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en/docs/chapter_computational_complexity/space_complexity.md
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@@ -316,6 +316,12 @@ The relevant code is as follows:
}
```
=== "Kotlin"
```kotlin title=""
```
=== "Zig"
```zig title=""
@@ -462,6 +468,12 @@ Consider the following code, the term "worst-case" in worst-case space complexit
}
```
=== "Kotlin"
```kotlin title=""
```
=== "Zig"
```zig title=""
@@ -482,9 +494,10 @@ Consider the following code, the term "worst-case" in worst-case space complexit
for _ in range(n):
function()
def recur(n: int) -> int:
def recur(n: int):
"""Recursion O(n)"""""
if n == 1: return
if n == 1:
return
return recur(n - 1)
```
@@ -699,6 +712,12 @@ Consider the following code, the term "worst-case" in worst-case space complexit
}
```
=== "Kotlin"
```kotlin title=""
```
=== "Zig"
```zig title=""
@@ -725,7 +744,7 @@ $$
<p align="center"> Figure 2-16 &nbsp; Common Types of Space Complexity </p>
### 1. &nbsp; Constant Order $O(1)$
### 1. &nbsp; Constant Order $O(1)$ {data-toc-label="Constant Order"}
Constant order is common in constants, variables, objects that are independent of the size of input data $n$.
@@ -1031,6 +1050,33 @@ Note that memory occupied by initializing variables or calling functions in a lo
}
```
=== "Kotlin"
```kotlin title="space_complexity.kt"
/* 函数 */
fun function(): Int {
// 执行某些操作
return 0
}
/* 常数阶 */
fun constant(n: Int) {
// 常量、变量、对象占用 O(1) 空间
val a = 0
var b = 0
val nums = Array(10000) { 0 }
val node = ListNode(0)
// 循环中的变量占用 O(1) 空间
for (i in 0..<n) {
val c = 0
}
// 循环中的函数占用 O(1) 空间
for (i in 0..<n) {
function()
}
}
```
=== "Zig"
```zig title="space_complexity.zig"
@@ -1070,7 +1116,7 @@ Note that memory occupied by initializing variables or calling functions in a lo
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20function%28%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%87%BD%E6%95%B0%22%22%22%0A%20%20%20%20%23%20%E6%89%A7%E8%A1%8C%E6%9F%90%E4%BA%9B%E6%93%8D%E4%BD%9C%0A%20%20%20%20return%200%0A%0Adef%20constant%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E5%B8%B8%E6%95%B0%E9%98%B6%22%22%22%0A%20%20%20%20%23%20%E5%B8%B8%E9%87%8F%E3%80%81%E5%8F%98%E9%87%8F%E3%80%81%E5%AF%B9%E8%B1%A1%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20a%20%3D%200%0A%20%20%20%20nums%20%3D%20%5B0%5D%20*%2010%0A%20%20%20%20node%20%3D%20ListNode%280%29%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E4%B8%AD%E7%9A%84%E5%8F%98%E9%87%8F%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20c%20%3D%200%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E4%B8%AD%E7%9A%84%E5%87%BD%E6%95%B0%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20function%28%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E5%B8%B8%E6%95%B0%E9%98%B6%0A%20%20%20%20constant%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20function%28%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%87%BD%E6%95%B0%22%22%22%0A%20%20%20%20%23%20%E6%89%A7%E8%A1%8C%E6%9F%90%E4%BA%9B%E6%93%8D%E4%BD%9C%0A%20%20%20%20return%200%0A%0Adef%20constant%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E5%B8%B8%E6%95%B0%E9%98%B6%22%22%22%0A%20%20%20%20%23%20%E5%B8%B8%E9%87%8F%E3%80%81%E5%8F%98%E9%87%8F%E3%80%81%E5%AF%B9%E8%B1%A1%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20a%20%3D%200%0A%20%20%20%20nums%20%3D%20%5B0%5D%20*%2010%0A%20%20%20%20node%20%3D%20ListNode%280%29%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E4%B8%AD%E7%9A%84%E5%8F%98%E9%87%8F%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20c%20%3D%200%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E4%B8%AD%E7%9A%84%E5%87%BD%E6%95%B0%E5%8D%A0%E7%94%A8%20O%281%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20function%28%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E5%B8%B8%E6%95%B0%E9%98%B6%0A%20%20%20%20constant%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
