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/CMakeLists.txt

@@ -15,3 +15,4 @@ add_subdirectory(chapter_graph)
 add_subdirectory(chapter_sorting)
 add_subdirectory(chapter_searching)
 add_subdirectory(chapter_backtracking)
+add_subdirectory(chapter_dynamic_programming)

codes/cpp/chapter_dynamic_programming/CMakeLists.txt

@@ -0,0 +1,5 @@
+add_executable(climbing_stairs_backtrack climbing_stairs_backtrack.cpp)
+add_executable(climbing_stairs_dfs climbing_stairs_dfs.cpp)
+add_executable(climbing_stairs_dfs_mem climbing_stairs_dfs_mem.cpp)
+add_executable(climbing_stairs_dp climbing_stairs_dp.cpp)
+add_executable(min_cost_climbing_stairs_dp min_cost_climbing_stairs_dp.cpp)

codes/cpp/chapter_dynamic_programming/climbing_stairs_backtrack.cpp

@@ -0,0 +1,43 @@
+
+/**
+ * File: climbing_stairs_backtrack.cpp
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* 回溯 */
+void backtrack(vector<int> &choices, int state, int n, vector<int> &res) {
+    // 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 (auto &choice : choices) {
+        // 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);
+        // 回退
+    }
+}
+
+/* 爬楼梯：回溯 */
+int climbingStairsBacktrack(int n) {
+    vector<int> choices = {1, 2}; // 可选择向上爬 1 或 2 阶
+    int state = 0;                // 从第 0 阶开始爬
+    vector<int> res = {0};        // 使用 res[0] 记录方案数量
+    backtrack(choices, state, n, res);
+    return res[0];
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsBacktrack(n);
+    cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
+
+    return 0;
+}

codes/cpp/chapter_dynamic_programming/climbing_stairs_dfs.cpp

@@ -0,0 +1,32 @@
+/**
+ * File: climbing_stairs_dfs.cpp
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* 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);
+    cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
+
+    return 0;
+}

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;
+}

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;
+}

codes/cpp/chapter_dynamic_programming/min_cost_climbing_stairs_dp.cpp

@@ -0,0 +1,53 @@
+/**
+ * File: min_cost_climbing_stairs_dp.cpp
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* 爬楼梯最小代价：动态规划 */
+int minCostClimbingStairsDP(vector<int> &cost) {
+    int n = cost.size() - 1;
+    if (n == 1 || n == 2)
+        return cost[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] = cost[1];
+    dp[2] = cost[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] = min(dp[i - 1], dp[i - 2]) + cost[i];
+    }
+    return dp[n];
+}
+
+/* 爬楼梯最小代价：状态压缩后的动态规划 */
+int minCostClimbingStairsDPComp(vector<int> &cost) {
+    int n = cost.size() - 1;
+    if (n == 1 || n == 2)
+        return cost[n];
+    int a = cost[1], b = cost[2];
+    for (int i = 3; i <= n; i++) {
+        int tmp = b;
+        b = min(a, tmp) + cost[i];
+        a = tmp;
+    }
+    return b;
+}
+
+/* Driver Code */
+int main() {
+    vector<int> cost = {0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1};
+    cout << "输入楼梯的代价列表为 ";
+    printVector(cost);
+
+    int res = minCostClimbingStairsDP(cost);
+    cout << "爬完楼梯的最低代价为 " << res << endl;
+
+    res = minCostClimbingStairsDPComp(cost);
+    cout << "爬完楼梯的最低代价为 " << res << endl;
+
+    return 0;
+}

codes/java/chapter_dynamic_programming/climbing_stairs_backtrack.java

@@ -0,0 +1,44 @@
+/**
+ * File: climbing_stairs_backtrack.java
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+package chapter_dynamic_programming;
+
+import java.util.*;
+
+public class climbing_stairs_backtrack {
+    /* 回溯 */
+    public static void backtrack(List<Integer> choices, int state, int n, List<Integer> res) {
+        // 当爬到第 n 阶时，方案数量加 1
+        if (state == n)
+            res.set(0, res.get(0) + 1);
+        // 遍历所有选择
+        for (Integer choice : choices) {
+            // 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);
+            // 回退
+        }
+    }
+
+    /* 爬楼梯：回溯 */
+    public static int climbingStairsBacktrack(int n) {
+        List<Integer> choices = Arrays.asList(1, 2); // 可选择向上爬 1 或 2 阶
+        int state = 0; // 从第 0 阶开始爬
+        List<Integer> res = new ArrayList<>();
+        res.add(0); // 使用 res[0] 记录方案数量
+        backtrack(choices, state, n, res);
+        return res.get(0);
+    }
+
+    public static void main(String[] args) {
+        int n = 9;
+
+        int res = climbingStairsBacktrack(n);
+        System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res));
+    }
+}

