2023-02-16 03:39:01 +08:00
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# 空间复杂度
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2022-11-22 17:47:26 +08:00
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2023-08-20 14:51:39 +08:00
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「空间复杂度 space complexity」用于衡量算法占用内存空间随着数据量变大时的增长趋势。这个概念与时间复杂度非常类似,只需将“运行时间”替换为“占用内存空间”。
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2022-11-22 17:47:26 +08:00
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2023-02-16 03:39:01 +08:00
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## 算法相关空间
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2022-11-22 17:47:26 +08:00
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2023-08-20 14:51:39 +08:00
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算法在运行过程中使用的内存空间主要包括以下几种。
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2022-11-22 17:47:26 +08:00
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2023-08-08 23:16:33 +08:00
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- **输入空间**:用于存储算法的输入数据。
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2023-08-20 14:51:39 +08:00
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- **暂存空间**:用于存储算法在运行过程中的变量、对象、函数上下文等数据。
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2023-08-08 23:16:33 +08:00
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- **输出空间**:用于存储算法的输出数据。
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2022-11-22 17:47:26 +08:00
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2023-08-08 23:16:33 +08:00
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一般情况下,空间复杂度的统计范围是“暂存空间”加上“输出空间”。
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2022-11-22 17:47:26 +08:00
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2023-08-20 14:51:39 +08:00
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暂存空间可以进一步划分为三个部分。
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2022-11-22 17:47:26 +08:00
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2023-08-08 23:16:33 +08:00
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- **暂存数据**:用于保存算法运行过程中的各种常量、变量、对象等。
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- **栈帧空间**:用于保存调用函数的上下文数据。系统在每次调用函数时都会在栈顶部创建一个栈帧,函数返回后,栈帧空间会被释放。
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- **指令空间**:用于保存编译后的程序指令,在实际统计中通常忽略不计。
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2022-11-22 17:47:26 +08:00
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2023-12-02 06:21:34 +08:00
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在分析一段程序的空间复杂度时,**我们通常统计暂存数据、栈帧空间和输出数据三部分**,如下图所示。
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2022-11-22 17:47:26 +08:00
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2023-02-26 18:18:34 +08:00
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![算法使用的相关空间](space_complexity.assets/space_types.png)
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2022-11-22 17:47:26 +08:00
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2023-12-02 06:21:34 +08:00
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相关代码如下:
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2023-09-04 03:19:08 +08:00
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=== "Python"
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```python title=""
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class Node:
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"""类"""
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def __init__(self, x: int):
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2023-09-20 01:53:44 +08:00
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self.val: int = x # 节点值
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2023-09-19 14:41:02 +08:00
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self.next: Node | None = None # 指向下一节点的引用
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2023-09-04 03:19:08 +08:00
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def function() -> int:
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"""函数"""
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# 执行某些操作...
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return 0
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def algorithm(n) -> int: # 输入数据
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A = 0 # 暂存数据(常量,一般用大写字母表示)
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b = 0 # 暂存数据(变量)
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node = Node(0) # 暂存数据(对象)
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c = function() # 栈帧空间(调用函数)
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return A + b + c # 输出数据
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```
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=== "C++"
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```cpp title=""
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/* 结构体 */
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struct Node {
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int val;
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Node *next;
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Node(int x) : val(x), next(nullptr) {}
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};
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/* 函数 */
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int func() {
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// 执行某些操作...
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return 0;
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}
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int algorithm(int n) { // 输入数据
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const int a = 0; // 暂存数据(常量)
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int b = 0; // 暂存数据(变量)
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Node* node = new Node(0); // 暂存数据(对象)
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int c = func(); // 栈帧空间(调用函数)
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return a + b + c; // 输出数据
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}
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```
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2022-11-22 17:47:26 +08:00
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=== "Java"
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```java title=""
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/* 类 */
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class Node {
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int val;
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Node next;
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Node(int x) { val = x; }
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}
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2022-12-26 13:15:09 +08:00
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/* 函数 */
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2022-11-22 17:47:26 +08:00
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int function() {
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2023-08-21 03:06:53 +08:00
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// 执行某些操作...
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2022-11-22 17:47:26 +08:00
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return 0;
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}
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int algorithm(int n) { // 输入数据
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final int a = 0; // 暂存数据(常量)
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int b = 0; // 暂存数据(变量)
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Node node = new Node(0); // 暂存数据(对象)
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int c = function(); // 栈帧空间(调用函数)
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return a + b + c; // 输出数据
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}
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```
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2023-09-04 03:19:08 +08:00
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=== "C#"
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2022-11-22 17:47:26 +08:00
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2023-09-04 03:19:08 +08:00
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```csharp title=""
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/* 类 */
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2023-11-26 23:18:44 +08:00
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class Node(int x) {
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int val = x;
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2023-09-04 03:19:08 +08:00
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Node next;
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}
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2022-11-27 04:20:30 +08:00
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2022-12-26 13:15:09 +08:00
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/* 函数 */
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2023-10-08 01:33:46 +08:00
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int Function() {
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2023-08-21 03:06:53 +08:00
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// 执行某些操作...
