Feature/chapter dynamic programming swift (#608)

* feat: add Swift codes for intro_to_dynamic_programming article

* feat: add Swift codes for dp_problem_features article

* feat: add Swift codes for dp_solution_pipeline article

* feat: add Swift codes for knapsack_problem article

* feat: add Swift codes for unbounded_knapsack_problem article

* feat: add Swift codes for edit_distance_problem article

codes/swift/Package.swift

@@ -72,6 +72,19 @@ let package = Package(
         .executable(name: "subset_sum_i", targets: ["subset_sum_i"]),
         .executable(name: "subset_sum_ii", targets: ["subset_sum_ii"]),
         .executable(name: "n_queens", targets: ["n_queens"]),
+        // chapter_dynamic_programming
+        .executable(name: "climbing_stairs_backtrack", targets: ["climbing_stairs_backtrack"]),
+        .executable(name: "climbing_stairs_dfs", targets: ["climbing_stairs_dfs"]),
+        .executable(name: "climbing_stairs_dfs_mem", targets: ["climbing_stairs_dfs_mem"]),
+        .executable(name: "climbing_stairs_dp", targets: ["climbing_stairs_dp"]),
+        .executable(name: "min_cost_climbing_stairs_dp", targets: ["min_cost_climbing_stairs_dp"]),
+        .executable(name: "climbing_stairs_constraint_dp", targets: ["climbing_stairs_constraint_dp"]),
+        .executable(name: "min_path_sum", targets: ["min_path_sum"]),
+        .executable(name: "knapsack", targets: ["knapsack"]),
+        .executable(name: "unbounded_knapsack", targets: ["unbounded_knapsack"]),
+        .executable(name: "coin_change", targets: ["coin_change"]),
+        .executable(name: "coin_change_ii", targets: ["coin_change_ii"]),
+        .executable(name: "edit_distance", targets: ["edit_distance"]),
     ],
     targets: [
         // helper
@@ -144,5 +157,18 @@ let package = Package(
         .executableTarget(name: "subset_sum_i", path: "chapter_backtracking", sources: ["subset_sum_i.swift"]),
         .executableTarget(name: "subset_sum_ii", path: "chapter_backtracking", sources: ["subset_sum_ii.swift"]),
         .executableTarget(name: "n_queens", path: "chapter_backtracking", sources: ["n_queens.swift"]),
+        // chapter_dynamic_programming
+        .executableTarget(name: "climbing_stairs_backtrack", path: "chapter_dynamic_programming", sources: ["climbing_stairs_backtrack.swift"]),
+        .executableTarget(name: "climbing_stairs_dfs", path: "chapter_dynamic_programming", sources: ["climbing_stairs_dfs.swift"]),
+        .executableTarget(name: "climbing_stairs_dfs_mem", path: "chapter_dynamic_programming", sources: ["climbing_stairs_dfs_mem.swift"]),
+        .executableTarget(name: "climbing_stairs_dp", path: "chapter_dynamic_programming", sources: ["climbing_stairs_dp.swift"]),
+        .executableTarget(name: "min_cost_climbing_stairs_dp", path: "chapter_dynamic_programming", sources: ["min_cost_climbing_stairs_dp.swift"]),
+        .executableTarget(name: "climbing_stairs_constraint_dp", path: "chapter_dynamic_programming", sources: ["climbing_stairs_constraint_dp.swift"]),
+        .executableTarget(name: "min_path_sum", path: "chapter_dynamic_programming", sources: ["min_path_sum.swift"]),
+        .executableTarget(name: "knapsack", path: "chapter_dynamic_programming", sources: ["knapsack.swift"]),
+        .executableTarget(name: "unbounded_knapsack", path: "chapter_dynamic_programming", sources: ["unbounded_knapsack.swift"]),
+        .executableTarget(name: "coin_change", path: "chapter_dynamic_programming", sources: ["coin_change.swift"]),
+        .executableTarget(name: "coin_change_ii", path: "chapter_dynamic_programming", sources: ["coin_change_ii.swift"]),
+        .executableTarget(name: "edit_distance", path: "chapter_dynamic_programming", sources: ["edit_distance.swift"]),
     ]
 )