codes/java/chapter_dynamic_programming/climbing_stairs_dfs.java

@@ -0,0 +1,31 @@
+/**
+ * File: climbing_stairs_dfs.java
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+package chapter_dynamic_programming;
+
+public class climbing_stairs_dfs {
+    /* 搜索 */
+    public static 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 */
+    public static int climbingStairsDFS(int n) {
+        return dfs(n);
+    }
+
+    public static void main(String[] args) {
+        int n = 9;
+
+        int res = climbingStairsDFS(n);
+        System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res));
+    }
+}

codes/java/chapter_dynamic_programming/climbing_stairs_dfs_mem.java

@@ -0,0 +1,41 @@
+/**
+ * File: climbing_stairs_dfs_mem.java
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+package chapter_dynamic_programming;
+
+import java.util.Arrays;
+
+public class climbing_stairs_dfs_mem {
+    /* 记忆化搜索 */
+    public static 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 */
+    public static 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 = new int[n + 1];
+        Arrays.fill(mem, -1);
+        return dfs(n, mem);
+    }
+
+    public static void main(String[] args) {
+        int n = 9;
+
+        int res = climbingStairsDFSMem(n);
+        System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res));
+    }
+}

codes/java/chapter_dynamic_programming/climbing_stairs_dp.java

@@ -0,0 +1,48 @@
+/**
+ * File: climbing_stairs_dp.java
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+package chapter_dynamic_programming;
+
+public class climbing_stairs_dp {
+    /* 爬楼梯：动态规划 */
+    public static 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
+        int[] dp = new int[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 */
+    public static 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;
+    }
+
+    public static void main(String[] args) {
+        int n = 9;
+
+        int res = climbingStairsDP(n);
+        System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res));
+
+        res = climbingStairsDPComp(n);
+        System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res));
+    }
+}

codes/java/chapter_dynamic_programming/min_cost_climbing_stairs_dp.java

@@ -0,0 +1,53 @@
+/**
+ * File: min_cost_climbing_stairs_dp.java
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+package chapter_dynamic_programming;
+
+import java.util.Arrays;
+
+public class min_cost_climbing_stairs_dp {
+    /* 爬楼梯最小代价：动态规划 */
+    public static int minCostClimbingStairsDP(int[] cost) {
+        int n = cost.length - 1;
+        if (n == 1 || n == 2)
+            return cost[n];
+        // 初始化 dp 列表，用于存储子问题的解
+        int[] dp = new int[n + 1];
+        // 初始状态：预设最小子问题的解
+        dp[1] = cost[1];
+        dp[2] = cost[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] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];
+        }
+        return dp[n];
+    }
+
+    /* 爬楼梯最小代价：状态压缩后的动态规划 */
+    public static int minCostClimbingStairsDPComp(int[] cost) {
+        int n = cost.length - 1;
+        if (n == 1 || n == 2)
+            return cost[n];
+        int a = cost[1], b = cost[2];
+        for (int i = 3; i <= n; i++) {
+            int tmp = b;
+            b = Math.min(a, tmp) + cost[i];
+            a = tmp;
+        }
+        return b;
+    }
+
+    public static void main(String[] args) {
+        int[] cost = { 0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1 };
+        System.out.println(String.format("输入楼梯的代价列表为 %s", Arrays.toString(cost)));
+
+        int res = minCostClimbingStairsDP(cost);
+        System.out.println(String.format("爬完楼梯的最低代价为 %d", res));
+
+        res = minCostClimbingStairsDPComp(cost);
+        System.out.println(String.format("爬完楼梯的最低代价为 %d", res));
+    }
+}

codes/python/chapter_dynamic_programming/climbing_stairs_backtrack.py

@@ -0,0 +1,37 @@
+"""
+File: climbing_stairs_backtrack.py
+Created Time: 2023-06-30
+Author: Krahets (krahets@163.com)
+"""
+
+
+def backtrack(choices: list[int], state: int, n: int, res: list[int]) -> int:
+    """回溯"""
+    # 当爬到第 n 阶时，方案数量加 1
+    if state == n:
+        res[0] += 1
+    # 遍历所有选择
+    for choice in choices:
+        # 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)
+        # 回退
+
+
+def climbing_stairs_backtrack(n: int) -> int:
+    """爬楼梯：回溯"""
+    choices = [1, 2]  # 可选择向上爬 1 或 2 阶
+    state = 0  # 从第 0 阶开始爬
+    res = [0]  # 使用 res[0] 记录方案数量
+    backtrack(choices, state, n, res)
+    return res[0]
+
+
+"""Driver Code"""
+if __name__ == "__main__":
+    n = 9
+
+    res = climbing_stairs_backtrack(n)
+    print(f"爬 {n} 阶楼梯共有 {res} 种方案")