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2022-11-27 04:20:30 +08:00
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return 0;
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}
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2022-11-22 17:47:26 +08:00
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2023-10-08 01:33:46 +08:00
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int Algorithm(int n) { // 输入数据
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2022-11-27 04:20:30 +08:00
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const int a = 0; // 暂存数据(常量)
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int b = 0; // 暂存数据(变量)
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2023-11-26 23:18:44 +08:00
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Node node = new(0); // 暂存数据(对象)
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2023-10-08 01:33:46 +08:00
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int c = Function(); // 栈帧空间(调用函数)
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2022-11-27 04:20:30 +08:00
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return a + b + c; // 输出数据
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}
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2022-11-22 17:47:26 +08:00
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```
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2022-12-03 01:31:29 +08:00
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=== "Go"
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```go title=""
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2022-12-15 16:41:47 +08:00
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/* 结构体 */
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2023-01-08 20:29:13 +08:00
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type node struct {
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2022-12-15 16:41:47 +08:00
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val int
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2023-01-08 20:29:13 +08:00
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next *node
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2022-12-15 16:41:47 +08:00
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}
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2023-01-08 20:29:13 +08:00
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/* 创建 node 结构体 */
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func newNode(val int) *node {
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return &node{val: val}
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2022-12-15 16:41:47 +08:00
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}
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2022-12-26 13:15:09 +08:00
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/* 函数 */
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2022-12-15 16:41:47 +08:00
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func function() int {
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2023-08-21 03:06:53 +08:00
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// 执行某些操作...
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2022-12-15 16:41:47 +08:00
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return 0
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}
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2022-12-03 01:31:29 +08:00
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2022-12-15 16:41:47 +08:00
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func algorithm(n int) int { // 输入数据
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const a = 0 // 暂存数据(常量)
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b := 0 // 暂存数据(变量)
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newNode(0) // 暂存数据(对象)
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c := function() // 栈帧空间(调用函数)
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return a + b + c // 输出数据
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}
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2022-12-03 01:31:29 +08:00
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```
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2023-09-04 03:19:08 +08:00
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=== "Swift"
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```swift title=""
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/* 类 */
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class Node {
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var val: Int
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var next: Node?
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init(x: Int) {
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val = x
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}
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}
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/* 函数 */
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func function() -> Int {
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// 执行某些操作...
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return 0
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}
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func algorithm(n: Int) -> Int { // 输入数据
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let a = 0 // 暂存数据(常量)
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var b = 0 // 暂存数据(变量)
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let node = Node(x: 0) // 暂存数据(对象)
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let c = function() // 栈帧空间(调用函数)
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return a + b + c // 输出数据
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}
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```
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2023-07-26 15:35:38 +08:00
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=== "JS"
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2022-12-03 01:31:29 +08:00
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2023-02-08 04:27:55 +08:00
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```javascript title=""
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2023-02-06 01:24:22 +08:00
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/* 类 */
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class Node {
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val;
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next;
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constructor(val) {
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2023-04-09 04:32:17 +08:00
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this.val = val === undefined ? 0 : val; // 节点值
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this.next = null; // 指向下一节点的引用
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2023-02-06 01:24:22 +08:00
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}
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}
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/* 函数 */
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function constFunc() {
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2023-08-21 03:06:53 +08:00
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// 执行某些操作
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2023-02-06 01:24:22 +08:00
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return 0;
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}
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2022-12-03 01:31:29 +08:00
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2023-02-06 01:24:22 +08:00
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function algorithm(n) { // 输入数据
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const a = 0; // 暂存数据(常量)
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2023-03-13 22:53:53 +08:00
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let b = 0; // 暂存数据(变量)
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2023-02-06 01:24:22 +08:00
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const node = new Node(0); // 暂存数据(对象)
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const c = constFunc(); // 栈帧空间(调用函数)
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return a + b + c; // 输出数据
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}
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2022-12-03 01:31:29 +08:00
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```
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2023-07-26 15:35:38 +08:00
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=== "TS"
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2022-12-03 01:31:29 +08:00
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```typescript title=""
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2023-02-06 01:24:22 +08:00
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/* 类 */
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class Node {
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val: number;
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next: Node | null;
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constructor(val?: number) {
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2023-04-09 04:32:17 +08:00
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this.val = val === undefined ? 0 : val; // 节点值
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this.next = null; // 指向下一节点的引用
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2023-02-06 01:24:22 +08:00
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}
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}
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/* 函数 */
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function constFunc(): number {
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2023-08-21 03:06:53 +08:00
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// 执行某些操作
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2023-02-06 01:24:22 +08:00
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return 0;
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}
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2022-12-03 01:31:29 +08:00
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2023-02-06 01:24:22 +08:00
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function algorithm(n: number): number { // 输入数据
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const a = 0; // 暂存数据(常量)
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2023-03-13 22:53:53 +08:00
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let b = 0; // 暂存数据(变量)
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2023-02-06 01:24:22 +08:00
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const node = new Node(0); // 暂存数据(对象)
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const c = constFunc(); // 栈帧空间(调用函数)
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return a + b + c; // 输出数据
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}
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2022-12-03 01:31:29 +08:00
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```
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2023-06-02 02:40:26 +08:00
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=== "Dart"
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```dart title=""
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2023-06-02 14:56:29 +08:00
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/* 类 */
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class Node {
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int val;
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Node next;
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Node(this.val, [this.next]);
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}
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/* 函数 */
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int function() {
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2023-08-21 03:06:53 +08:00
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// 执行某些操作...