### 2. &nbsp; Linear Order $O(n)$
### 2. &nbsp; Linear Order $O(n)$ {data-toc-label="Linear Order"}
Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to $n$:
@@ -1307,6 +1353,26 @@ Linear order is common in arrays, linked lists, stacks, queues, etc., where the
}
```
=== "Kotlin"
```kotlin title="space_complexity.kt"
/* 线性阶 */
fun linear(n: Int) {
// 长度为 n 的数组占用 O(n) 空间
val nums = Array(n) { 0 }
// 长度为 n 的列表占用 O(n) 空间
val nodes = mutableListOf<ListNode>()
for (i in 0..<n) {
nodes.add(ListNode(i))
}
// 长度为 n 的哈希表占用 O(n) 空间
val map = mutableMapOf<Int, String>()
for (i in 0..<n) {
map[i] = i.toString()
}
}
```
=== "Zig"
```zig title="space_complexity.zig"
@@ -1471,6 +1537,18 @@ As shown below, this function's recursive depth is $n$, meaning there are $n$ in
}
```
=== "Kotlin"
```kotlin title="space_complexity.kt"
/* 线性阶（递归实现） */
fun linearRecur(n: Int) {
println("递归 n = $n")
if (n == 1)
return
linearRecur(n - 1)
}
```
=== "Zig"
```zig title="space_complexity.zig"
@@ -1491,7 +1569,7 @@ As shown below, this function's recursive depth is $n$, meaning there are $n$ in
<p align="center"> Figure 2-17 &nbsp; Recursive Function Generating Linear Order Space Complexity </p>
### 3. &nbsp; Quadratic Order $O(n^2)$
### 3. &nbsp; Quadratic Order $O(n^2)$ {data-toc-label="Quadratic Order"}
Quadratic order is common in matrices and graphs, where the number of elements is quadratic to $n$:
@@ -1686,6 +1764,25 @@ Quadratic order is common in matrices and graphs, where the number of elements i
}
```
=== "Kotlin"
```kotlin title="space_complexity.kt"
/* 平方阶 */
fun quadratic(n: Int) {
// 矩阵占用 O(n^2) 空间
val numMatrix: Array<Array<Int>?> = arrayOfNulls(n)
// 二维列表占用 O(n^2) 空间
val numList: MutableList<MutableList<Int>> = arrayListOf()
for (i in 0..<n) {
val tmp = mutableListOf<Int>()
for (j in 0..<n) {
tmp.add(0)
}
numList.add(tmp)
}
}
```
=== "Zig"
```zig title="space_complexity.zig"
@@ -1861,6 +1958,20 @@ As shown below, the recursive depth of this function is $n$, and in each recursi
}
```
=== "Kotlin"
```kotlin title="space_complexity.kt"
/* 平方阶（递归实现） */
tailrec fun quadraticRecur(n: Int): Int {
if (n <= 0)
return 0
// 数组 nums 长度为 n, n-1, ..., 2, 1
val nums = Array(n) { 0 }
println("递归 n = $n 中的 nums 长度 = ${nums.size}")
return quadraticRecur(n - 1)
}
```
=== "Zig"
```zig title="space_complexity.zig"
@@ -1882,7 +1993,7 @@ As shown below, the recursive depth of this function is $n$, and in each recursi
<p align="center"> Figure 2-18 &nbsp; Recursive Function Generating Quadratic Order Space Complexity </p>
### 4. &nbsp; Exponential Order $O(2^n)$
### 4. &nbsp; Exponential Order $O(2^n)$ {data-toc-label="Exponential Order"}
Exponential order is common in binary trees. Observe the below image, a "full binary tree" with $n$ levels has $2^n - 1$ nodes, occupying $O(2^n)$ space:
@@ -2039,6 +2150,20 @@ Exponential order is common in binary trees. Observe the below image, a "full bi
}
```
=== "Kotlin"
```kotlin title="space_complexity.kt"
/* 指数阶（建立满二叉树） */
fun buildTree(n: Int): TreeNode? {
if (n == 0)
return null
val root = TreeNode(0)
root.left = buildTree(n - 1)
root.right = buildTree(n - 1)
return root
}
```
=== "Zig"
```zig title="space_complexity.zig"
@@ -2062,7 +2187,7 @@ Exponential order is common in binary trees. Observe the below image, a "full bi
<p align="center"> Figure 2-19 &nbsp; Full Binary Tree Generating Exponential Order Space Complexity </p>
### 5. &nbsp; Logarithmic Order $O(\log n)$
### 5. &nbsp; Logarithmic Order $O(\log n)$ {data-toc-label="Logarithmic Order"}
Logarithmic order is common in divide-and-conquer algorithms. For example, in merge sort, an array of length $n$ is recursively divided in half each round, forming a recursion tree of height $\log n$, using $O(\log n)$ stack frame space.