codes/python/chapter_dynamic_programming/climbing_stairs_dfs.py

@@ -0,0 +1,28 @@
+"""
+File: climbing_stairs_dfs.py
+Created Time: 2023-06-30
+Author: Krahets (krahets@163.com)
+"""
+
+
+def dfs(i: int) -> int:
+    """搜索"""
+    # 已知 dp[1] 和 dp[2] ，返回之
+    if i == 1 or i == 2:
+        return i
+    # dp[i] = dp[i-1] + dp[i-2]
+    count = dfs(i - 1) + dfs(i - 2)
+    return count
+
+
+def climbing_stairs_dfs(n: int) -> int:
+    """爬楼梯：搜索"""
+    return dfs(n)
+
+
+"""Driver Code"""
+if __name__ == "__main__":
+    n = 9
+
+    res = climbing_stairs_dfs(n)
+    print(f"爬 {n} 阶楼梯共有 {res} 种方案")

codes/python/chapter_dynamic_programming/climbing_stairs_dfs_mem.py

@@ -0,0 +1,35 @@
+"""
+File: climbing_stairs_dfs_mem.py
+Created Time: 2023-06-30
+Author: Krahets (krahets@163.com)
+"""
+
+
+def dfs(i: int, mem: list[int]) -> int:
+    """记忆化搜索"""
+    # 已知 dp[1] 和 dp[2] ，返回之
+    if i == 1 or 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]
+    count = dfs(i - 1, mem) + dfs(i - 2, mem)
+    # 记录 dp[i]
+    mem[i] = count
+    return count
+
+
+def climbing_stairs_dfs_mem(n: int) -> int:
+    """爬楼梯：记忆化搜索"""
+    # mem[i] 记录爬到第 i 阶的方案总数，-1 代表无记录
+    mem = [-1] * (n + 1)
+    return dfs(n, mem)
+
+
+"""Driver Code"""
+if __name__ == "__main__":
+    n = 9
+
+    res = climbing_stairs_dfs_mem(n)
+    print(f"爬 {n} 阶楼梯共有 {res} 种方案")

codes/python/chapter_dynamic_programming/climbing_stairs_dp.py

@@ -0,0 +1,40 @@
+"""
+File: climbing_stairs_dp.py
+Created Time: 2023-06-30
+Author: Krahets (krahets@163.com)
+"""
+
+
+def climbing_stairs_dp(n: int) -> int:
+    """爬楼梯：动态规划"""
+    if n == 1 or n == 2:
+        return n
+    # 初始化 dp 列表，用于存储子问题的解
+    dp = [0] * (n + 1)
+    # 初始状态：预设最小子问题的解
+    dp[1], dp[2] = 1, 2
+    # 状态转移：从较小子问题逐步求解较大子问题
+    for i in range(3, n + 1):
+        dp[i] = dp[i - 1] + dp[i - 2]
+    return dp[n]
+
+
+def climbing_stairs_dp_comp(n: int) -> int:
+    """爬楼梯：状态压缩后的动态规划"""
+    if n == 1 or n == 2:
+        return n
+    a, b = 1, 2
+    for _ in range(3, n + 1):
+        a, b = b, a + b
+    return b
+
+
+"""Driver Code"""
+if __name__ == "__main__":
+    n = 9
+
+    res = climbing_stairs_dp(n)
+    print(f"爬 {n} 阶楼梯共有 {res} 种方案")
+
+    res = climbing_stairs_dp_comp(n)
+    print(f"爬 {n} 阶楼梯共有 {res} 种方案")

codes/python/chapter_dynamic_programming/min_cost_climbing_stairs_dp.py

@@ -0,0 +1,43 @@
+"""
+File: min_cost_climbing_stairs_dp.py
+Created Time: 2023-06-30
+Author: Krahets (krahets@163.com)
+"""
+
+
+def min_cost_climbing_stairs_dp(cost: list[int]) -> int:
+    """爬楼梯最小代价：动态规划"""
+    n = len(cost) - 1
+    if n == 1 or n == 2:
+        return cost[n]
+    # 初始化 dp 列表，用于存储子问题的解
+    dp = [0] * (n + 1)
+    # 初始状态：预设最小子问题的解
+    dp[1], dp[2] = cost[1], cost[2]
+    # 状态转移：从较小子问题逐步求解较大子问题
+    for i in range(3, n + 1):
+        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]
+    return dp[n]
+
+
+def min_cost_climbing_stairs_dp_comp(cost: list[int]) -> int:
+    """爬楼梯最小代价：状态压缩后的动态规划"""
+    n = len(cost) - 1
+    if n == 1 or n == 2:
+        return cost[n]
+    a, b = cost[1], cost[2]
+    for i in range(3, n + 1):
+        a, b = b, min(a, b) + cost[i]
+    return b
+
+
+"""Driver Code"""
+if __name__ == "__main__":
+    cost = [0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1]
+    print(f"输入楼梯的代价列表为 {cost}")
+
+    res = min_cost_climbing_stairs_dp(cost)
+    print(f"爬完楼梯的最低代价为 {res}")
+
+    res = min_cost_climbing_stairs_dp_comp(cost)
+    print(f"爬完楼梯的最低代价为 {res}")