codes/swift/chapter_dynamic_programming/climbing_stairs_backtrack.swift

@@ -0,0 +1,42 @@
+/**
+ * File: climbing_stairs_backtrack.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 回溯 */
+func backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {
+    // 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
+        }
+        backtrack(choices: choices, state: state + choice, n: n, res: &res)
+    }
+}
+
+/* 爬楼梯：回溯 */
+func climbingStairsBacktrack(n: Int) -> Int {
+    let choices = [1, 2] // 可选择向上爬 1 或 2 阶
+    let state = 0 // 从第 0 阶开始爬
+    var res: [Int] = []
+    res.append(0) // 使用 res[0] 记录方案数量
+    backtrack(choices: choices, state: state, n: n, res: &res)
+    return res[0]
+}
+
+@main
+enum ClimbingStairsBacktrack {
+    /* Driver Code */
+    static func main() {
+        let n = 9
+
+        let res = climbingStairsBacktrack(n: n)
+        print("爬 \(n) 阶楼梯共有 \(res) 种方案")
+    }
+}

codes/swift/chapter_dynamic_programming/climbing_stairs_constraint_dp.swift

@@ -0,0 +1,36 @@
+/**
+ * File: climbing_stairs_constraint_dp.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 带约束爬楼梯：动态规划 */
+func climbingStairsConstraintDP(n: Int) -> Int {
+    if n == 1 || n == 2 {
+        return n
+    }
+    // 初始化 dp 表，用于存储子问题的解
+    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)
+    // 初始状态：预设最小子问题的解
+    dp[1][1] = 1
+    dp[1][2] = 0
+    dp[2][1] = 0
+    dp[2][2] = 1
+    // 状态转移：从较小子问题逐步求解较大子问题
+    for i in stride(from: 3, through: n, by: 1) {
+        dp[i][1] = dp[i - 1][2]
+        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]
+    }
+    return dp[n][1] + dp[n][2]
+}
+
+@main
+enum ClimbingStairsConstraintDP {
+    /* Driver Code */
+    static func main() {
+        let n = 9
+
+        let res = climbingStairsConstraintDP(n: n)
+        print("爬 \(n) 阶楼梯共有 \(res) 种方案")
+    }
+}

codes/swift/chapter_dynamic_programming/climbing_stairs_dfs.swift

@@ -0,0 +1,32 @@
+/**
+ * File: climbing_stairs_dfs.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 搜索 */
+func dfs(i: Int) -> Int {
+    // 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]
+    let count = dfs(i: i - 1) + dfs(i: i - 2)
+    return count
+}
+
+/* 爬楼梯：搜索 */
+func climbingStairsDFS(n: Int) -> Int {
+    dfs(i: n)
+}
+
+@main
+enum ClimbingStairsDFS {
+    /* Driver Code */
+    static func main() {
+        let n = 9
+
+        let res = climbingStairsDFS(n: n)
+        print("爬 \(n) 阶楼梯共有 \(res) 种方案")
+    }
+}

codes/swift/chapter_dynamic_programming/climbing_stairs_dfs_mem.swift

@@ -0,0 +1,40 @@
+/**
+ * File: climbing_stairs_dfs_mem.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 记忆化搜索 */
+func dfs(i: Int, mem: inout [Int]) -> Int {
+    // 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]
+    let count = dfs(i: i - 1, mem: &mem) + dfs(i: i - 2, mem: &mem)
+    // 记录 dp[i]
+    mem[i] = count
+    return count
+}
+
+/* 爬楼梯：记忆化搜索 */
+func climbingStairsDFSMem(n: Int) -> Int {
+    // mem[i] 记录爬到第 i 阶的方案总数，-1 代表无记录
+    var mem = Array(repeating: -1, count: n + 1)
+    return dfs(i: n, mem: &mem)
+}
+
+@main
+enum ClimbingStairsDFSMem {
+    /* Driver Code */
+    static func main() {
+        let n = 9
+
+        let res = climbingStairsDFSMem(n: n)
+        print("爬 \(n) 阶楼梯共有 \(res) 种方案")
+    }
+}

