diff --git a/codes/c/chapter_computational_complexity/time_complexity.c b/codes/c/chapter_computational_complexity/time_complexity.c new file mode 100644 index 00000000..8df06a75 --- /dev/null +++ b/codes/c/chapter_computational_complexity/time_complexity.c @@ -0,0 +1,175 @@ +/* + * File: time_complexity.c + * Created Time: 2023-01-03 + * Author: sjinzh (sjinzh@gmail.com) + */ + +#include "../include/include.h" + +/* 常数阶 */ +int constant(int n) { + int count = 0; + int size = 100000; + int i = 0; + for (int i = 0; i < size; i++) { + count ++; + } + return count; +} + +/* 线性阶 */ +int linear(int n) { + int count = 0; + for (int i = 0; i < n; i++) { + count ++; + } + return count; +} + +/* 线性阶(遍历数组) */ +int arrayTraversal(int *nums, int n) { + int count = 0; + // 循环次数与数组长度成正比 + for (int i = 0; i < n; i++) { + count ++; + } + return count; +} + +/* 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; +} + +/* 平方阶(冒泡排序) */ +int bubbleSort(int *nums, int n) { + int count = 0; // 计数器 + // 外循环:待排序元素数量为 n-1, n-2, ..., 1 + for (int i = n - 1; i > 0; i--) { + // 内循环:冒泡操作 + for (int j = 0; j < i; j++) { + // 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(循环实现) */ +int exponential(int n) { + int count = 0; + int bas = 1; + // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1) + for (int i = 0; i < n; i++) { + for (int j = 0; j < bas; j++) { + count++; + } + bas *= 2; + } + // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1 + return count; +} + +/* 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(循环实现) */ +int logarithmic(float n) { + int count = 0; + while (n > 1) { + n = n / 2; + count++; + } + return count; +} + +/* 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 */ +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; +} + +/* 阶乘阶(递归实现) */ +int factorialRecur(int n) { + if (n == 0) return 1; + int count = 0; + for (int i = 0; i < n; i++) { + count += factorialRecur(n - 1); + } + return count; +} + +/* Driver Code */ +int main(int argc, char *argv[]) { + // 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; + printf("输入数据大小 n = %d\n", n); + + int count = constant(n); + printf("常数阶的计算操作数量 = %d\n", count); + + count = linear(n); + printf("线性阶的计算操作数量 = %d\n", count); + // 分配堆区内存(创建一维可变长数组:数组中元素数量为n,元素类型为int) + int *nums = (int *)malloc(n * sizeof(int)); + count = arrayTraversal(nums, n); + printf("线性阶(遍历数组)的计算操作数量 = %d\n", count); + + count = quadratic(n); + printf("平方阶的计算操作数量 = %d\n", count); + for (int i = 0; i < n; i++) { + nums[i] = n - i; // [n,n-1,...,2,1] + } + count = bubbleSort(nums, n); + printf("平方阶(冒泡排序)的计算操作数量 = %d\n", count); + + count = exponential(n); + printf("指数阶(循环实现)的计算操作数量 = %d\n", count); + count = expRecur(n); + printf("指数阶(递归实现)的计算操作数量 = %d\n", count); + + count = logarithmic(n); + printf("对数阶(循环实现)的计算操作数量 = %d\n", count); + count = logRecur(n); + printf("对数阶(递归实现)的计算操作数量 = %d\n", count); + + count = linearLogRecur(n); + printf("线性对数阶(递归实现)的计算操作数量 = %d\n", count); + + count = factorialRecur(n); + printf("阶乘阶(递归实现)的计算操作数量 = %d\n", count); + + // 释放堆区内存 + if (nums != NULL) { + free(nums); + nums = NULL; + } + getchar(); + return 0; +} + diff --git a/codes/c/chapter_computational_complexity/worst_best_time_complexity.c b/codes/c/chapter_computational_complexity/worst_best_time_complexity.c new file mode 100644 index 00000000..88ed4dd2 --- /dev/null +++ b/codes/c/chapter_computational_complexity/worst_best_time_complexity.c @@ -0,0 +1,55 @@ +/* + * File: worst_best_time_complexity.c + * Created Time: 2023-01-03 + * Author: sjinzh (sjinzh@gmail.com) + */ + +#include "../include/include.h" + +/* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */ +int *randomNumbers(int n) { + // 分配堆区内存(创建一维可变长数组:数组中元素数量为n,元素类型为int) + int *nums = (int *)malloc(n * sizeof(int)); + // 生成数组 nums = { 1, 2, 3, ..., n } + for (int i = 0; i < n; i++) { + nums[i] = i + 1; + } + // 随机打乱数组元素 + for (int i = n - 1; i > 0; i--) { + int j = rand() % (i + 1); + int temp = nums[i]; + nums[i] = nums[j]; + nums[j] = temp; + } + return nums; +} + +/* 查找数组 nums 中数字 1 所在索引 */ +int findOne(int *nums, int n) { + for (int i = 0; i < n; i++) { + if (nums[i] == 1) return i; + } + return -1; +} + +/* Driver Code */ +int main(int argc, char *argv[]) { + // 初始化随机数种子 + srand((unsigned int)time(NULL)); + for (int i = 0; i < 10; i++) { + int n = 100; + int *nums = randomNumbers(n); + int index = findOne(nums, n); + printf("\n数组 [ 1, 2, ..., n ] 被打乱后 = "); + printArray(nums, n); + printf("数字 1 的索引为 %d\n", index); + // 释放堆区内存 + if (nums != NULL) { + free(nums); + nums = NULL; + } + } + getchar(); + return 0; +} + diff --git a/codes/c/include/include.h b/codes/c/include/include.h index 02858b67..c567aa2e 100644 --- a/codes/c/include/include.h +++ b/codes/c/include/include.h @@ -1,12 +1,15 @@ -/* - * File: include.h - * Created Time: 2022-12-20 - * Author: MolDuM (moldum@163.com) - */ - -#include -#include -#include -#include - -#include "PrintUtil.h" +/* + * File: include.h + * Created Time: 2022-12-20 + * Author: MolDuM (moldum@163.com) + */ + +#include +#include +#include +#include +#include + +#include "PrintUtil.h" + + diff --git a/docs/chapter_computational_complexity/time_complexity.md b/docs/chapter_computational_complexity/time_complexity.md index 539c9b5c..42bf2c25 100644 --- a/docs/chapter_computational_complexity/time_complexity.md +++ b/docs/chapter_computational_complexity/time_complexity.md @@ -91,7 +91,16 @@ $$ === "C" ```c title="" - + // 在某运行平台下 + void algorithm(int n) { + int a = 2; // 1 ns + a = a + 1; // 1 ns + a = a * 2; // 10 ns + // 循环 n 次 + for (int i = 0; i < n; i++) { // 1 ns ,每轮都要执行 i++ + printf("%d", 0); // 5 ns + } + } ``` === "C#" @@ -232,7 +241,22 @@ $$ === "C" ```c title="" - + // 算法 A 时间复杂度:常数阶 + void algorithm_A(int n) { + printf("%d", 0); + } + // 算法 B 时间复杂度:线性阶 + void algorithm_B(int n) { + for (int i = 0; i < n; i++) { + printf("%d", 0); + } + } + // 算法 C 时间复杂度:常数阶 + void algorithm_C(int n) { + for (int i = 0; i < 1000000; i++) { + printf("%d", 0); + } + } ``` === "C#" @@ -373,7 +397,15 @@ $$ === "C" ```c title="" - + void algorithm(int n) { + int a = 1; // +1 + a = a + 1; // +1 + a = a * 2; // +1 + // 循环 n 次 + for (int i = 0; i < n; i++) { // +1(每轮都执行 i ++) + printf("%d", 0); // +1 + } + } ``` === "C#" @@ -542,7 +574,20 @@ $$ === "C" ```c 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++) { + printf("%d", 0); + } + // +n*n(技巧 3) + for (int i = 0; i < 2 * n; i++) { + for (int j = 0; j < n + 1; j++) { + printf("%d", 0); + } + } + } ``` === "C#" @@ -697,7 +742,16 @@ $$ === "C" ```c title="time_complexity.c" - + /* 常数阶 */ + int constant(int n) { + int count = 0; + int size = 100000; + int i = 0; + for (int i = 0; i < size; i++) { + count ++; + } + return count; + } ``` === "C#" @@ -795,7 +849,14 @@ $$ === "C" ```c title="time_complexity.c" - + /* 线性阶 */ + int linear(int n) { + int count = 0; + for (int i = 0; i < n; i++) { + count ++; + } + return count; + } ``` === "C#" @@ -899,7 +960,15 @@ $$ === "C" ```c