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2023-06-02 14:56:29 +08:00
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return 0;
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}
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2023-06-02 02:40:26 +08:00
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2023-06-02 14:56:29 +08:00
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int algorithm(int n) { // 输入数据
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const int a = 0; // 暂存数据(常量)
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int b = 0; // 暂存数据(变量)
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Node node = Node(0); // 暂存数据(对象)
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int c = function(); // 栈帧空间(调用函数)
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return a + b + c; // 输出数据
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}
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2023-06-02 02:40:26 +08:00
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```
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2023-07-26 11:00:53 +08:00
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=== "Rust"
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```rust title=""
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2023-09-06 16:58:20 +08:00
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use std::rc::Rc;
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use std::cell::RefCell;
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|
|
|
|
|
|
|
|
|
/* 结构体 */
|
|
|
|
|
struct Node {
|
|
|
|
|
val: i32,
|
|
|
|
|
next: Option<Rc<RefCell<Node>>>,
|
|
|
|
|
}
|
2023-07-26 11:00:53 +08:00
|
|
|
|
|
2023-09-06 16:58:20 +08:00
|
|
|
|
/* 创建 Node 结构体 */
|
|
|
|
|
impl Node {
|
|
|
|
|
fn new(val: i32) -> Self {
|
|
|
|
|
Self { val: val, next: None }
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 函数 */
|
|
|
|
|
fn function() -> i32 {
|
|
|
|
|
// 执行某些操作...
|
|
|
|
|
return 0;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
fn algorithm(n: i32) -> i32 { // 输入数据
|
|
|
|
|
const a: i32 = 0; // 暂存数据(常量)
|
|
|
|
|
let mut b = 0; // 暂存数据(变量)
|
|
|
|
|
let node = Node::new(0); // 暂存数据(对象)
|
|
|
|
|
let c = function(); // 栈帧空间(调用函数)
|
|
|
|
|
return a + b + c; // 输出数据
|
|
|
|
|
}
|
2023-07-26 11:00:53 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "C"
|
|
|
|
|
|
|
|
|
|
```c title=""
|
|
|
|
|
/* 函数 */
|
|
|
|
|
int func() {
|
|
|
|
|
// 执行某些操作...
|
|
|
|
|
return 0;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
int algorithm(int n) { // 输入数据
|
|
|
|
|
const int a = 0; // 暂存数据(常量)
|
|
|
|
|
int b = 0; // 暂存数据(变量)
|
|
|
|
|
int c = func(); // 栈帧空间(调用函数)
|
|
|
|
|
return a + b + c; // 输出数据
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
2024-03-25 03:15:36 +08:00
|
|
|
|
=== "Kotlin"
|
|
|
|
|
|
|
|
|
|
```kotlin title=""
|
2024-03-28 17:08:39 +08:00
|
|
|
|
/* 类 */
|
|
|
|
|
class Node(var _val: Int) {
|
|
|
|
|
var next: Node? = null
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 函数 */
|
|
|
|
|
fun function(): Int {
|
|
|
|
|
// 执行某些操作...
