Translate all code to English (#1836)

* Review the EN heading format.

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en/codes/rust/chapter_dynamic_programming/climbing_stairs_backtrack.rs

@@ -0,0 +1,41 @@
+/*
+ * File: climbing_stairs_backtrack.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Backtracking */
+fn backtrack(choices: &[i32], state: i32, n: i32, res: &mut [i32]) {
+    // When climbing to the n-th stair, add 1 to the solution count
+    if state == n {
+        res[0] = res[0] + 1;
+    }
+    // Traverse all choices
+    for &choice in choices {
+        // Pruning: not allowed to go beyond the n-th stair
+        if state + choice > n {
+            continue;
+        }
+        // Attempt: make choice, update state
+        backtrack(choices, state + choice, n, res);
+        // Backtrack
+    }
+}
+
+/* Climbing stairs: Backtracking */
+fn climbing_stairs_backtrack(n: usize) -> i32 {
+    let choices = vec![1, 2]; // Can choose to climb up 1 or 2 stairs
+    let state = 0; // Start climbing from the 0-th stair
+    let mut res = Vec::new();
+    res.push(0); // Use res[0] to record the solution count
+    backtrack(&choices, state, n as i32, &mut res);
+    res[0]
+}
+
+/* Driver Code */
+pub fn main() {
+    let n: usize = 9;
+
+    let res = climbing_stairs_backtrack(n);
+    println!("Climbing {n} stairs has {res} solutions");
+}

en/codes/rust/chapter_dynamic_programming/climbing_stairs_constraint_dp.rs

@@ -0,0 +1,33 @@
+/*
+ * File: climbing_stairs_constraint_dp.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Climbing stairs with constraint: Dynamic programming */
+fn climbing_stairs_constraint_dp(n: usize) -> i32 {
+    if n == 1 || n == 2 {
+        return 1;
+    };
+    // Initialize dp table, used to store solutions to subproblems
+    let mut dp = vec![vec![-1; 3]; n + 1];
+    // Initial state: preset the solution to the smallest subproblem
+    dp[1][1] = 1;
+    dp[1][2] = 0;
+    dp[2][1] = 0;
+    dp[2][2] = 1;
+    // State transition: gradually solve larger subproblems from smaller ones
+    for i in 3..=n {
+        dp[i][1] = dp[i - 1][2];
+        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];
+    }
+    dp[n][1] + dp[n][2]
+}
+
+/* Driver Code */
+pub fn main() {
+    let n: usize = 9;
+
+    let res = climbing_stairs_constraint_dp(n);
+    println!("Climbing {n} stairs has {res} solutions");
+}

en/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs.rs

@@ -0,0 +1,29 @@
+/*
+ * File: climbing_stairs_dfs.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Search */
+fn dfs(i: usize) -> i32 {
+    // Known dp[1] and dp[2], return them
+    if i == 1 || i == 2 {
+        return i as i32;
+    }
+    // dp[i] = dp[i-1] + dp[i-2]
+    let count = dfs(i - 1) + dfs(i - 2);
+    count
+}
+
+/* Climbing stairs: Search */
+fn climbing_stairs_dfs(n: usize) -> i32 {
+    dfs(n)
+}
+
+/* Driver Code */
+pub fn main() {
+    let n: usize = 9;
+
+    let res = climbing_stairs_dfs(n);
+    println!("Climbing {n} stairs has {res} solutions");
+}

en/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs_mem.rs

@@ -0,0 +1,37 @@
+/*
+ * File: climbing_stairs_dfs_mem.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Memoization search */
+fn dfs(i: usize, mem: &mut [i32]) -> i32 {
+    // Known dp[1] and dp[2], return them
+    if i == 1 || i == 2 {
+        return i as i32;
+    }
+    // If record dp[i] exists, return it directly
+    if mem[i] != -1 {
+        return mem[i];
+    }
+    // dp[i] = dp[i-1] + dp[i-2]
+    let count = dfs(i - 1, mem) + dfs(i - 2, mem);
+    // Record dp[i]
+    mem[i] = count;
+    count
+}
+
+/* Climbing stairs: Memoization search */
+fn climbing_stairs_dfs_mem(n: usize) -> i32 {
+    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record
+    let mut mem = vec![-1; n + 1];
+    dfs(n, &mut mem)
+}
+
+/* Driver Code */
+pub fn main() {
+    let n: usize = 9;
+
+    let res = climbing_stairs_dfs_mem(n);
+    println!("Climbing {n} stairs has {res} solutions");
+}