title="time_complexity.c" - + /* 线性阶(遍历数组) */ + int arrayTraversal(int *nums, int n) { + int count = 0; + // 循环次数与数组长度成正比 + for (int i = 0; i < n; i++) { + count ++; + } + return count; + } ``` === "C#" @@ -1012,7 +1081,17 @@ $$ === "C" ```c title="time_complexity.c" - + /* 平方阶 */ + int quadratic(int n) { + int count = 0; + // 循环次数与数组长度成平方关系 + for (int i = 0; i < n; i++) { + for (int j = 0; j < n; j++) { + count ++; + } + } + return count; + } ``` === "C#" @@ -1163,7 +1242,23 @@ $$ === "C" ```c title="time_complexity.c" - + /* 平方阶(冒泡排序) */ + int bubbleSort(int *nums, int n) { + int count = 0; // 计数器 + // 外循环:待排序元素数量为 n-1, n-2, ..., 1 + for (int i = n - 1; i > 0; i--) { + // 内循环:冒泡操作 + for (int j = 0; j < i; j++) { + // 交换 nums[j] 与 nums[j + 1] + int tmp = nums[j]; + nums[j] = nums[j + 1]; + nums[j + 1] = tmp; + count += 3; // 元素交换包含 3 个单元操作 + } + + } + return count; + } ``` === "C#" @@ -1309,7 +1404,20 @@ $$ === "C" ```c title="time_complexity.c" - + /* 指数阶(循环实现) */ + int exponential(int n) { + int count = 0; + int bas = 1; + // cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1) + for (int i = 0; i < n; i++) { + for (int j = 0; j < bas; j++) { + count++; + } + bas *= 2; + } + // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1 + return count; + } ``` === "C#" @@ -1414,7 +1522,11 @@ $$ === "C" ```c title="time_complexity.c" - + /* 指数阶(递归实现) */ + int expRecur(int n) { + if (n == 1) return 1; + return expRecur(n - 1) + expRecur(n - 1) + 1; + } ``` === "C#" @@ -1517,7 +1629,15 @@ $$ === "C" ```c title="time_complexity.c" - + /* 对数阶(循环实现) */ + int logarithmic(float n) { + int count = 0; + while (n > 1) { + n = n / 2; + count++; + } + return count; + } ``` === "C#" @@ -1613,7 +1733,11 @@ $$ === "C" ```c title="time_complexity.c" - + /* 对数阶(递归实现) */ + int logRecur(float n) { + if (n <= 1) return 0; + return logRecur(n / 2) + 1; + } ``` === "C#" @@ -1720,7 +1844,16 @@ $$ === "C" ```c title="time_complexity.c" - + /* 线性对数阶 */ + int linearLogRecur(float n) { + if (n <= 1) return 1; + int count = linearLogRecur(n / 2) + + linearLogRecur(n / 2); + for (int i = 0; i < n; i++) { + count ++; + } + return count; + } ``` === "C#" @@ -1845,7 +1978,15 @@ $$ === "C" ```c title="time_complexity.c" - + /* 阶乘阶(递归实现) */ + int factorialRecur(int n) { + if (n == 0) return 1; + int count = 0; + for (int i = 0; i < n; i++) { + count += factorialRecur(n - 1); + } + return count; + } ``` === "C#" @@ -2061,7 +2202,52 @@ $$ === "C" ```c title="worst_best_time_complexity.c" + /* 生成一个数组,元素为 { 1, 2, ..., n },顺序被打乱 */ + int *randomNumbers(int n) { + // 分配堆区内存(创建一维可变长数组:数组中元素数量为n,元素类型为int) + int *nums = (int *)malloc(n * sizeof(int)); + // 生成数组 nums = { 1, 2, 3, ..., n } + for (int i = 0; i < n; i++) { + nums[i] = i + 1; + } + // 随机打乱数组元素 + for (int i = n - 1; i > 0; i--) { + int j = rand() % (i + 1); + int temp = nums[i]; + nums[i] = nums[j]; + nums[j] = temp; + } + return nums; + } + /* 查找数组 nums 中数字 1 所在索引 */ + int findOne(int *nums, int n) { + for (int i = 0; i < n; i++) { + if (nums[i] == 1) return i; + } + return -1; + } + + /* Driver Code */ + int main(int argc, char *argv[]) { + // 初始化随机数种子 + srand((unsigned int)time(NULL)); + for (int i = 0; i < 10; i++) { + int n = 100; + int *nums = randomNumbers(n); + int index = findOne(nums, n); + printf("\n数组 [ 1, 2, ..., n ] 被打乱后 = "); + printArray(nums, n); + printf("数字 1 的索引为 %d\n", index); + // 释放堆区内存 + if (nums != NULL) { + free(nums); + nums = NULL; + } + } + getchar(); + return 0; + } ``` === "C#"