codes/swift/chapter_dynamic_programming/climbing_stairs_dp.swift

@@ -0,0 +1,49 @@
+/**
+ * File: climbing_stairs_dp.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 爬楼梯：动态规划 */
+func climbingStairsDP(n: Int) -> Int {
+    if n == 1 || n == 2 {
+        return n
+    }
+    // 初始化 dp 表，用于存储子问题的解
+    var dp = Array(repeating: 0, count: 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 i in stride(from: 3, through: n, by: 1) {
+        dp[i] = dp[i - 1] + dp[i - 2]
+    }
+    return dp[n]
+}
+
+/* 爬楼梯：状态压缩后的动态规划 */
+func climbingStairsDPComp(n: Int) -> Int {
+    if n == 1 || n == 2 {
+        return n
+    }
+    var a = 1
+    var b = 2
+    for _ in stride(from: 3, through: n, by: 1) {
+        (a, b) = (b, a + b)
+    }
+    return b
+}
+
+@main
+enum ClimbingStairsDP {
+    /* Driver Code */
+    static func main() {
+        let n = 9
+
+        var res = climbingStairsDP(n: n)
+        print("爬 \(n) 阶楼梯共有 \(res) 种方案")
+
+        res = climbingStairsDPComp(n: n)
+        print("爬 \(n) 阶楼梯共有 \(res) 种方案")
+    }
+}

codes/swift/chapter_dynamic_programming/coin_change.swift

@@ -0,0 +1,69 @@
+/**
+ * File: coin_change.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 零钱兑换：动态规划 */
+func coinChangeDP(coins: [Int], amt: Int) -> Int {
+    let n = coins.count
+    let MAX = amt + 1
+    // 初始化 dp 表
+    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)
+    // 状态转移：首行首列
+    for a in stride(from: 1, through: amt, by: 1) {
+        dp[0][a] = MAX
+    }
+    // 状态转移：其余行列
+    for i in stride(from: 1, through: n, by: 1) {
+        for a in stride(from: 1, through: amt, by: 1) {
+            if coins[i - 1] > a {
+                // 若超过背包容量，则不选硬币 i
+                dp[i][a] = dp[i - 1][a]
+            } else {
+                // 不选和选硬币 i 这两种方案的较小值
+                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)
+            }
+        }
+    }
+    return dp[n][amt] != MAX ? dp[n][amt] : -1
+}
+
+/* 零钱兑换：状态压缩后的动态规划 */
+func coinChangeDPComp(coins: [Int], amt: Int) -> Int {
+    let n = coins.count
+    let MAX = amt + 1
+    // 初始化 dp 表
+    var dp = Array(repeating: MAX, count: amt + 1)
+    dp[0] = 0
+    // 状态转移
+    for i in stride(from: 1, through: n, by: 1) {
+        for a in stride(from: 1, through: amt, by: 1) {
+            if coins[i - 1] > a {
+                // 若超过背包容量，则不选硬币 i
+                dp[a] = dp[a]
+            } else {
+                // 不选和选硬币 i 这两种方案的较小值
+                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)
+            }
+        }
+    }
+    return dp[amt] != MAX ? dp[amt] : -1
+}
+
+@main
+enum CoinChange {
+    /* Driver Code */
+    static func main() {
+        let coins = [1, 2, 5]
+        let amt = 4
+
+        // 动态规划
+        var res = coinChangeDP(coins: coins, amt: amt)
+        print("凑到目标金额所需的最少硬币数量为 \(res)")
+
+        // 状态压缩后的动态规划
+        res = coinChangeDPComp(coins: coins, amt: amt)
+        print("凑到目标金额所需的最少硬币数量为 \(res)")
+    }
+}