en/codes/rust/chapter_dynamic_programming/climbing_stairs_dp.rs

@@ -0,0 +1,48 @@
+/*
+ * File: climbing_stairs_dp.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Climbing stairs: Dynamic programming */
+fn climbing_stairs_dp(n: usize) -> i32 {
+    // Known dp[1] and dp[2], return them
+    if n == 1 || n == 2 {
+        return n as i32;
+    }
+    // Initialize dp table, used to store solutions to subproblems
+    let mut dp = vec![-1; n + 1];
+    // Initial state: preset the solution to the smallest subproblem
+    dp[1] = 1;
+    dp[2] = 2;
+    // State transition: gradually solve larger subproblems from smaller ones
+    for i in 3..=n {
+        dp[i] = dp[i - 1] + dp[i - 2];
+    }
+    dp[n]
+}
+
+/* Climbing stairs: Space-optimized dynamic programming */
+fn climbing_stairs_dp_comp(n: usize) -> i32 {
+    if n == 1 || n == 2 {
+        return n as i32;
+    }
+    let (mut a, mut b) = (1, 2);
+    for _ in 3..=n {
+        let tmp = b;
+        b = a + b;
+        a = tmp;
+    }
+    b
+}
+
+/* Driver Code */
+pub fn main() {
+    let n: usize = 9;
+
+    let res = climbing_stairs_dp(n);
+    println!("Climbing {n} stairs has {res} solutions");
+
+    let res = climbing_stairs_dp_comp(n);
+    println!("Climbing {n} stairs has {res} solutions");
+}

en/codes/rust/chapter_dynamic_programming/coin_change.rs

@@ -0,0 +1,75 @@
+/*
+ * File: coin_change.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Coin change: Dynamic programming */
+fn coin_change_dp(coins: &[i32], amt: usize) -> i32 {
+    let n = coins.len();
+    let max = amt + 1;
+    // Initialize dp table
+    let mut dp = vec![vec![0; amt + 1]; n + 1];
+    // State transition: first row and first column
+    for a in 1..=amt {
+        dp[0][a] = max;
+    }
+    // State transition: rest of the rows and columns
+    for i in 1..=n {
+        for a in 1..=amt {
+            if coins[i - 1] > a as i32 {
+                // If exceeds target amount, don't select coin i
+                dp[i][a] = dp[i - 1][a];
+            } else {
+                // The smaller value between not selecting and selecting coin i
+                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);
+            }
+        }
+    }
+    if dp[n][amt] != max {
+        return dp[n][amt] as i32;
+    } else {
+        -1
+    }
+}
+
+/* Coin change: Space-optimized dynamic programming */
+fn coin_change_dp_comp(coins: &[i32], amt: usize) -> i32 {
+    let n = coins.len();
+    let max = amt + 1;
+    // Initialize dp table
+    let mut dp = vec![0; amt + 1];
+    dp.fill(max);
+    dp[0] = 0;
+    // State transition
+    for i in 1..=n {
+        for a in 1..=amt {
+            if coins[i - 1] > a as i32 {
+                // If exceeds target amount, don't select coin i
+                dp[a] = dp[a];
+            } else {
+                // The smaller value between not selecting and selecting coin i
+                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);
+            }
+        }
+    }
+    if dp[amt] != max {
+        return dp[amt] as i32;
+    } else {
+        -1
+    }
+}
+
+/* Driver Code */
+pub fn main() {
+    let coins = [1, 2, 5];
+    let amt: usize = 4;
+
+    // Dynamic programming
+    let res = coin_change_dp(&coins, amt);
+    println!("Minimum coins needed to make target amount is {res}");
+
+    // Space-optimized dynamic programming
+    let res = coin_change_dp_comp(&coins, amt);
+    println!("Minimum coins needed to make target amount is {res}");
+}