|
|
|
|
|
return 0
|
|
|
|
|
}
|
2024-03-25 03:15:36 +08:00
|
|
|
|
|
2024-03-28 17:08:39 +08:00
|
|
|
|
fun algorithm(n: Int): Int { // 输入数据
|
|
|
|
|
val a = 0 // 暂存数据(常量)
|
|
|
|
|
var b = 0 // 暂存数据(变量)
|
|
|
|
|
val node = Node(0) // 暂存数据(对象)
|
|
|
|
|
val c = function() // 栈帧空间(调用函数)
|
|
|
|
|
return a + b + c // 输出数据
|
|
|
|
|
}
|
2024-03-25 03:15:36 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Zig"
|
|
|
|
|
|
|
|
|
|
```zig title=""
|
|
|
|
|
|
|
|
|
|
```
|
|
|
|
|
|
2023-02-16 03:39:01 +08:00
|
|
|
|
## 推算方法
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-08-20 14:51:39 +08:00
|
|
|
|
空间复杂度的推算方法与时间复杂度大致相同,只需将统计对象从“操作数量”转为“使用空间大小”。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-08-20 23:28:30 +08:00
|
|
|
|
而与时间复杂度不同的是,**我们通常只关注最差空间复杂度**。这是因为内存空间是一项硬性要求,我们必须确保在所有输入数据下都有足够的内存空间预留。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-08-08 23:16:33 +08:00
|
|
|
|
观察以下代码,最差空间复杂度中的“最差”有两层含义。
|
|
|
|
|
|
2023-12-02 06:21:34 +08:00
|
|
|
|
1. **以最差输入数据为准**:当 $n < 10$ 时,空间复杂度为 $O(1)$ ;但当 $n > 10$ 时,初始化的数组 `nums` 占用 $O(n)$ 空间,因此最差空间复杂度为 $O(n)$ 。
|
|
|
|
|
2. **以算法运行中的峰值内存为准**:例如,程序在执行最后一行之前,占用 $O(1)$ 空间;当初始化数组 `nums` 时,程序占用 $O(n)$ 空间,因此最差空间复杂度为 $O(n)$ 。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Python"
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```python title=""
|
|
|
|
|
def algorithm(n: int):
|
|
|
|
|
a = 0 # O(1)
|
|
|
|
|
b = [0] * 10000 # O(1)
|
|
|
|
|
if n > 10:
|
|
|
|
|
nums = [0] * n # O(n)
|
2022-11-22 17:47:26 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C++"
|
|
|
|
|
|
|
|
|
|
```cpp title=""
|
2022-11-27 04:20:30 +08:00
|
|
|
|
void algorithm(int n) {
|
|
|
|
|
int a = 0; // O(1)
|
|
|
|
|
vector<int> b(10000); // O(1)
|
|
|
|
|
if (n > 10)
|
|
|
|
|
vector<int> nums(n); // O(n)
|
|
|
|
|
}
|
2022-11-22 17:47:26 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Java"
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```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#"
|
|
|
|
|
|
|
|
|
|
```csharp title=""
|
2023-10-08 01:33:46 +08:00
|
|
|
|
void Algorithm(int n) {
|
2023-09-04 03:19:08 +08:00
|
|
|
|
int a = 0; // O(1)
|
|
|
|
|
int[] b = new int[10000]; // O(1)
|
|
|
|
|
if (n > 10) {
|
|
|
|
|
int[] nums = new int[n]; // O(n)
|
|
|
|
|
}
|
|
|
|
|
}
|
2022-11-22 17:47:26 +08:00
|
|
|
|
```
|
|
|
|
|
|
2022-12-03 01:31:29 +08:00
|
|
|
|
=== "Go"
|
|
|
|
|
|
|
|
|
|
```go title=""
|
2022-12-15 22:38:18 +08:00
|
|
|
|
func algorithm(n int) {
|
2022-12-30 15:57:15 +08:00
|
|
|
|
a := 0 // O(1)
|
|
|
|
|
b := make([]int, 10000) // O(1)
|
2022-12-15 16:41:47 +08:00
|
|
|
|
var nums []int
|
|
|
|
|
if n > 10 {
|
2022-12-30 15:57:15 +08:00
|
|
|
|
nums := make([]int, n) // O(n)
|
2022-12-15 16:41:47 +08:00
|
|
|
|
}
|
|
|
|
|
fmt.Println(a, b, nums)
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Swift"
|
|
|
|
|
|
|
|
|
|
```swift title=""
|
|
|
|
|
func algorithm(n: Int) {
|
|
|
|
|
let a = 0 // O(1)
|
|
|
|
|
let b = Array(repeating: 0, count: 10000) // O(1)
|
|
|
|
|
if n > 10 {
|
|
|
|
|
let nums = Array(repeating: 0, count: n) // O(n)
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
2023-07-26 15:35:38 +08:00
|
|
|
|
=== "JS"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-02-08 04:27:55 +08:00
|
|
|
|
```javascript title=""
|
2023-02-06 01:24:22 +08:00
|
|
|
|
function algorithm(n) {
|
|
|
|
|
const a = 0; // O(1)
|
|
|
|
|
const b = new Array(10000); // O(1)
|
|
|
|
|
if (n > 10) {
|
|
|
|
|
const nums = new Array(n); // O(n)
|
|
|
|
|
}
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-07-26 15:35:38 +08:00
|
|
|
|
=== "TS"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
|
|
|
|
```typescript title=""
|
2023-02-06 01:24:22 +08:00
|
|
|
|
function algorithm(n: number): void {
|
|
|
|
|
const a = 0; // O(1)
