feat: add the section of the introduction to dynamic programming (#571)

* add the section of the introduction to
dynamic programming

* add a code comments.

codes/cpp/chapter_dynamic_programming/climbing_stairs_dp.cpp

@@ -0,0 +1,49 @@
+/**
+ * File: climbing_stairs_dp.cpp
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* 爬楼梯：动态规划 */
+int climbingStairsDP(int n) {
+    if (n == 1 || n == 2)
+        return n;
+    // 初始化 dp 列表，用于存储子问题的解
+    vector<int> dp(n + 1);
+    // 初始状态：预设最小子问题的解
+    dp[1] = 1;
+    dp[2] = 2;
+    // 状态转移：从较小子问题逐步求解较大子问题
+    for (int i = 3; i <= n; i++) {
+        dp[i] = dp[i - 1] + dp[i - 2];
+    }
+    return dp[n];
+}
+
+/* 爬楼梯：状态压缩后的动态规划 */
+int climbingStairsDPComp(int n) {
+    if (n == 1 || n == 2)
+        return n;
+    int a = 1, b = 2;
+    for (int i = 3; i <= n; i++) {
+        int tmp = b;
+        b = a + b;
+        a = tmp;
+    }
+    return b;
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsDP(n);
+    cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
+
+    res = climbingStairsDPComp(n);
+    cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
+
+    return 0;
+}