codes/swift/chapter_dynamic_programming/coin_change_ii.swift

@@ -0,0 +1,67 @@
+/**
+ * File: coin_change_ii.swift
+ * Created Time: 2023-07-16
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 零钱兑换 II：动态规划 */
+func coinChangeIIDP(coins: [Int], amt: Int) -> Int {
+    let n = coins.count
+    // 初始化 dp 表
+    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)
+    // 初始化首列
+    for i in stride(from: 0, through: n, by: 1) {
+        dp[i][0] = 1
+    }
+    // 状态转移
+    for i in stride(from: 1, through: n, by: 1) {
+        for a in stride(from: 1, through: amt, by: 1) {
+            if coins[i - 1] > a {
+                // 若超过背包容量，则不选硬币 i
+                dp[i][a] = dp[i - 1][a]
+            } else {
+                // 不选和选硬币 i 这两种方案之和
+                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]
+            }
+        }
+    }
+    return dp[n][amt]
+}
+
+/* 零钱兑换 II：状态压缩后的动态规划 */
+func coinChangeIIDPComp(coins: [Int], amt: Int) -> Int {
+    let n = coins.count
+    // 初始化 dp 表
+    var dp = Array(repeating: 0, count: amt + 1)
+    dp[0] = 1
+    // 状态转移
+    for i in stride(from: 1, through: n, by: 1) {
+        for a in stride(from: 1, through: amt, by: 1) {
+            if coins[i - 1] > a {
+                // 若超过背包容量，则不选硬币 i
+                dp[a] = dp[a]
+            } else {
+                // 不选和选硬币 i 这两种方案之和
+                dp[a] = dp[a] + dp[a - coins[i - 1]]
+            }
+        }
+    }
+    return dp[amt]
+}
+
+@main
+enum CoinChangeII {
+    /* Driver Code */
+    static func main() {
+        let coins = [1, 2, 5]
+        let amt = 5
+
+        // 动态规划
+        var res = coinChangeIIDP(coins: coins, amt: amt)
+        print("凑出目标金额的硬币组合数量为 \(res)")
+
+        // 状态压缩后的动态规划
+        res = coinChangeIIDPComp(coins: coins, amt: amt)
+        print("凑出目标金额的硬币组合数量为 \(res)")
+    }
+}