|
|
|
|
|
const b = new Array(10000); // O(1)
|
|
|
|
|
if (n > 10) {
|
|
|
|
|
const nums = new Array(n); // O(n)
|
|
|
|
|
}
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Dart"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```dart title=""
|
2023-04-18 20:21:31 +08:00
|
|
|
|
void algorithm(int n) {
|
2023-09-04 03:19:08 +08:00
|
|
|
|
int a = 0; // O(1)
|
|
|
|
|
List<int> b = List.filled(10000, 0); // O(1)
|
|
|
|
|
if (n > 10) {
|
|
|
|
|
List<int> nums = List.filled(n, 0); // O(n)
|
|
|
|
|
}
|
2023-04-18 20:21:31 +08:00
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Rust"
|
|
|
|
|
|
|
|
|
|
```rust title=""
|
2023-09-06 16:58:20 +08:00
|
|
|
|
fn algorithm(n: i32) {
|
|
|
|
|
let a = 0; // O(1)
|
|
|
|
|
let b = [0; 10000]; // O(1)
|
|
|
|
|
if n > 10 {
|
|
|
|
|
let nums = vec![0; n as usize]; // O(n)
|
|
|
|
|
}
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "C"
|
2023-01-01 21:29:45 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```c title=""
|
|
|
|
|
void algorithm(int n) {
|
|
|
|
|
int a = 0; // O(1)
|
|
|
|
|
int b[10000]; // O(1)
|
|
|
|
|
if (n > 10)
|
|
|
|
|
int nums[n] = {0}; // O(n)
|
2023-01-01 21:29:45 +08:00
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
2024-03-25 03:15:36 +08:00
|
|
|
|
=== "Kotlin"
|
|
|
|
|
|
|
|
|
|
```kotlin title=""
|
2024-03-28 17:08:39 +08:00
|
|
|
|
fun algorithm(n: Int) {
|
|
|
|
|
val a = 0 // O(1)
|
|
|
|
|
val b = IntArray(10000) // O(1)
|
|
|
|
|
if (n > 10) {
|
|
|
|
|
val nums = IntArray(n) // O(n)
|
|
|
|
|
}
|
|
|
|
|
}
|
2024-03-25 03:15:36 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-02-01 22:03:04 +08:00
|
|
|
|
=== "Zig"
|
|
|
|
|
|
|
|
|
|
```zig title=""
|
|
|
|
|
|
|
|
|
|
```
|
|
|
|
|
|
2023-12-02 06:21:34 +08:00
|
|
|
|
**在递归函数中,需要注意统计栈帧空间**。观察以下代码:
|
2023-06-02 02:40:26 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Python"
|
2023-07-26 11:00:53 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```python title=""
|
|
|
|
|
def function() -> int:
|
|
|
|
|
# 执行某些操作
|
|
|
|
|
return 0
|
|
|
|
|
|
|
|
|
|
def loop(n: int):
|
2023-12-28 18:06:09 +08:00
|
|
|
|
"""循环的空间复杂度为 O(1)"""
|
2023-09-04 03:19:08 +08:00
|
|
|
|
for _ in range(n):
|
|
|
|
|
function()
|
2023-07-26 11:00:53 +08:00
|
|
|
|
|
2024-03-25 01:53:21 +08:00
|
|
|
|
def recur(n: int):
|
2023-12-28 18:06:09 +08:00
|
|
|
|
"""递归的空间复杂度为 O(n)"""
|
2024-03-25 01:53:21 +08:00
|
|
|
|
if n == 1:
|
|
|
|
|
return
|
2023-09-04 03:19:08 +08:00
|
|
|
|
return recur(n - 1)
|
2023-07-26 11:00:53 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "C++"
|
2023-08-08 23:16:33 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```cpp title=""
|
|
|
|
|
int func() {
|
|
|
|
|
// 执行某些操作
|
|
|
|
|
return 0;
|
|
|
|
|
}
|
|
|
|
|
/* 循环 O(1) */
|
|
|
|
|
void loop(int n) {
|
|
|
|
|
for (int i = 0; i < n; i++) {
|
|
|
|
|
func();
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
/* 递归 O(n) */
|
|
|
|
|
void recur(int n) {
|
|
|
|
|
if (n == 1) return;
|
|
|
|
|
return recur(n - 1);
|
|
|
|
|
}
|
|
|
|
|
```
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
|
|
|
|
=== "Java"
|
|
|
|
|
|
|
|
|
|
```java title=""
|
|
|
|
|
int function() {
|
2023-08-21 03:06:53 +08:00
|
|
|
|
// 执行某些操作
|
2022-11-22 17:47:26 +08:00
|
|
|
|
return 0;
|
|
|
|
|
}
|
2022-11-25 20:12:20 +08:00
|
|
|
|
/* 循环 O(1) */
|
2022-11-22 17:47:26 +08:00
|
|
|
|
void loop(int n) {
|
|
|
|
|
for (int i = 0; i < n; i++) {
|
|
|
|
|
function();
|
|
|
|
|
}
|
|
|
|
|
}
|
2022-11-25 20:12:20 +08:00
|
|
|
|
/* 递归 O(n) */
|
2022-11-22 17:47:26 +08:00
|
|
|
|
void recur(int n) {
|
|
|
|
|
if (n == 1) return;
|
|
|
|
|
return recur(n - 1);
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "C#"
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```csharp title=""
|
