github/hello-algo
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Create csharp time_complexity.cs

Browse Source 
Update csharp time_complexity.md
This commit is contained in:
SayoKun 2022-12-19 11:11:55 +08:00
parent 5baa642a36
commit a9ddc70da5
3 changed files with 499 additions and 15 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

2
.gitignore vendored
View File

@ -13,3 +13,5 @@ docs/overrides/
# python files
__pycache__
/.vs
/codes/csharp/.cr/personal/FavoritesList

277
codes/csharp/chapter_computational_complexity/time_complexity.cs Normal file
View File

@ -0,0 +1,277 @@
// File: LinkedList.cs
// Created Time: 2022-12-19
// Author: SayoKun (373484252@qq.com)
using System;
using System.Linq;
namespace hello_algo.chapter_computational_complexity
{
public class time_complexity
{
/// <summary>
/// 常数阶
/// </summary>
/// <param name="n"></param>
/// <returns></returns>
int constant(int n)
{
int count = 0;
int size = 100000;
for (int i = 0; i < size; i++)
count++;
return count;
}
/// <summary>
/// 线性阶
/// </summary>
/// <param name="n"></param>
/// <returns></returns>
int linear(int n)
{
int count = 0;
for (int i = 0; i < n; i++)
count++;
return count;
}
/// <summary>
/// 线性阶（遍历数组）
/// </summary>
/// <param name="nums"></param>
/// <returns></returns>
int arrayTraversal(int[] nums)
{
int count = 0;
// 循环次数与数组长度成正比
foreach (int num in nums)
{
count++;
}
return count;
}
/// <summary>
/// 平方阶
/// </summary>
/// <param name="n"></param>
/// <returns></returns>
int quadratic(int n)
{
int count = 0;
// 循环次数与数组长度成平方关系
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
count++;
}
}
return count;
}
/// <summary>
/// 平方阶（冒泡排序）
/// </summary>
/// <param name="nums"></param>
/// <returns></returns>
int bubbleSort(int[] nums)
{
int count = 0; // 计数器
// 外循环：待排序元素数量为 n-1, n-2, ..., 1
for (int i = nums.Length - 1; i > 0; i--)
{
// 内循环：冒泡操作
for (int j = 0; j < i; j++)
{
if (nums[j] > nums[j + 1])
{
// 交换 nums[j] 与 nums[j + 1]
int tmp = nums[j];
nums[j] = nums[j + 1];
nums[j + 1] = tmp;
count += 3; // 元素交换包含 3 个单元操作
}
}
}
return count;
}
/// <summary>
/// 指数阶（循环实现）
/// </summary>
/// <param name="n"></param>
/// <returns></returns>
int exponential(int n)
{
int count = 0, baseNum = 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 < baseNum; j++)
{
count++;
}
baseNum *= 2;
}
// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
return count;
}
/// <summary>
/// 指数阶（递归实现）
/// </summary>
/// <param name="n"></param>
/// <returns></returns>
int expRecur(int n)
{
if (n == 1) return 1;
return expRecur(n - 1) + expRecur(n - 1) + 1;
}
/// <summary>
/// 对数阶（循环实现）
/// </summary>
/// <param name="n"></param>
/// <returns></returns>
int logarithmic(float n)
{
int count = 0;
while (n > 1)
{
n = n / 2;
count++;
}
return count;
}
/// <summary>
/// 对数阶（递归实现）
/// </summary>
/// <param name="n"></param>
/// <returns></returns>
int logRecur(float n)
{
if (n <= 1) return 0;
return logRecur(n / 2) + 1;
}
/// <summary>
/// 线性对数阶
/// </summary>
/// <param name="n"></param>
/// <returns></returns>
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;
}
/// <summary>
/// 阶乘阶（递归实现）
/// </summary>
/// <param name="n">递归数</param>
/// <returns></returns>
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;
}
/// <summary>
/// 生成一个数组，元素为 { 1, 2, ..., n }，顺序被打乱
/// </summary>
/// <param name="n">数组大小</param>
/// <returns></returns>
int[] randomNumbers(int n)
{
int[] nums = new int[n];
// 生成数组 nums = { 1, 2, 3, ..., n }
for (int i = 0; i < n; i++)
{
nums[i] = i + 1;
}
// 随机打乱数组元素
nums = nums.OrderBy(num => System.Random.Shared.Next()).ToArray();
return nums;
}
/// <summary>
/// 查找数组 nums 中数字 1 所在索引
/// </summary>
/// <param name="nums">索引数组</param>
/// <returns></returns>
int findOne(in Span<int> nums) => nums.IndexOf(1);
void worstBestTimeComplexity()
{
for (int i = 0; i < 10; i++)
{
int n = 100;
int[] nums = randomNumbers(n);
int index = findOne(nums);
System.Console.WriteLine($"打乱后的数组为 [{string.Join(",", nums)}]");
System.Console.WriteLine($"数字 1 的索引为 [{index}]");
}
}
/// <summary>
/// Driver Code
/// </summary>
public void main()
{
// 可以修改 n 运行，体会一下各种复杂度的操作数量变化趋势
int n = 8;
System.Console.WriteLine("输入数据大小 n = " + n);
int count = constant(n);
System.Console.WriteLine("常数阶的计算操作数量 = " + count);
count = linear(n);
System.Console.WriteLine("线性阶的计算操作数量 = " + count);
count = arrayTraversal(new int[n]);
System.Console.WriteLine("线性阶（遍历数组）的计算操作数量 = " + count);
count = quadratic(n);
System.Console.WriteLine("平方阶的计算操作数量 = " + count);
int[] nums = new int[n];
for (int i = 0; i < n; i++)
nums[i] = n - i; // [n,n-1,...,2,1]
count = bubbleSort(nums);
System.Console.WriteLine("平方阶（冒泡排序）的计算操作数量 = " + count);
count = exponential(n);
System.Console.WriteLine("指数阶（循环实现）的计算操作数量 = " + count);
count = expRecur(n);
System.Console.WriteLine("指数阶（递归实现）的计算操作数量 = " + count);
count = logarithmic((float)n);
System.Console.WriteLine("对数阶（循环实现）的计算操作数量 = " + count);
count = logRecur((float)n);
System.Console.WriteLine("对数阶（递归实现）的计算操作数量 = " + count);
count = linearLogRecur((float)n);
System.Console.WriteLine("线性对数阶（递归实现）的计算操作数量 = " + count);
count = factorialRecur(n);
System.Console.WriteLine("阶乘阶（递归实现）的计算操作数量 = " + count);
}
}
}