en/codes/rust/chapter_dynamic_programming/coin_change_ii.rs

@@ -0,0 +1,64 @@
+/*
+ * File: coin_change_ii.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Coin change II: Dynamic programming */
+fn coin_change_ii_dp(coins: &[i32], amt: usize) -> i32 {
+    let n = coins.len();
+    // Initialize dp table
+    let mut dp = vec![vec![0; amt + 1]; n + 1];
+    // Initialize first column
+    for i in 0..=n {
+        dp[i][0] = 1;
+    }
+    // State transition
+    for i in 1..=n {
+        for a in 1..=amt {
+            if coins[i - 1] > a as i32 {
+                // If exceeds target amount, don't select coin i
+                dp[i][a] = dp[i - 1][a];
+            } else {
+                // Sum of the two options: not selecting and selecting coin i
+                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];
+            }
+        }
+    }
+    dp[n][amt]
+}
+
+/* Coin change II: Space-optimized dynamic programming */
+fn coin_change_ii_dp_comp(coins: &[i32], amt: usize) -> i32 {
+    let n = coins.len();
+    // Initialize dp table
+    let mut dp = vec![0; amt + 1];
+    dp[0] = 1;
+    // State transition
+    for i in 1..=n {
+        for a in 1..=amt {
+            if coins[i - 1] > a as i32 {
+                // If exceeds target amount, don't select coin i
+                dp[a] = dp[a];
+            } else {
+                // Sum of the two options: not selecting and selecting coin i
+                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];
+            }
+        }
+    }
+    dp[amt]
+}
+
+/* Driver Code */
+pub fn main() {
+    let coins = [1, 2, 5];
+    let amt: usize = 5;
+
+    // Dynamic programming
+    let res = coin_change_ii_dp(&coins, amt);
+    println!("Number of coin combinations to make target amount is {res}");
+
+    // Space-optimized dynamic programming
+    let res = coin_change_ii_dp_comp(&coins, amt);
+    println!("Number of coin combinations to make target amount is {res}");
+}