2023-10-08 01:33:46 +08:00
|
|
|
|
int Function() {
|
2023-08-21 03:06:53 +08:00
|
|
|
|
// 执行某些操作
|
2022-11-27 04:20:30 +08:00
|
|
|
|
return 0;
|
|
|
|
|
}
|
|
|
|
|
/* 循环 O(1) */
|
2023-10-08 01:33:46 +08:00
|
|
|
|
void Loop(int n) {
|
2022-11-27 04:20:30 +08:00
|
|
|
|
for (int i = 0; i < n; i++) {
|
2023-10-08 01:33:46 +08:00
|
|
|
|
Function();
|
2022-11-27 04:20:30 +08:00
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
/* 递归 O(n) */
|
2023-10-08 01:33:46 +08:00
|
|
|
|
int Recur(int n) {
|
2023-09-04 03:19:08 +08:00
|
|
|
|
if (n == 1) return 1;
|
2023-10-08 01:33:46 +08:00
|
|
|
|
return Recur(n - 1);
|
2022-11-27 04:20:30 +08:00
|
|
|
|
}
|
2022-11-22 17:47:26 +08:00
|
|
|
|
```
|
|
|
|
|
|
2022-12-03 01:31:29 +08:00
|
|
|
|
=== "Go"
|
|
|
|
|
|
|
|
|
|
```go title=""
|
2022-12-15 16:41:47 +08:00
|
|
|
|
func function() int {
|
2023-08-21 03:06:53 +08:00
|
|
|
|
// 执行某些操作
|
2022-12-15 16:41:47 +08:00
|
|
|
|
return 0
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 循环 O(1) */
|
|
|
|
|
func loop(n int) {
|
|
|
|
|
for i := 0; i < n; i++ {
|
|
|
|
|
function()
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 递归 O(n) */
|
|
|
|
|
func recur(n int) {
|
|
|
|
|
if n == 1 {
|
|
|
|
|
return
|
|
|
|
|
}
|
|
|
|
|
recur(n - 1)
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Swift"
|
|
|
|
|
|
|
|
|
|
```swift title=""
|
|
|
|
|
@discardableResult
|
|
|
|
|
func function() -> Int {
|
|
|
|
|
// 执行某些操作
|
|
|
|
|
return 0
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 循环 O(1) */
|
|
|
|
|
func loop(n: Int) {
|
|
|
|
|
for _ in 0 ..< n {
|
|
|
|
|
function()
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 递归 O(n) */
|
|
|
|
|
func recur(n: Int) {
|
|
|
|
|
if n == 1 {
|
|
|
|
|
return
|
|
|
|
|
}
|
|
|
|
|
recur(n: n - 1)
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
2023-07-26 15:35:38 +08:00
|
|
|
|
=== "JS"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-02-08 04:27:55 +08:00
|
|
|
|
```javascript title=""
|
2023-02-06 01:24:22 +08:00
|
|
|
|
function constFunc() {
|
2023-08-21 03:06:53 +08:00
|
|
|
|
// 执行某些操作
|
2023-02-06 01:24:22 +08:00
|
|
|
|
return 0;
|
|
|
|
|
}
|
|
|
|
|
/* 循环 O(1) */
|
|
|
|
|
function loop(n) {
|
|
|
|
|
for (let i = 0; i < n; i++) {
|
|
|
|
|
constFunc();
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
/* 递归 O(n) */
|
|
|
|
|
function recur(n) {
|
|
|
|
|
if (n === 1) return;
|
|
|
|
|
return recur(n - 1);
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-07-26 15:35:38 +08:00
|
|
|
|
=== "TS"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
|
|
|
|
```typescript title=""
|
2023-02-06 01:24:22 +08:00
|
|
|
|
function constFunc(): number {
|
2023-08-21 03:06:53 +08:00
|
|
|
|
// 执行某些操作
|
2023-02-06 01:24:22 +08:00
|
|
|
|
return 0;
|
|
|
|
|
}
|
|
|
|
|
/* 循环 O(1) */
|
|
|
|
|
function loop(n: number): void {
|
|
|
|
|
for (let i = 0; i < n; i++) {
|
|
|
|
|
constFunc();
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
/* 递归 O(n) */
|
|
|
|
|
function recur(n: number): void {
|
|
|
|
|
if (n === 1) return;
|
|
|
|
|
return recur(n - 1);
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Dart"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```dart title=""
|
|
|
|
|
int function() {
|
|
|
|
|
// 执行某些操作
|
|
|
|
|
return 0;
|
2023-04-18 20:21:31 +08:00
|
|
|
|
}
|
|
|
|
|
/* 循环 O(1) */
|
|
|
|
|
void loop(int n) {
|
2023-09-04 03:19:08 +08:00
|
|
|
|
for (int i = 0; i < n; i++) {
|
|
|
|
|
function();
|
|
|
|
|
}
|
2023-04-18 20:21:31 +08:00
|
|
|
|
}
|
|
|
|
|
/* 递归 O(n) */
|
|
|
|
|
void recur(int n) {
|
2023-09-04 03:19:08 +08:00
|
|
|
|
if (n == 1) return;
|
|
|
|
|
return recur(n - 1);
|
2023-04-18 20:21:31 +08:00
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
=== "Rust"
|
