Update time complexity and space complexity.

codes/cpp/chapter_computational_complexity/time_complexity.cpp

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+/*
+ * File: time_complexity.cpp
+ * Created Time: 2022-11-25
+ * Author: Krahets (krahets@163.com)
+ */
+
+#include "../include/include.hpp"
+
+/* 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 */
+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(vector<int>& nums) {
+    int count = 0;
+    // 循环次数与数组长度成正比
+    for (int num : nums) {
+        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(vector<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.size() - 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（循环实现） */
+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;
+}
+
+/* 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;
+    // 从 1 个分裂出 n 个
+    for (int i = 0; i < n; i++) {
+        count += factorialRecur(n - 1);
+    }
+    return count;
+}
+
+
+/* Driver Code */
+int 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;
+    cout << "输入数据大小 n = " << n << endl;
+
+    int count = constant(n);
+    cout << "常数阶的计算操作数量 = " << count << endl;
+
+    count = linear(n);
+    cout << "线性阶的计算操作数量 = " << count << endl;
+    vector<int> arr(n);
+    count = arrayTraversal(arr);
+    cout << "线性阶（遍历数组）的计算操作数量 = " << count << endl;
+
+    count = quadratic(n);
+    cout << "平方阶的计算操作数量 = " << count << endl;
+    vector<int> nums(n);
+    for (int i = 0; i < n; i++)
+        nums[i] = n - i;  // [n,n-1,...,2,1]
+    count = bubbleSort(nums);
+    cout << "平方阶（冒泡排序）的计算操作数量 = " << count << endl;
+
+    count = exponential(n);
+    cout << "指数阶（循环实现）的计算操作数量 = " << count << endl;
+    count = expRecur(n);
+    cout << "指数阶（递归实现）的计算操作数量 = " << count << endl;
+
+    count = logarithmic((float) n);
+    cout << "对数阶（循环实现）的计算操作数量 = " << count << endl;
+    count = logRecur((float) n);
+    cout << "对数阶（递归实现）的计算操作数量 = " << count << endl;
+
+    count = linearLogRecur((float) n);
+    cout << "线性对数阶（递归实现）的计算操作数量 = " << count << endl;
+
+    count = factorialRecur(n);
+    cout << "阶乘阶（递归实现）的计算操作数量 = " << count << endl;
+
+    return 0;
+}