en/codes/rust/chapter_dynamic_programming/edit_distance.rs

@@ -0,0 +1,145 @@
+/*
+ * File: edit_distance.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Edit distance: Brute-force search */
+fn edit_distance_dfs(s: &str, t: &str, i: usize, j: usize) -> i32 {
+    // If both s and t are empty, return 0
+    if i == 0 && j == 0 {
+        return 0;
+    }
+    // If s is empty, return length of t
+    if i == 0 {
+        return j as i32;
+    }
+    // If t is empty, return length of s
+    if j == 0 {
+        return i as i32;
+    }
+    // If two characters are equal, skip both characters
+    if s.chars().nth(i - 1) == t.chars().nth(j - 1) {
+        return edit_distance_dfs(s, t, i - 1, j - 1);
+    }
+    // Minimum edit steps = minimum edit steps of insert, delete, replace + 1
+    let insert = edit_distance_dfs(s, t, i, j - 1);
+    let delete = edit_distance_dfs(s, t, i - 1, j);
+    let replace = edit_distance_dfs(s, t, i - 1, j - 1);
+    // Return minimum edit steps
+    std::cmp::min(std::cmp::min(insert, delete), replace) + 1
+}
+
+/* Edit distance: Memoization search */
+fn edit_distance_dfs_mem(s: &str, t: &str, mem: &mut Vec<Vec<i32>>, i: usize, j: usize) -> i32 {
+    // If both s and t are empty, return 0
+    if i == 0 && j == 0 {
+        return 0;
+    }
+    // If s is empty, return length of t
+    if i == 0 {
+        return j as i32;
+    }
+    // If t is empty, return length of s
+    if j == 0 {
+        return i as i32;
+    }
+    // If there's a record, return it directly
+    if mem[i][j] != -1 {
+        return mem[i][j];
+    }
+    // If two characters are equal, skip both characters
+    if s.chars().nth(i - 1) == t.chars().nth(j - 1) {
+        return edit_distance_dfs_mem(s, t, mem, i - 1, j - 1);
+    }
+    // Minimum edit steps = minimum edit steps of insert, delete, replace + 1
+    let insert = edit_distance_dfs_mem(s, t, mem, i, j - 1);
+    let delete = edit_distance_dfs_mem(s, t, mem, i - 1, j);
+    let replace = edit_distance_dfs_mem(s, t, mem, i - 1, j - 1);
+    // Record and return minimum edit steps
+    mem[i][j] = std::cmp::min(std::cmp::min(insert, delete), replace) + 1;
+    mem[i][j]
+}
+
+/* Edit distance: Dynamic programming */
+fn edit_distance_dp(s: &str, t: &str) -> i32 {
+    let (n, m) = (s.len(), t.len());
+    let mut dp = vec![vec![0; m + 1]; n + 1];
+    // State transition: first row and first column
+    for i in 1..=n {
+        dp[i][0] = i as i32;
+    }
+    for j in 1..m {
+        dp[0][j] = j as i32;
+    }
+    // State transition: rest of the rows and columns
+    for i in 1..=n {
+        for j in 1..=m {
+            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {
+                // If two characters are equal, skip both characters
+                dp[i][j] = dp[i - 1][j - 1];
+            } else {
+                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1
+                dp[i][j] =
+                    std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;
+            }
+        }
+    }
+    dp[n][m]
+}
+
+/* Edit distance: Space-optimized dynamic programming */
+fn edit_distance_dp_comp(s: &str, t: &str) -> i32 {
+    let (n, m) = (s.len(), t.len());
+    let mut dp = vec![0; m + 1];
+    // State transition: first row
+    for j in 1..m {
+        dp[j] = j as i32;
+    }
+    // State transition: rest of the rows
+    for i in 1..=n {
+        // State transition: first column
+        let mut leftup = dp[0]; // Temporarily store dp[i-1, j-1]
+        dp[0] = i as i32;
+        // State transition: rest of the columns
+        for j in 1..=m {
+            let temp = dp[j];
+            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {
+                // If two characters are equal, skip both characters
+                dp[j] = leftup;
+            } else {
+                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1
+                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;
+            }
+            leftup = temp; // Update for next round's dp[i-1, j-1]
+        }
+    }
+    dp[m]
+}
+
+/* Driver Code */
+pub fn main() {
+    let s = "bag";
+    let t = "pack";
+    let (n, m) = (s.len(), t.len());
+
+    // Brute-force search
+    let res = edit_distance_dfs(s, t, n, m);
+    println!("Changing {s} to {t} requires minimum {res} edits");
+
+    // Memoization search
+    let mut mem = vec![vec![0; m + 1]; n + 1];
+    for row in mem.iter_mut() {
+        row.fill(-1);
+    }
+    let res = edit_distance_dfs_mem(s, t, &mut mem, n, m);
+    println!("Changing {s} to {t} requires minimum {res} edits");
+
+    // Dynamic programming
+    let res = edit_distance_dp(s, t);
+    println!("Changing {s} to {t} requires minimum {res} edits");
+
+    // Space-optimized dynamic programming
+    let res = edit_distance_dp_comp(s, t);
+    println!("Changing {s} to {t} requires minimum {res} edits");
+}