235
docs/chapter_computational_complexity/time_complexity.md
View File

@ -97,7 +97,18 @@ $$
=== "C#"
```csharp title=""
// 在某运行平台下
void algorithm(int n)
{
int a = 2; // 1 ns
a = a + 1; // 1 ns
a = a * 2; // 10 ns
// 循环 n 次
for (int i = 0; i < n; i++)
{ // 1 ns ，每轮都要执行 i++
System.Console.WriteLine(0); // 5 ns
}
}
```
但实际上， **统计算法的运行时间既不合理也不现实。** 首先，我们不希望预估时间和运行平台绑定，毕竟算法需要跑在各式各样的平台之上。其次，我们很难获知每一种操作的运行时间，这为预估过程带来了极大的难度。
@ -212,7 +223,27 @@ $$
=== "C#"
```csharp title=""
// 算法 A 时间复杂度：常数阶
void algorithm_A(int n)
{
System.Console.WriteLine(n);
}
// 算法 B 时间复杂度：线性阶
void algorithm_B(int n)
{
for (int i = 0; i < n; i++)
{
System.Console.WriteLine(i);
}
}
// 算法 C 时间复杂度：常数阶
void algorithm_C(int n)
{
for (int i = 0; i < 1000000; i++)
{
System.Console.WriteLine(i);
}
}
```
![time_complexity_first_example](time_complexity.assets/time_complexity_first_example.png)
@ -310,7 +341,17 @@ $$
=== "C#"
```csharp title=""
void algorithm(int n)
{
int a = 1; // +1
a = a + 1; // +1
a = a * 2; // +1
// 循环 n 次
for (int i = 0; i < n; i++)
{ // +1（每轮都执行 i ++）
System.Console.WriteLine(0); // +1
}
}
```
$T(n)$ 是个一次函数，说明时间增长趋势是线性的，因此易得时间复杂度是线性阶。
@ -457,7 +498,24 @@ $$
=== "C#"
```csharp 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.Console.WriteLine(0);
}
// +n*n（技巧 3）
for (int i = 0; i < 2 * n; i++)
{
for (int j = 0; j < n + 1; j++)
{
System.Console.WriteLine(0);
}
}
}
```
### 2. 判断渐近上界
@ -576,7 +634,15 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 常数阶 */
int constant(int n)
{
int count = 0;
int size = 100000;
for (int i = 0; i < size; i++)
count++;
return count;
}
```
### 线性阶 $O(n)$
@ -652,7 +718,14 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 线性阶 */
int linear(int n)
{
int count = 0;
for (int i = 0; i < n; i++)
count++;
return count;
}
```
「遍历数组」和「遍历链表」等操作，时间复杂度都为 $O(n)$ ，其中 $n$ 为数组或链表的长度。
@ -736,7 +809,17 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 线性阶（遍历数组） */
int arrayTraversal(int[] nums)
{
int count = 0;
// 循环次数与数组长度成正比
foreach (int num in nums)
{
count++;
}
return count;
}
```
### 平方阶 $O(n^2)$
@ -825,7 +908,20 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 平方阶 */
int quadratic(int n)
{
int count = 0;
// 循环次数与数组长度成平方关系
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
count++;
}
}
return count;
}
```
![time_complexity_constant_linear_quadratic](time_complexity.assets/time_complexity_constant_linear_quadratic.png)
@ -947,7 +1043,28 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 平方阶（冒泡排序） */
int bubbleSort(int[] nums)
{
int count = 0; // 计数器
// 外循环：待排序元素数量为 n-1, n-2, ..., 1
for (int i = nums.Length - 1; i > 0; i--)
{
// 内循环：冒泡操作
for (int j = 0; j < i; j++)
{
