Add the initial EN translation for C++ code (#1346)

en/codes/cpp/chapter_dynamic_programming/CMakeLists.txt

@@ -0,0 +1,10 @@
+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)
+add_executable(min_path_sum min_path_sum.cpp)
+add_executable(unbounded_knapsack unbounded_knapsack.cpp)
+add_executable(coin_change coin_change.cpp)
+add_executable(coin_change_ii coin_change_ii.cpp)
+add_executable(edit_distance edit_distance.cpp)

en/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"
+
+/* Backtracking */
+void backtrack(vector<int> &choices, int state, int n, vector<int> &res) {
+    // When climbing to the nth step, add 1 to the number of solutions
+    if (state == n)
+        res[0]++;
+    // Traverse all choices
+    for (auto &choice : choices) {
+        // Pruning: do not allow climbing beyond the nth step
+        if (state + choice > n)
+            continue;
+        // Attempt: make a choice, update the state
+        backtrack(choices, state + choice, n, res);
+        // Retract
+    }
+}
+
+/* Climbing stairs: Backtracking */
+int climbingStairsBacktrack(int n) {
+    vector<int> choices = {1, 2}; // Can choose to climb up 1 step or 2 steps
+    int state = 0;                // Start climbing from the 0th step
+    vector<int> res = {0};        // Use res[0] to record the number of solutions
+    backtrack(choices, state, n, res);
+    return res[0];
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsBacktrack(n);
+    cout << "There are " << res << " solutions to climb " << n << " stairs" << endl;
+
+    return 0;
+}

en/codes/cpp/chapter_dynamic_programming/climbing_stairs_constraint_dp.cpp

@@ -0,0 +1,37 @@
+/**
+ * File: climbing_stairs_constraint_dp.cpp
+ * Created Time: 2023-07-01
+ * Author: krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* Constrained climbing stairs: Dynamic programming */
+int climbingStairsConstraintDP(int n) {
+    if (n == 1 || n == 2) {
+        return 1;
+    }
+    // Initialize dp table, used to store subproblem solutions
+    vector<vector<int>> dp(n + 1, vector<int>(3, 0));
+    // Initial state: preset the smallest subproblem solution
+    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 (int i = 3; i <= n; i++) {
+        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];
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsConstraintDP(n);
+    cout << "There are " << res << " solutions to climb " << n << " stairs" << endl;
+
+    return 0;
+}

en/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"
+
+/* Search */
+int dfs(int i) {
+    // Known dp[1] and dp[2], return them
+    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;
+}
+
+/* Climbing stairs: Search */
+int climbingStairsDFS(int n) {
+    return dfs(n);
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsDFS(n);
+    cout << "There are " << res << " solutions to climb " << n << " stairs" << endl;
+
+    return 0;
+}

en/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"
+
+/* Memoized search */
+int dfs(int i, vector<int> &mem) {
+    // Known dp[1] and dp[2], return them
+    if (i == 1 || i == 2)
+        return i;
+    // If there is a record for dp[i], return it
+    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);
+    // Record dp[i]
+    mem[i] = count;
+    return count;
+}
+
+/* Climbing stairs: Memoized search */
+int climbingStairsDFSMem(int n) {
+    // mem[i] records the total number of solutions for climbing to the ith step, -1 means no record
+    vector<int> mem(n + 1, -1);
+    return dfs(n, mem);
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsDFSMem(n);
+    cout << "There are " << res << " solutions to climb " << n << " stairs" << endl;
+
+    return 0;
+}

en/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"
+
+/* Climbing stairs: Dynamic programming */
+int climbingStairsDP(int n) {
+    if (n == 1 || n == 2)
+        return n;
+    // Initialize dp table, used to store subproblem solutions
+    vector<int> dp(n + 1);
+    // Initial state: preset the smallest subproblem solution
+    dp[1] = 1;
+    dp[2] = 2;
+    // State transition: gradually solve larger subproblems from smaller ones
+    for (int i = 3; i <= n; i++) {
+        dp[i] = dp[i - 1] + dp[i - 2];
+    }
+    return dp[n];
+}
+
+/* Climbing stairs: Space-optimized dynamic programming */
+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 << "There are " << res << " solutions to climb " << n << " stairs" << endl;
+
+    res = climbingStairsDPComp(n);
+    cout << "There are " << res << " solutions to climb " << n << " stairs" << endl;
+
+    return 0;
+}