en/codes/rust/chapter_dynamic_programming/knapsack.rs

@@ -0,0 +1,113 @@
+/*
+ * File: knapsack.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* 0-1 knapsack: Brute-force search */
+fn knapsack_dfs(wgt: &[i32], val: &[i32], i: usize, c: usize) -> i32 {
+    // If all items have been selected or knapsack has no remaining capacity, return value 0
+    if i == 0 || c == 0 {
+        return 0;
+    }
+    // If exceeds knapsack capacity, can only choose not to put it in
+    if wgt[i - 1] > c as i32 {
+        return knapsack_dfs(wgt, val, i - 1, c);
+    }
+    // Calculate the maximum value of not putting in and putting in item i
+    let no = knapsack_dfs(wgt, val, i - 1, c);
+    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];
+    // Return the larger value of the two options
+    std::cmp::max(no, yes)
+}
+
+/* 0-1 knapsack: Memoization search */
+fn knapsack_dfs_mem(wgt: &[i32], val: &[i32], mem: &mut Vec<Vec<i32>>, i: usize, c: usize) -> i32 {
+    // If all items have been selected or knapsack has no remaining capacity, return value 0
+    if i == 0 || c == 0 {
+        return 0;
+    }
+    // If there's a record, return it directly
+    if mem[i][c] != -1 {
+        return mem[i][c];
+    }
+    // If exceeds knapsack capacity, can only choose not to put it in
+    if wgt[i - 1] > c as i32 {
+        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);
+    }
+    // Calculate the maximum value of not putting in and putting in item i
+    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);
+    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];
+    // Record and return the larger value of the two options
+    mem[i][c] = std::cmp::max(no, yes);
+    mem[i][c]
+}
+
+/* 0-1 knapsack: Dynamic programming */
+fn knapsack_dp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {
+    let n = wgt.len();
+    // Initialize dp table
+    let mut dp = vec![vec![0; cap + 1]; n + 1];
+    // State transition
+    for i in 1..=n {
+        for c in 1..=cap {
+            if wgt[i - 1] > c as i32 {
+                // If exceeds knapsack capacity, don't select item i
+                dp[i][c] = dp[i - 1][c];
+            } else {
+                // The larger value between not selecting and selecting item i
+                dp[i][c] = std::cmp::max(
+                    dp[i - 1][c],
+                    dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1],
+                );
+            }
+        }
+    }
+    dp[n][cap]
+}
+
+/* 0-1 knapsack: Space-optimized dynamic programming */
+fn knapsack_dp_comp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {
+    let n = wgt.len();
+    // Initialize dp table
+    let mut dp = vec![0; cap + 1];
+    // State transition
+    for i in 1..=n {
+        // Traverse in reverse order
+        for c in (1..=cap).rev() {
+            if wgt[i - 1] <= c as i32 {
+                // The larger value between not selecting and selecting item i
+                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);
+            }
+        }
+    }
+    dp[cap]
+}
+
+/* Driver Code */
+pub fn main() {
+    let wgt = [10, 20, 30, 40, 50];
+    let val = [50, 120, 150, 210, 240];
+    let cap: usize = 50;
+    let n = wgt.len();
+
+    // Brute-force search
+    let res = knapsack_dfs(&wgt, &val, n, cap);
+    println!("Maximum item value not exceeding knapsack capacity is {res}");
+
+    // Memoization search
+    let mut mem = vec![vec![0; cap + 1]; n + 1];
+    for row in mem.iter_mut() {
+        row.fill(-1);
+    }
+    let res = knapsack_dfs_mem(&wgt, &val, &mut mem, n, cap);
+    println!("Maximum item value not exceeding knapsack capacity is {res}");
+
+    // Dynamic programming
+    let res = knapsack_dp(&wgt, &val, cap);
+    println!("Maximum item value not exceeding knapsack capacity is {res}");
+
+    // Space-optimized dynamic programming
+    let res = knapsack_dp_comp(&wgt, &val, cap);
+    println!("Maximum item value not exceeding knapsack capacity is {res}");
+}