if (nums[j] > nums[j + 1])
{
// 交换 nums[j] 与 nums[j + 1]
int tmp = nums[j];
nums[j] = nums[j + 1];
nums[j + 1] = tmp;
count += 3; // 元素交换包含 3 个单元操作
}
}
}
return count;
}
```
### 指数阶 $O(2^n)$
@ -1048,7 +1165,22 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 指数阶（循环实现） */
int exponential(int n)
{
int count = 0, baseNum = 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 < baseNum; j++)
{
count++;
}
baseNum *= 2;
}
// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
return count;
}
```
![time_complexity_exponential](time_complexity.assets/time_complexity_exponential.png)
@ -1119,7 +1251,12 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 指数阶（递归实现） */
int expRecur(int n)
{
if (n == 1) return 1;
return expRecur(n - 1) + expRecur(n - 1) + 1;
}
```
### 对数阶 $O(\log n)$
@ -1205,7 +1342,17 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 对数阶（循环实现） */
int logarithmic(float n)
{
int count = 0;
while (n > 1)
{
n = n / 2;
count++;
}
return count;
}
```
![time_complexity_logarithmic](time_complexity.assets/time_complexity_logarithmic.png)
@ -1276,7 +1423,12 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 对数阶（递归实现） */
int logRecur(float n)
{
if (n <= 1) return 0;
return logRecur(n / 2) + 1;
}
```
### 线性对数阶 $O(n \log n)$
@ -1366,7 +1518,18 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 线性对数阶 */
int linearLogRecur(float n)
{
if (n <= 1) return 1;
int count = linearLogRecur(n / 2) +
linearLogRecur(n / 2);
for (int i = 0; i < n; i++)
{
count++;
}
return count;
}
```
![time_complexity_logarithmic_linear](time_complexity.assets/time_complexity_logarithmic_linear.png)
@ -1464,7 +1627,18 @@ $$
=== "C#"
```csharp title="time_complexity.cs"
/* 阶乘阶（递归实现） */
int factorialRecur(int n)
{
if (n == 0) return 1;
int count = 0;
// 从 1 个分裂出 n 个
for (int i = 0; i < n; i++)
{
count += factorialRecur(n - 1);
}
return count;
}
```
![time_complexity_factorial](time_complexity.assets/time_complexity_factorial.png)
@ -1652,7 +1826,38 @@ $$
=== "C#"
```csharp title="worst_best_time_complexity.cs"
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)
{
int[] nums = new int[n];
// 生成数组 nums = { 1, 2, 3, ..., n }
for (int i = 0; i < n; i++)
{
nums[i] = i + 1;
}
// 随机打乱数组元素
nums = nums.OrderBy(num => System.Random.Shared.Next()).ToArray();
return nums;
}
/* 查找数组 nums 中数字 1 所在索引 */
static int findOne(in Span<int> nums) => nums.IndexOf(1);
/* Driver Code */
public static void main()
{
for (int i = 0; i < 10; i++)
{
int n = 100;
int[] nums = randomNumbers(n);
int index = findOne(nums);
System.Console.WriteLine($"打乱后的数组为 [{string.Join(",", nums)}]");
System.Console.WriteLine($"数字 1 的索引为 [{index}]");
}
}
}
```
!!! tip