +## 修改代码 + +若发现代码源文件有错误,可以本地修改并提交 Pull Request : + +1. 登录 GitHub ,并 Fork [本仓库](https://github.com/krahets/hello-algo) 至个人账号; +2. 使用 Git 克隆 Fork 的仓库至本地(Git 安装教程见上节 “编程环境安装” ); +3. 在本地修改 `.java` , `.cpp` , `.py` 文件中的代码,并运行测试;测试完毕后,请同步修改 Markdown 文件中的对应代码; +5. 将本地更新 Commit ,并 Push 至远程仓库; +6. 刷新仓库网页,点击 “Create pull request” 按钮发起拉取请求即可; + +(TODO:教学视频) + +## 创作新内容 + +「修改代码」的流程是完整的,还可以用来 **重写某章节、新增章节、翻译代码至其他编程语言** 等。非常欢迎您和我一同来创作本书! diff --git a/docs/chapter_introduction/index.assets/learning_route.png b/docs/chapter_introduction/index.assets/learning_route.png new file mode 100644 index 0000000000000000000000000000000000000000..7423808d62faf0c17f0a7f753b405a934c187ce0 GIT binary patch literal 81816 zcmeFZWmH^E)&|;i8YgH6cXxMpcMk*)?u1|gg1c+u?u1}Tu;38fAwY141c%_{p3Hpj zd^35!U-#F&YYi;cIdq>>vTIlEy`TM5MXIaHJw+iz0f9hI738HgK_GbG7CeCj1}@@c zig3UM?vr64=$Anod{?*NI_ac+Y9c;XT&y=zKpWigb;CL>J%JsAhrNvVwY*m zAijfxg~c3~@iz;#3tz6)A2GW`<3}-HHy?;==>oeDh%C5FIbmqvL@r5rN%|t@6mf5| z{ZP%x-K95UPm^kHMITx_UR709Rkr3H9aO%ZHJl~kby{jl&Vo>r!y(IygRv6={&}1K z10pz^*Y&~vYmLDNyUT;w?{n`AJNzHQo)Te>lkhrIV>=Oh(N}&ivy=ai_kbeGsS}DV zv9tKQKc9~YdH$HM(<)OnYH{CR{9wO3mcvw}cUi%rE6`PMIjmc)9|%Hk&hl5l@>JyW zq*EjLSN(rACC~~-Jh`-KeTQ!qvvaxC^LU}bT1CY`3dCm8?mL)7gZ8tqc0oSQWQ@d# zI25OL_;qpgG{t{*1t`jfG*B^+z$e*6o6$DR%#Bm+Xtqi(o>YWTKrh(mnb7Xd`L4yu z$N;CI{OZK#rUiNKU$M(gnJ3O$)uc)PBapuu1c%HmNnSDRg2_p|kB3Nfzz1Q+dA-0M zb@cT;+1#X#Rt;;w>f2ur-sd}51^ND{U>FGg>5EHZAI1MI;D6+7APIV>baDOk3+>Ha z0cebeUGDDcxZ`_Gt8Q*h6v(*QZ951ab)u!f-?cYvuCL2^Q6AmXvheobN9mts8svbY z(z*GVRx$h~Z=DF?SK9jc*}vRg>`%Wl{L~E({Sr!WH#*Bkh(DR7oS%I0R4!MyCS)`7 ztf+*|AH!><@S85UJ-7OsE+mLkCR7b4WUosFxSS9X9pK5?tX;=CUu_6@ovita zp2$H?vISmU&uPe;_-nM?7h1X2u1Cb4)5x_nxvZI-xa$3{M*ffdCWOHe^Ft8@Z%#1~ zJNAruiRDd*DFSY`Q5Av_o+>gpa0Ko|%h}IWBVQ6(bQJja5~Pz_`T4aJwZs3e z=i3m;*E>l#hl5UM$X~4D(e*{CEG9n>=&P#8OS5Y7`4R0ufg6EqzZzzD9$0osfVAG1 zuA3y8@@-R$VNLNZ?I)oBss9P$!^m4nTuX_xl-wRSu}|Tm5q9_vISFHC4p%?huyIg>>a_Os5 z@^Q1J*C~MB8+>34jmE4Sc79<~LUkiIFn{^?p{wOVf@<4k7iw0OnkqbmVbrJ70}844 z-Z)lIbSrCPEAAe0-9RKYvVQ#jZ{uWIg^h$o7?iRaM62@1KZUD_w=f*)NC;<@XcM~y{Y9v4ncYW(V5303*xv`jQ5qzX*=x(6Bt1&fv>IXYI$-R^)z>f;#3SX(Mn#Z=#ztF;EJMX&?%Bjjt7i<+t{k>ok*;?L0&e+m>u7$D62r;xVE^Fgd%{J8P{)D!jJMIn%5DE&1=|MRA}08sa^ zE7@r1pOw!5Mb*F!^!%-)|90a~8UA-P{!`iit5pA;;(x6#|D7BEog4o!A?p9N8~?Q% z|C|f|e{)7ADu@@@@v!Nc0%f9cfw0%yb3Q?M;1OBb&zem&R=79T$z51eI&I6fQzjh6;<636@ z7Ju`hpRYYpnBe*|5P4a6ZuCYzMP9Sc_1-B-H}*OHe6<2V2SF1gybkjLa8Di|R)2|< zWC+!?oehaxEpc5<0v`8_Lcqq)qt@doTT6<{WLOt6ncfA?U_Pn0jjo+d_M$g!LyYmS~^$7brmqNO?23tn09)|$7GoCB* zN~CR=wUwQaxXY%s>_m{7JTMysZ#P@1(~NNh^#0in`I5&sN9T?QkE^5Qj@#MyxM01b zsSKISV&XXhAo?ccFo*Od_*V^eX4`yP_Ncn}DhEEy2(Z#RygAD9L~ay>gLkQ-;*mg} zQOZ|JXGQD5zD+Z9$`JNG>kNFNHPZZQm$Px0wJ=@wXR?~;4bVvTY{S|_)%7oNByTn% z>89Le_h{9K@g=_;>9wDw&IJ8B^6bb8|1A`1@D%{M03$f<#IGuJ*w}+1ChR>$6nbl! 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