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_dfs_mem.cpp

@@ -0,0 +1,39 @@
+/**
+ * File: climbing_stairs_dfs_mem.cpp
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* 记忆化搜索 */
+int dfs(int i, vector<int> &mem) {
+    // 已知 dp[1] 和 dp[2] ，返回之
+    if (i == 1 || i == 2)
+        return i;
+    // 若存在记录 dp[i] ，则直接返回之
+    if (mem[i] != -1)
+        return mem[i];
+    // dp[i] = dp[i-1] + dp[i-2]
+    int count = dfs(i - 1, mem) + dfs(i - 2, mem);
+    // 记录 dp[i]
+    mem[i] = count;
+    return count;
+}
+
+/* 爬楼梯：记忆化搜索 */
+int climbingStairsDFSMem(int n) {
+    // mem[i] 记录爬到第 i 阶的方案总数，-1 代表无记录
+    vector<int> mem(n + 1, -1);
+    return dfs(n, mem);
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsDFSMem(n);
+    cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
+
+    return 0;
+}