en/codes/rust/chapter_dynamic_programming/min_cost_climbing_stairs_dp.rs

@@ -0,0 +1,52 @@
+/*
+ * File: min_cost_climbing_stairs_dp.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+use std::cmp;
+
+/* Minimum cost climbing stairs: Dynamic programming */
+fn min_cost_climbing_stairs_dp(cost: &[i32]) -> i32 {
+    let n = cost.len() - 1;
+    if n == 1 || n == 2 {
+        return cost[n];
+    }
+    // Initialize dp table, used to store solutions to subproblems
+    let mut dp = vec![-1; n + 1];
+    // Initial state: preset the solution to the smallest subproblem
+    dp[1] = cost[1];
+    dp[2] = cost[2];
+    // State transition: gradually solve larger subproblems from smaller ones
+    for i in 3..=n {
+        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];
+    }
+    dp[n]
+}
+
+/* Minimum cost climbing stairs: Space-optimized dynamic programming */
+fn min_cost_climbing_stairs_dp_comp(cost: &[i32]) -> i32 {
+    let n = cost.len() - 1;
+    if n == 1 || n == 2 {
+        return cost[n];
+    };
+    let (mut a, mut b) = (cost[1], cost[2]);
+    for i in 3..=n {
+        let tmp = b;
+        b = cmp::min(a, tmp) + cost[i];
+        a = tmp;
+    }
+    b
+}
+
+/* Driver Code */
+pub fn main() {
+    let cost = [0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1];
+    println!("Input stair cost list is {:?}", &cost);
+
+    let res = min_cost_climbing_stairs_dp(&cost);
+    println!("Minimum cost to climb stairs is {res}");
+
+    let res = min_cost_climbing_stairs_dp_comp(&cost);
+    println!("Minimum cost to climb stairs is {res}");
+}

en/codes/rust/chapter_dynamic_programming/min_path_sum.rs

@@ -0,0 +1,120 @@
+/*
+ * File: min_path_sum.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Minimum path sum: Brute-force search */
+fn min_path_sum_dfs(grid: &Vec<Vec<i32>>, i: i32, j: i32) -> i32 {
+    // If it's the top-left cell, terminate the search
+    if i == 0 && j == 0 {
+        return grid[0][0];
+    }
+    // If row or column index is out of bounds, return +∞ cost
+    if i < 0 || j < 0 {
+        return i32::MAX;
+    }
+    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)
+    let up = min_path_sum_dfs(grid, i - 1, j);
+    let left = min_path_sum_dfs(grid, i, j - 1);
+    // Return the minimum path cost from top-left to (i, j)
+    std::cmp::min(left, up) + grid[i as usize][j as usize]
+}
+
+/* Minimum path sum: Memoization search */
+fn min_path_sum_dfs_mem(grid: &Vec<Vec<i32>>, mem: &mut Vec<Vec<i32>>, i: i32, j: i32) -> i32 {
+    // If it's the top-left cell, terminate the search
+    if i == 0 && j == 0 {
+        return grid[0][0];
+    }
+    // If row or column index is out of bounds, return +∞ cost
+    if i < 0 || j < 0 {
+        return i32::MAX;
+    }
+    // If there's a record, return it directly
+    if mem[i as usize][j as usize] != -1 {
+        return mem[i as usize][j as usize];
+    }
+    // Minimum path cost for left and upper cells
+    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);
+    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);
+    // Record and return the minimum path cost from top-left to (i, j)
+    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];
+    mem[i as usize][j as usize]
+}
+
+/* Minimum path sum: Dynamic programming */
+fn min_path_sum_dp(grid: &Vec<Vec<i32>>) -> i32 {
+    let (n, m) = (grid.len(), grid[0].len());
+    // Initialize dp table
+    let mut dp = vec![vec![0; m]; n];
+    dp[0][0] = grid[0][0];
+    // State transition: first row
+    for j in 1..m {
+        dp[0][j] = dp[0][j - 1] + grid[0][j];
+    }
+    // State transition: first column
+    for i in 1..n {
+        dp[i][0] = dp[i - 1][0] + grid[i][0];
+    }
+    // State transition: rest of the rows and columns
+    for i in 1..n {
+        for j in 1..m {
+            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
+        }
+    }
+    dp[n - 1][m - 1]
+}
+
+/* Minimum path sum: Space-optimized dynamic programming */
+fn min_path_sum_dp_comp(grid: &Vec<Vec<i32>>) -> i32 {
+    let (n, m) = (grid.len(), grid[0].len());
+    // Initialize dp table
+    let mut dp = vec![0; m];
+    // State transition: first row
+    dp[0] = grid[0][0];
+    for j in 1..m {
+        dp[j] = dp[j - 1] + grid[0][j];
+    }
+    // State transition: rest of the rows
+    for i in 1..n {
+        // State transition: first column
+        dp[0] = dp[0] + grid[i][0];
+        // State transition: rest of the columns
+        for j in 1..m {
+            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];
+        }
+    }
+    dp[m - 1]
+}
+
+/* Driver Code */
+pub fn main() {
+    let grid = vec![
+        vec![1, 3, 1, 5],
+        vec![2, 2, 4, 2],
+        vec![5, 3, 2, 1],
+        vec![4, 3, 5, 2],
+    ];
+    let (n, m) = (grid.len(), grid[0].len());
+
+    // Brute-force search
+    let res = min_path_sum_dfs(&grid, n as i32 - 1, m as i32 - 1);
+    println!("Minimum path sum from top-left to bottom-right is {res}");
+
+    // Memoization search
+    let mut mem = vec![vec![0; m]; n];
+    for row in mem.iter_mut() {
+        row.fill(-1);
+    }
+    let res = min_path_sum_dfs_mem(&grid, &mut mem, n as i32 - 1, m as i32 - 1);
+    println!("Minimum path sum from top-left to bottom-right is {res}");
+
+    // Dynamic programming
+    let res = min_path_sum_dp(&grid);
+    println!("Minimum path sum from top-left to bottom-right is {res}");
+
+    // Space-optimized dynamic programming
+    let res = min_path_sum_dp_comp(&grid);
+    println!("Minimum path sum from top-left to bottom-right is {res}");
+}

