feat: Add C code for the section of intro to DP (#771)

* Add dynamic programming C language code

* Add dynamic programming chapter C language code

* Modifying errors in files

codes/c/chapter_dynamic_programming/climbing_stairs_backtrack.c

@@ -0,0 +1,47 @@
+/**
+ * File: climbing_stairs_backtrack.c
+ * Created Time: 2023-09-22
+ * Author: huawuque404 (huawuque404@163.com)
+ */
+
+#include "../utils/common.h"
+
+/* 回溯 */
+void backtrack(int *choices, int state, int n, int *res, int len) {
+    // 当爬到第 n 阶时，方案数量加 1
+    if (state == n)
+        res[0]++;
+    // 遍历所有选择
+    for (int i = 0; i < len; i++) {
+        int choice = choices[i];
+        // 剪枝：不允许越过第 n 阶
+        if (state + choice > n)
+            break;
+        // 尝试：做出选择，更新状态
+        backtrack(choices, state + choice, n, res, len);
+        // 回退
+    }
+}
+
+/* 爬楼梯：回溯 */
+int climbingStairsBacktrack(int n) {
+    int choices[2] = {1, 2}; // 可选择向上爬 1 或 2 阶
+    int state = 0;           // 从第 0 阶开始爬
+    int *res = (int *)malloc(sizeof(int));
+    *res = 0;                // 使用 res[0] 记录方案数量
+    int len = sizeof(choices) / sizeof(int);
+    backtrack(choices, state, n, res, len);
+    int result = *res;
+    free(res);
+    return result;
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsBacktrack(n);
+    printf("爬 %d 阶楼梯共有 %d 种方案\n", n, res);
+
+    return 0;
+}

codes/c/chapter_dynamic_programming/climbing_stairs_dfs.c

@@ -0,0 +1,32 @@
+/**
+ * File: climbing_stairs_dfs.c
+ * Created Time: 2023-09-19
+ * Author: huawuque404 (huawuque404@163.com)
+ */
+
+#include "../utils/common.h"
+
+/* 搜索 */
+int dfs(int i) {
+    // 已知 dp[1] 和 dp[2] ，返回之
+    if (i == 1 || i == 2)
+        return i;
+    // dp[i] = dp[i-1] + dp[i-2]
+    int count = dfs(i - 1) + dfs(i - 2);
+    return count;
+}
+
+/* 爬楼梯：搜索 */
+int climbingStairsDFS(int n) {
+    return dfs(n);
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsDFS(n);
+    printf("爬 %d 阶楼梯共有 %d 种方案\n", n, res);
+
+    return 0;
+}

codes/c/chapter_dynamic_programming/climbing_stairs_dfs_mem.c

@@ -0,0 +1,44 @@
+/**
+ * File: climbing_stairs_dfs_mem.c
+ * Created Time: 2023-09-19
+ * Author: huawuque404 (huawuque404@163.com)
+ */
+
+#include "../utils/common.h"
+
+/* 记忆化搜索 */
+int dfs(int i, 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
+    int *mem = (int *)malloc((n + 1) * sizeof(int));
+    for (int i = 0; i <= n; i++) {
+        mem[i] = -1;
+    }
+    int result = dfs(n, mem);
+    free(mem);
+    return result;
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+    
+    int res = climbingStairsDFSMem(n);
+    printf("爬 %d 阶楼梯共有 %d 种方案\n", n, res);
+
+    return 0;
+}

codes/c/chapter_dynamic_programming/climbing_stairs_dp.c

@@ -0,0 +1,51 @@
+/**
+ * File: climbing_stairs_dp.c
+ * Created Time: 2023-09-19
+ * Author: huawuque404 (huawuque404@163.com)
+ */
+
+#include "../utils/common.h"
+
+/* 爬楼梯：动态规划 */
+int climbingStairsDP(int n) {
+    if (n == 1 || n == 2)
+        return n;
+    // 初始化 dp 表，用于存储子问题的解
+    int *dp = (int *)malloc((n + 1) * sizeof(int));
+    // 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];
+    }
+    int result = dp[n];
+    free(dp);
+    return result;
+}
+
+/* 爬楼梯：空间优化后的动态规划 */
+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);
+    printf("爬 %d 阶楼梯共有 %d 种方案\n", n, res);
+
+    res = climbingStairsDPComp(n);
+    printf("爬 %d 阶楼梯共有 %d 种方案\n", n, res);
+
+    return 0;
+}