codes/swift/chapter_dynamic_programming/edit_distance.swift

@@ -0,0 +1,147 @@
+/**
+ * File: edit_distance.swift
+ * Created Time: 2023-07-16
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 编辑距离：暴力搜索 */
+func editDistanceDFS(s: String, t: String, i: Int, j: Int) -> Int {
+    // 若 s 和 t 都为空，则返回 0
+    if i == 0, j == 0 {
+        return 0
+    }
+    // 若 s 为空，则返回 t 长度
+    if i == 0 {
+        return j
+    }
+    // 若 t 为空，则返回 s 长度
+    if j == 0 {
+        return i
+    }
+    // 若两字符相等，则直接跳过此两字符
+    if s.utf8CString[i - 1] == t.utf8CString[j - 1] {
+        return editDistanceDFS(s: s, t: t, i: i - 1, j: j - 1)
+    }
+    // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1
+    let insert = editDistanceDFS(s: s, t: t, i: i, j: j - 1)
+    let delete = editDistanceDFS(s: s, t: t, i: i - 1, j: j)
+    let replace = editDistanceDFS(s: s, t: t, i: i - 1, j: j - 1)
+    // 返回最少编辑步数
+    return min(min(insert, delete), replace) + 1
+}
+
+/* 编辑距离：记忆化搜索 */
+func editDistanceDFSMem(s: String, t: String, mem: inout [[Int]], i: Int, j: Int) -> Int {
+    // 若 s 和 t 都为空，则返回 0
+    if i == 0, j == 0 {
+        return 0
+    }
+    // 若 s 为空，则返回 t 长度
+    if i == 0 {
+        return j
+    }
+    // 若 t 为空，则返回 s 长度
+    if j == 0 {
+        return i
+    }
+    // 若已有记录，则直接返回之
+    if mem[i][j] != -1 {
+        return mem[i][j]
+    }
+    // 若两字符相等，则直接跳过此两字符
+    if s.utf8CString[i - 1] == t.utf8CString[j - 1] {
+        return editDistanceDFS(s: s, t: t, i: i - 1, j: j - 1)
+    }
+    // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1
+    let insert = editDistanceDFS(s: s, t: t, i: i, j: j - 1)
+    let delete = editDistanceDFS(s: s, t: t, i: i - 1, j: j)
+    let replace = editDistanceDFS(s: s, t: t, i: i - 1, j: j - 1)
+    // 记录并返回最少编辑步数
+    mem[i][j] = min(min(insert, delete), replace) + 1
+    return mem[i][j]
+}
+
+/* 编辑距离：动态规划 */
+func editDistanceDP(s: String, t: String) -> Int {
+    let n = s.utf8CString.count
+    let m = t.utf8CString.count
+    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)
+    // 状态转移：首行首列
+    for i in stride(from: 1, through: n, by: 1) {
+        dp[i][0] = i
+    }
+    for j in stride(from: 1, through: m, by: 1) {
+        dp[0][j] = j
+    }
+    // 状态转移：其余行列
+    for i in stride(from: 1, through: n, by: 1) {
+        for j in stride(from: 1, through: m, by: 1) {
+            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {
+                // 若两字符相等，则直接跳过此两字符
+                dp[i][j] = dp[i - 1][j - 1]
+            } else {
+                // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1
+                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1
+            }
+        }
+    }
+    return dp[n][m]
+}
+
+/* 编辑距离：状态压缩后的动态规划 */
+func editDistanceDPComp(s: String, t: String) -> Int {
+    let n = s.utf8CString.count
+    let m = t.utf8CString.count
+    var dp = Array(repeating: 0, count: m + 1)
+    // 状态转移：首行
+    for j in stride(from: 1, through: m, by: 1) {
+        dp[j] = j
+    }
+    // 状态转移：其余行
+    for i in stride(from: 1, through: n, by: 1) {
+        // 状态转移：首列
+        var leftup = dp[0] // 暂存 dp[i-1, j-1]
+        dp[0] = i
+        // 状态转移：其余列
+        for j in stride(from: 1, through: m, by: 1) {
+            let temp = dp[j]
+            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {
+                // 若两字符相等，则直接跳过此两字符
+                dp[j] = leftup
+            } else {
+                // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1
+                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1
+            }
+            leftup = temp // 更新为下一轮的 dp[i-1, j-1]
+        }
+    }
+    return dp[m]
+}
+
+@main
+enum EditDistance {
+    /* Driver Code */
+    static func main() {
+        let s = "bag"
+        let t = "pack"
+        let n = s.utf8CString.count
+        let m = t.utf8CString.count
+
+        // 暴力搜索
+        var res = editDistanceDFS(s: s, t: t, i: n, j: m)
+        print("将 \(s) 更改为 \(t) 最少需要编辑 \(res) 步")
+
+        // 记忆化搜索
+        var mem = Array(repeating: Array(repeating: -1, count: m + 1), count: n + 1)
+        res = editDistanceDFSMem(s: s, t: t, mem: &mem, i: n, j: m)
+        print("将 \(s) 更改为 \(t) 最少需要编辑 \(res) 步")
+
+        // 动态规划
+        res = editDistanceDP(s: s, t: t)
+        print("将 \(s) 更改为 \(t) 最少需要编辑 \(res) 步")
+
+        // 状态压缩后的动态规划
+        res = editDistanceDPComp(s: s, t: t)
+        print("将 \(s) 更改为 \(t) 最少需要编辑 \(res) 步")
+    }
+}