en/codes/cpp/chapter_dynamic_programming/coin_change.cpp

@@ -0,0 +1,70 @@
+/**
+ * File: coin_change.cpp
+ * Created Time: 2023-07-11
+ * Author: krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* Coin change: Dynamic programming */
+int coinChangeDP(vector<int> &coins, int amt) {
+    int n = coins.size();
+    int MAX = amt + 1;
+    // Initialize dp table
+    vector<vector<int>> dp(n + 1, vector<int>(amt + 1, 0));
+    // State transition: first row and first column
+    for (int a = 1; a <= amt; a++) {
+        dp[0][a] = MAX;
+    }
+    // State transition: the rest of the rows and columns
+    for (int i = 1; i <= n; i++) {
+        for (int a = 1; a <= amt; a++) {
+            if (coins[i - 1] > a) {
+                // If exceeding the target amount, do not choose coin i
+                dp[i][a] = dp[i - 1][a];
+            } else {
+                // The smaller value between not choosing and choosing coin i
+                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);
+            }
+        }
+    }
+    return dp[n][amt] != MAX ? dp[n][amt] : -1;
+}
+
+/* Coin change: Space-optimized dynamic programming */
+int coinChangeDPComp(vector<int> &coins, int amt) {
+    int n = coins.size();
+    int MAX = amt + 1;
+    // Initialize dp table
+    vector<int> dp(amt + 1, MAX);
+    dp[0] = 0;
+    // State transition
+    for (int i = 1; i <= n; i++) {
+        for (int a = 1; a <= amt; a++) {
+            if (coins[i - 1] > a) {
+                // If exceeding the target amount, do not choose coin i
+                dp[a] = dp[a];
+            } else {
+                // The smaller value between not choosing and choosing coin i
+                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);
+            }
+        }
+    }
+    return dp[amt] != MAX ? dp[amt] : -1;
+}
+
+/* Driver code */
+int main() {
+    vector<int> coins = {1, 2, 5};
+    int amt = 4;
+
+    // Dynamic programming
+    int res = coinChangeDP(coins, amt);
+    cout << "The minimum number of coins required to make up the target amount is " << res << endl;
+
+    // Space-optimized dynamic programming
+    res = coinChangeDPComp(coins, amt);
+    cout << "The minimum number of coins required to make up the target amount is " << res << endl;
+
+    return 0;
+}

en/codes/cpp/chapter_dynamic_programming/coin_change_ii.cpp

@@ -0,0 +1,68 @@
+/**
+ * File: coin_change_ii.cpp
+ * Created Time: 2023-07-11
+ * Author: krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* Coin change II: Dynamic programming */
+int coinChangeIIDP(vector<int> &coins, int amt) {
+    int n = coins.size();
+    // Initialize dp table
+    vector<vector<int>> dp(n + 1, vector<int>(amt + 1, 0));
+    // Initialize first column
+    for (int i = 0; i <= n; i++) {
+        dp[i][0] = 1;
+    }
+    // State transition
+    for (int i = 1; i <= n; i++) {
+        for (int a = 1; a <= amt; a++) {
+            if (coins[i - 1] > a) {
+                // If exceeding the target amount, do not choose coin i
+                dp[i][a] = dp[i - 1][a];
+            } else {
+                // The sum of the two options of not choosing and choosing coin i
+                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];
+            }
+        }
+    }
+    return dp[n][amt];
+}
+
+/* Coin change II: Space-optimized dynamic programming */
+int coinChangeIIDPComp(vector<int> &coins, int amt) {
+    int n = coins.size();
+    // Initialize dp table
+    vector<int> dp(amt + 1, 0);
+    dp[0] = 1;
+    // State transition
+    for (int i = 1; i <= n; i++) {
+        for (int a = 1; a <= amt; a++) {
+            if (coins[i - 1] > a) {
+                // If exceeding the target amount, do not choose coin i
+                dp[a] = dp[a];
+            } else {
+                // The sum of the two options of not choosing and choosing coin i
+                dp[a] = dp[a] + dp[a - coins[i - 1]];
+            }
+        }
+    }
+    return dp[amt];
+}
+
+/* Driver code */
+int main() {
+    vector<int> coins = {1, 2, 5};
+    int amt = 5;
+
+    // Dynamic programming
+    int res = coinChangeIIDP(coins, amt);
+    cout << "The number of coin combinations to make up the target amount is " << res << endl;
+
+    // Space-optimized dynamic programming
+    res = coinChangeIIDPComp(coins, amt);
+    cout << "The number of coin combinations to make up the target amount is " << res << endl;
+
+    return 0;
+}