2022-12-03 01:31:29 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```rust title=""
|
2023-09-06 16:58:20 +08:00
|
|
|
|
fn function() -> i32 {
|
|
|
|
|
// 执行某些操作
|
|
|
|
|
return 0;
|
|
|
|
|
}
|
|
|
|
|
/* 循环 O(1) */
|
|
|
|
|
fn loop(n: i32) {
|
|
|
|
|
for i in 0..n {
|
|
|
|
|
function();
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
/* 递归 O(n) */
|
2024-03-31 03:06:41 +08:00
|
|
|
|
fn recur(n: i32) {
|
2023-09-06 16:58:20 +08:00
|
|
|
|
if n == 1 {
|
|
|
|
|
return;
|
|
|
|
|
}
|
|
|
|
|
recur(n - 1);
|
|
|
|
|
}
|
2023-09-04 03:19:08 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C"
|
|
|
|
|
|
|
|
|
|
```c title=""
|
|
|
|
|
int func() {
|
2023-08-21 03:06:53 +08:00
|
|
|
|
// 执行某些操作
|
2022-12-23 15:42:02 +08:00
|
|
|
|
return 0;
|
|
|
|
|
}
|
|
|
|
|
/* 循环 O(1) */
|
2023-07-24 19:50:00 +08:00
|
|
|
|
void loop(int n) {
|
|
|
|
|
for (int i = 0; i < n; i++) {
|
2023-09-04 03:19:08 +08:00
|
|
|
|
func();
|
2022-12-23 15:42:02 +08:00
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
/* 递归 O(n) */
|
2023-09-04 03:19:08 +08:00
|
|
|
|
void recur(int n) {
|
|
|
|
|
if (n == 1) return;
|
2022-12-23 15:42:02 +08:00
|
|
|
|
return recur(n - 1);
|
|
|
|
|
}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2024-03-25 03:15:36 +08:00
|
|
|
|
=== "Kotlin"
|
|
|
|
|
|
|
|
|
|
```kotlin title=""
|
2024-03-28 17:08:39 +08:00
|
|
|
|
fun function(): Int {
|
|
|
|
|
// 执行某些操作
|
|
|
|
|
return 0
|
|
|
|
|
}
|
|
|
|
|
/* 循环 O(1) */
|
|
|
|
|
fun loop(n: Int) {
|
|
|
|
|
for (i in 0..<n) {
|
|
|
|
|
function()
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
/* 递归 O(n) */
|
|
|
|
|
fun recur(n: Int) {
|
|
|
|
|
if (n == 1) return
|
|
|
|
|
return recur(n - 1)
|
|
|
|
|
}
|
2024-03-25 03:15:36 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-02-01 22:03:04 +08:00
|
|
|
|
=== "Zig"
|
|
|
|
|
|
|
|
|
|
```zig title=""
|
|
|
|
|
|
|
|
|
|
```
|
|
|
|
|
|
2023-12-02 06:21:34 +08:00
|
|
|
|
函数 `loop()` 和 `recur()` 的时间复杂度都为 $O(n)$ ,但空间复杂度不同。
|
|
|
|
|
|
|
|
|
|
- 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 $O(1)$ 。
|
|
|
|
|
- 递归函数 `recur()` 在运行过程中会同时存在 $n$ 个未返回的 `recur()` ,从而占用 $O(n)$ 的栈帧空间。
|
|
|
|
|
|
2023-02-16 03:39:01 +08:00
|
|
|
|
## 常见类型
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-08-20 14:51:39 +08:00
|
|
|
|
设输入数据大小为 $n$ ,下图展示了常见的空间复杂度类型(从低到高排列)。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
|
|
|
|
$$
|
|
|
|
|
\begin{aligned}
|
|
|
|
|
O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
|
|
|
|
\text{常数阶} < \text{对数阶} < \text{线性阶} < \text{平方阶} < \text{指数阶}
|
|
|
|
|
\end{aligned}
|
|
|
|
|
$$
|
|
|
|
|
|
2023-08-20 14:51:39 +08:00
|
|
|
|
![常见的空间复杂度类型](space_complexity.assets/space_complexity_common_types.png)
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2024-03-23 21:04:14 +08:00
|
|
|
|
### 常数阶 $O(1)$ {data-toc-label="常数阶"}
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
|
|
|
|
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。
|
|
|
|
|
|
2023-08-21 19:33:45 +08:00
|
|
|
|
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,因此不会累积占用空间,空间复杂度仍为 $O(1)$ :
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-10-17 01:06:00 +08:00
|
|
|
|
```src
|
|
|
|
|
[file]{space_complexity}-[class]{}-[func]{constant}
|
|
|
|
|
```
|
2023-07-26 11:00:53 +08:00
|
|
|
|
|
2024-03-23 21:04:14 +08:00
|
|
|
|
### 线性阶 $O(n)$ {data-toc-label="线性阶"}
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-08-20 14:51:39 +08:00
|
|
|
|
线性阶常见于元素数量与 $n$ 成正比的数组、链表、栈、队列等:
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-10-17 01:06:00 +08:00
|
|
|
|
```src
|
|
|
|
|
[file]{space_complexity}-[class]{}-[func]{linear}
|
|
|
|
|
```
|
2023-02-01 22:03:04 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
如下图所示,此函数的递归深度为 $n$ ,即同时存在 $n$ 个未返回的 `linear_recur()` 函数,使用 $O(n)$ 大小的栈帧空间:
|