codes/swift/chapter_dynamic_programming/knapsack.swift

@@ -0,0 +1,110 @@
+/**
+ * File: knapsack.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 0-1 背包：暴力搜索 */
+func knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -> Int {
+    // 若已选完所有物品或背包无容量，则返回价值 0
+    if i == 0 || c == 0 {
+        return 0
+    }
+    // 若超过背包容量，则只能不放入背包
+    if wgt[i - 1] > c {
+        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)
+    }
+    // 计算不放入和放入物品 i 的最大价值
+    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)
+    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]
+    // 返回两种方案中价值更大的那一个
+    return max(no, yes)
+}
+
+/* 0-1 背包：记忆化搜索 */
+func knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -> Int {
+    // 若已选完所有物品或背包无容量，则返回价值 0
+    if i == 0 || c == 0 {
+        return 0
+    }
+    // 若已有记录，则直接返回
+    if mem[i][c] != -1 {
+        return mem[i][c]
+    }
+    // 若超过背包容量，则只能不放入背包
+    if wgt[i - 1] > c {
+        return knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: i - 1, c: c)
+    }
+    // 计算不放入和放入物品 i 的最大价值
+    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: i - 1, c: c)
+    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]
+    // 记录并返回两种方案中价值更大的那一个
+    mem[i][c] = max(no, yes)
+    return mem[i][c]
+}
+
+/* 0-1 背包：动态规划 */
+func knapsackDP(wgt: [Int], val: [Int], cap: Int) -> Int {
+    let n = wgt.count
+    // 初始化 dp 表
+    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)
+    // 状态转移
+    for i in stride(from: 1, through: n, by: 1) {
+        for c in stride(from: 1, through: cap, by: 1) {
+            if wgt[i - 1] > c {
+                // 若超过背包容量，则不选物品 i
+                dp[i][c] = dp[i - 1][c]
+            } else {
+                // 不选和选物品 i 这两种方案的较大值
+                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])
+            }
+        }
+    }
+    return dp[n][cap]
+}
+
+/* 0-1 背包：状态压缩后的动态规划 */
+func knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -> Int {
+    let n = wgt.count
+    // 初始化 dp 表
+    var dp = Array(repeating: 0, count: cap + 1)
+    // 状态转移
+    for i in stride(from: 1, through: n, by: 1) {
+        // 倒序遍历
+        for c in stride(from: cap, through: 1, by: -1) {
+            if wgt[i - 1] <= c {
+                // 不选和选物品 i 这两种方案的较大值
+                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])
+            }
+        }
+    }
+    return dp[cap]
+}
+
+@main
+enum Knapsack {
+    /* Driver Code */
+    static func main() {
+        let wgt = [10, 20, 30, 40, 50]
+        let val = [50, 120, 150, 210, 240]
+        let cap = 50
+        let n = wgt.count
+
+        // 暴力搜索
+        var res = knapsackDFS(wgt: wgt, val: val, i: n, c: cap)
+        print("不超过背包容量的最大物品价值为 \(res)")
+
+        // 记忆化搜索
+        var mem = Array(repeating: Array(repeating: -1, count: cap + 1), count: n + 1)
+        res = knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: n, c: cap)
+        print("不超过背包容量的最大物品价值为 \(res)")
+
+        // 动态规划
+        res = knapsackDP(wgt: wgt, val: val, cap: cap)
+        print("不超过背包容量的最大物品价值为 \(res)")
+
+        // 状态压缩后的动态规划
+        res = knapsackDPComp(wgt: wgt, val: val, cap: cap)
+        print("不超过背包容量的最大物品价值为 \(res)")
+    }
+}