en/codes/rust/chapter_dynamic_programming/unbounded_knapsack.rs

@@ -0,0 +1,60 @@
+/*
+ * File: unbounded_knapsack.rs
+ * Created Time: 2023-07-09
+ * Author: codingonion (coderonion@gmail.com)
+ */
+
+/* Unbounded knapsack: Dynamic programming */
+fn unbounded_knapsack_dp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {
+    let n = wgt.len();
+    // Initialize dp table
+    let mut dp = vec![vec![0; cap + 1]; n + 1];
+    // State transition
+    for i in 1..=n {
+        for c in 1..=cap {
+            if wgt[i - 1] > c as i32 {
+                // If exceeds knapsack capacity, don't select item i
+                dp[i][c] = dp[i - 1][c];
+            } else {
+                // The larger value between not selecting and selecting item i
+                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);
+            }
+        }
+    }
+    return dp[n][cap];
+}
+
+/* Unbounded knapsack: Space-optimized dynamic programming */
+fn unbounded_knapsack_dp_comp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {
+    let n = wgt.len();
+    // Initialize dp table
+    let mut dp = vec![0; cap + 1];
+    // State transition
+    for i in 1..=n {
+        for c in 1..=cap {
+            if wgt[i - 1] > c as i32 {
+                // If exceeds knapsack capacity, don't select item i
+                dp[c] = dp[c];
+            } else {
+                // The larger value between not selecting and selecting item i
+                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);
+            }
+        }
+    }
+    dp[cap]
+}
+
+/* Driver Code */
+pub fn main() {
+    let wgt = [1, 2, 3];
+    let val = [5, 11, 15];
+    let cap: usize = 4;
+
+    // Dynamic programming
+    let res = unbounded_knapsack_dp(&wgt, &val, cap);
+    println!("Maximum item value not exceeding knapsack capacity is {res}");
+
+    // Space-optimized dynamic programming
+    let res = unbounded_knapsack_dp_comp(&wgt, &val, cap);
+    println!("Maximum item value not exceeding knapsack capacity is {res}");
+}