2023-06-02 02:40:26 +08:00
|
|
|
|
|
2023-10-17 01:06:00 +08:00
|
|
|
|
```src
|
|
|
|
|
[file]{space_complexity}-[class]{}-[func]{linear_recur}
|
|
|
|
|
```
|
2023-02-01 22:03:04 +08:00
|
|
|
|
|
2023-02-26 18:18:34 +08:00
|
|
|
|
![递归函数产生的线性阶空间复杂度](space_complexity.assets/space_complexity_recursive_linear.png)
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2024-03-23 21:04:14 +08:00
|
|
|
|
### 平方阶 $O(n^2)$ {data-toc-label="平方阶"}
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-08-20 14:51:39 +08:00
|
|
|
|
平方阶常见于矩阵和图,元素数量与 $n$ 成平方关系:
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-10-17 01:06:00 +08:00
|
|
|
|
```src
|
|
|
|
|
[file]{space_complexity}-[class]{}-[func]{quadratic}
|
|
|
|
|
```
|
2023-02-01 22:03:04 +08:00
|
|
|
|
|
2023-09-04 03:19:08 +08:00
|
|
|
|
如下图所示,该函数的递归深度为 $n$ ,在每个递归函数中都初始化了一个数组,长度分别为 $n$、$n-1$、$\dots$、$2$、$1$ ,平均长度为 $n / 2$ ,因此总体占用 $O(n^2)$ 空间:
|
2023-06-02 02:40:26 +08:00
|
|
|
|
|
2023-10-17 01:06:00 +08:00
|
|
|
|
```src
|
|
|
|
|
[file]{space_complexity}-[class]{}-[func]{quadratic_recur}
|
|
|
|
|
```
|
2023-02-01 22:03:04 +08:00
|
|
|
|
|
2023-02-26 18:18:34 +08:00
|
|
|
|
![递归函数产生的平方阶空间复杂度](space_complexity.assets/space_complexity_recursive_quadratic.png)
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2024-03-23 21:04:14 +08:00
|
|
|
|
### 指数阶 $O(2^n)$ {data-toc-label="指数阶"}
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-12-14 02:42:53 +08:00
|
|
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指数阶常见于二叉树。观察下图,层数为 $n$ 的“满二叉树”的节点数量为 $2^n - 1$ ,占用 $O(2^n)$ 空间:
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2022-11-22 17:47:26 +08:00
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2023-10-17 01:06:00 +08:00
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```src
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[file]{space_complexity}-[class]{}-[func]{build_tree}
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```
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2023-02-26 18:18:34 +08:00
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![满二叉树产生的指数阶空间复杂度](space_complexity.assets/space_complexity_exponential.png)
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2024-03-23 21:04:14 +08:00
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### 对数阶 $O(\log n)$ {data-toc-label="对数阶"}
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2023-12-02 06:21:34 +08:00
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对数阶常见于分治算法。例如归并排序,输入长度为 $n$ 的数组,每轮递归将数组从中点处划分为两半,形成高度为 $\log n$ 的递归树,使用 $O(\log n)$ 栈帧空间。
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2022-11-22 17:47:26 +08:00
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2024-01-25 12:50:53 +08:00
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再例如将数字转化为字符串,输入一个正整数 $n$ ,它的位数为 $\lfloor \log_{10} n \rfloor + 1$ ,即对应字符串长度为 $\lfloor \log_{10} n \rfloor + 1$ ,因此空间复杂度为 $O(\log_{10} n + 1) = O(\log n)$ 。
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2023-04-17 18:22:18 +08:00
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## 权衡时间与空间
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2023-12-02 06:21:34 +08:00
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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达到最优。然而在实际情况中,同时优化时间复杂度和空间复杂度通常非常困难。
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2023-04-17 18:22:18 +08:00
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**降低时间复杂度通常需要以提升空间复杂度为代价,反之亦然**。我们将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,则称为“以时间换空间”。
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2023-12-02 06:21:34 +08:00
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选择哪种思路取决于我们更看重哪个方面。在大多数情况下,时间比空间更宝贵,因此“以空间换时间”通常是更常用的策略。当然,在数据量很大的情况下,控制空间复杂度也非常重要。
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