codes/swift/chapter_dynamic_programming/min_cost_climbing_stairs_dp.swift

@@ -0,0 +1,51 @@
+/**
+ * File: min_cost_climbing_stairs_dp.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 爬楼梯最小代价：动态规划 */
+func minCostClimbingStairsDP(cost: [Int]) -> Int {
+    let n = cost.count - 1
+    if n == 1 || n == 2 {
+        return cost[n]
+    }
+    // 初始化 dp 表，用于存储子问题的解
+    var dp = Array(repeating: 0, count: 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 i in stride(from: 3, through: n, by: 1) {
+        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 */
+func minCostClimbingStairsDPComp(cost: [Int]) -> Int {
+    let n = cost.count - 1
+    if n == 1 || n == 2 {
+        return cost[n]
+    }
+    var (a, b) = (cost[1], cost[2])
+    for i in stride(from: 3, through: n, by: 1) {
+        (a, b) = (b, min(a, b) + cost[i])
+    }
+    return b
+}
+
+@main
+enum MinCostClimbingStairsDP {
+    /* Driver Code */
+    static func main() {
+        let cost = [0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1]
+        print("输入楼梯的代价列表为 \(cost)")
+
+        var res = minCostClimbingStairsDP(cost: cost)
+        print("爬完楼梯的最低代价为 \(res)")
+
+        res = minCostClimbingStairsDPComp(cost: cost)
+        print("爬完楼梯的最低代价为 \(res)")
+    }
+}

codes/swift/chapter_dynamic_programming/min_path_sum.swift

@@ -0,0 +1,123 @@
+/**
+ * File: min_path_sum.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 最小路径和：暴力搜索 */
+func minPathSumDFS(grid: [[Int]], i: Int, j: Int) -> Int {
+    // 若为左上角单元格，则终止搜索
+    if i == 0, j == 0 {
+        return grid[0][0]
+    }
+    // 若行列索引越界，则返回 +∞ 代价
+    if i < 0 || j < 0 {
+        return .max
+    }
+    // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
+    let left = minPathSumDFS(grid: grid, i: i - 1, j: j)
+    let up = minPathSumDFS(grid: grid, i: i, j: j - 1)
+    // 返回从左上角到 (i, j) 的最小路径代价
+    return min(left, up) + grid[i][j]
+}
+
+/* 最小路径和：记忆化搜索 */
+func minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -> Int {
+    // 若为左上角单元格，则终止搜索
+    if i == 0, j == 0 {
+        return grid[0][0]
+    }
+    // 若行列索引越界，则返回 +∞ 代价
+    if i < 0 || j < 0 {
+        return .max
+    }
+    // 若已有记录，则直接返回
+    if mem[i][j] != -1 {
+        return mem[i][j]
+    }
+    // 左边和上边单元格的最小路径代价
+    let left = minPathSumDFSMem(grid: grid, mem: &mem, i: i - 1, j: j)
+    let up = minPathSumDFSMem(grid: grid, mem: &mem, i: i, j: j - 1)
+    // 记录并返回左上角到 (i, j) 的最小路径代价
+    mem[i][j] = min(left, up) + grid[i][j]
+    return mem[i][j]
+}
+
+/* 最小路径和：动态规划 */
+func minPathSumDP(grid: [[Int]]) -> Int {
+    let n = grid.count
+    let m = grid[0].count
+    // 初始化 dp 表
+    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)
+    dp[0][0] = grid[0][0]
+    // 状态转移：首行
+    for j in stride(from: 1, to: m, by: 1) {
+        dp[0][j] = dp[0][j - 1] + grid[0][j]
+    }
+    // 状态转移：首列
+    for i in stride(from: 1, to: n, by: 1) {
+        dp[i][0] = dp[i - 1][0] + grid[i][0]
+    }
+    // 状态转移：其余行列
+    for i in stride(from: 1, to: n, by: 1) {
+        for j in stride(from: 1, to: m, by: 1) {
+            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]
+        }
+    }
+    return dp[n - 1][m - 1]
+}
+
+/* 最小路径和：状态压缩后的动态规划 */
+func minPathSumDPComp(grid: [[Int]]) -> Int {
+    let n = grid.count
+    let m = grid[0].count
+    // 初始化 dp 表
+    var dp = Array(repeating: 0, count: m)
+    // 状态转移：首行
+    dp[0] = grid[0][0]
+    for j in stride(from: 1, to: m, by: 1) {
+        dp[j] = dp[j - 1] + grid[0][j]
+    }
+    // 状态转移：其余行
+    for i in stride(from: 1, to: n, by: 1) {
+        // 状态转移：首列
+        dp[0] = dp[0] + grid[i][0]
+        // 状态转移：其余列
+        for j in stride(from: 1, to: m, by: 1) {
+            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]
+        }
+    }
+    return dp[m - 1]
+}
+
+@main
+enum MinPathSum {
+    /* Driver Code */
+    static func main() {
+        let grid = [
+            [1, 3, 1, 5],
+            [2, 2, 4, 2],
+            [5, 3, 2, 1],
+            [4, 3, 5, 2],
+        ]
+        let n = grid.count
+        let m = grid[0].count
+
+        // 暴力搜索
+        var res = minPathSumDFS(grid: grid, i: n - 1, j: m - 1)
+        print("从左上角到右下角的做小路径和为 \(res)")
+
+        // 记忆化搜索
+        var mem = Array(repeating: Array(repeating: -1, count: m), count: n)
+        res = minPathSumDFSMem(grid: grid, mem: &mem, i: n - 1, j: m - 1)
+        print("从左上角到右下角的做小路径和为 \(res)")
+
+        // 动态规划
+        res = minPathSumDP(grid: grid)
+        print("从左上角到右下角的做小路径和为 \(res)")
+
+        // 状态压缩后的动态规划
+        res = minPathSumDPComp(grid: grid)
+        print("从左上角到右下角的做小路径和为 \(res)")
+    }
+}

codes/swift/chapter_dynamic_programming/unbounded_knapsack.swift

@@ -0,0 +1,63 @@
+/**
+ * File: unbounded_knapsack.swift
+ * Created Time: 2023-07-15
+ * Author: nuomi1 (nuomi1@qq.com)
+ */
+
+/* 完全背包：动态规划 */
+func unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -> Int {
+    let n = wgt.count
+    // 初始化 dp 表
+    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)
+    // 状态转移
+    for i in stride(from: 1, through: n, by: 1) {
+        for c in stride(from: 1, through: cap, by: 1) {
+            if wgt[i - 1] > c {
+                // 若超过背包容量，则不选物品 i
+                dp[i][c] = dp[i - 1][c]
+            } else {
+                // 不选和选物品 i 这两种方案的较大值
+                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])
+            }
+        }
+    }
+    return dp[n][cap]
+}
+
+/* 完全背包：状态压缩后的动态规划 */
+func unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -> Int {
+    let n = wgt.count
+    // 初始化 dp 表
+    var dp = Array(repeating: 0, count: cap + 1)
+    // 状态转移
+    for i in stride(from: 1, through: n, by: 1) {
+        for c in stride(from: 1, through: cap, by: 1) {
+            if wgt[i - 1] > c {
+                // 若超过背包容量，则不选物品 i
+                dp[c] = dp[c]
+            } else {
+                // 不选和选物品 i 这两种方案的较大值
+                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])
+            }
+        }
+    }
+    return dp[cap]
+}
+
+@main
+enum UnboundedKnapsack {
+    /* Driver Code */
+    static func main() {
+        let wgt = [1, 2, 3]
+        let val = [5, 11, 15]
+        let cap = 4
+
+        // 动态规划
+        var res = unboundedKnapsackDP(wgt: wgt, val: val, cap: cap)
+        print("不超过背包容量的最大物品价值为 \(res)")
+
+        // 状态压缩后的动态规划
+        res = unboundedKnapsackDPComp(wgt: wgt, val: val, cap: cap)
+        print("不超过背包容量的最大物品价值为 \(res)")
+    }
+}