en/codes/cpp/chapter_dynamic_programming/edit_distance.cpp

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+/**
+ * File: edit_distance.cpp
+ * Created Time: 2023-07-13
+ * Author: krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* Edit distance: Brute force search */
+int editDistanceDFS(string s, string t, int i, int j) {
+    // If both s and t are empty, return 0
+    if (i == 0 && j == 0)
+        return 0;
+    // If s is empty, return the length of t
+    if (i == 0)
+        return j;
+    // If t is empty, return the length of s
+    if (j == 0)
+        return i;
+    // If the two characters are equal, skip these two characters
+    if (s[i - 1] == t[j - 1])
+        return editDistanceDFS(s, t, i - 1, j - 1);
+    // The minimum number of edits = the minimum number of edits from three operations (insert, remove, replace) + 1
+    int insert = editDistanceDFS(s, t, i, j - 1);
+    int del = editDistanceDFS(s, t, i - 1, j);
+    int replace = editDistanceDFS(s, t, i - 1, j - 1);
+    // Return the minimum number of edits
+    return min(min(insert, del), replace) + 1;
+}
+
+/* Edit distance: Memoized search */
+int editDistanceDFSMem(string s, string t, vector<vector<int>> &mem, int i, int j) {
+    // If both s and t are empty, return 0
+    if (i == 0 && j == 0)
+        return 0;
+    // If s is empty, return the length of t
+    if (i == 0)
+        return j;
+    // If t is empty, return the length of s
+    if (j == 0)
+        return i;
+    // If there is a record, return it
+    if (mem[i][j] != -1)
+        return mem[i][j];
+    // If the two characters are equal, skip these two characters
+    if (s[i - 1] == t[j - 1])
+        return editDistanceDFSMem(s, t, mem, i - 1, j - 1);
+    // The minimum number of edits = the minimum number of edits from three operations (insert, remove, replace) + 1
+    int insert = editDistanceDFSMem(s, t, mem, i, j - 1);
+    int del = editDistanceDFSMem(s, t, mem, i - 1, j);
+    int replace = editDistanceDFSMem(s, t, mem, i - 1, j - 1);
+    // Record and return the minimum number of edits
+    mem[i][j] = min(min(insert, del), replace) + 1;
+    return mem[i][j];
+}
+
+/* Edit distance: Dynamic programming */
+int editDistanceDP(string s, string t) {
+    int n = s.length(), m = t.length();
+    vector<vector<int>> dp(n + 1, vector<int>(m + 1, 0));
+    // State transition: first row and first column
+    for (int i = 1; i <= n; i++) {
+        dp[i][0] = i;
+    }
+    for (int j = 1; j <= m; j++) {
+        dp[0][j] = j;
+    }
+    // State transition: the rest of the rows and columns
+    for (int i = 1; i <= n; i++) {
+        for (int j = 1; j <= m; j++) {
+            if (s[i - 1] == t[j - 1]) {
+                // If the two characters are equal, skip these two characters
+                dp[i][j] = dp[i - 1][j - 1];
+            } else {
+                // The minimum number of edits = the minimum number of edits from three operations (insert, remove, replace) + 1
+                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;
+            }
+        }
+    }
+    return dp[n][m];
+}
+
+/* Edit distance: Space-optimized dynamic programming */
+int editDistanceDPComp(string s, string t) {
+    int n = s.length(), m = t.length();
+    vector<int> dp(m + 1, 0);
+    // State transition: first row
+    for (int j = 1; j <= m; j++) {
+        dp[j] = j;
+    }
+    // State transition: the rest of the rows
+    for (int i = 1; i <= n; i++) {
+        // State transition: first column
+        int leftup = dp[0]; // Temporarily store dp[i-1, j-1]
+        dp[0] = i;
+        // State transition: the rest of the columns
+        for (int j = 1; j <= m; j++) {
+            int temp = dp[j];
+            if (s[i - 1] == t[j - 1]) {
+                // If the two characters are equal, skip these two characters
+                dp[j] = leftup;
+            } else {
+                // The minimum number of edits = the minimum number of edits from three operations (insert, remove, replace) + 1
+                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;
+            }
+            leftup = temp; // Update for the next round of dp[i-1, j-1]
+        }
+    }
+    return dp[m];
+}
+
+/* Driver Code */
+int main() {
+    string s = "bag";
+    string t = "pack";
+    int n = s.length(), m = t.length();
+
+    // Brute force search
+    int res = editDistanceDFS(s, t, n, m);
+    cout << "Changing " << s << " to " << t << " requires a minimum of " << res << " edits.\n";
+
+    // Memoized search
+    vector<vector<int>> mem(n + 1, vector<int>(m + 1, -1));
+    res = editDistanceDFSMem(s, t, mem, n, m);
+    cout << "Changing " << s << " to " << t << " requires a minimum of " << res << " edits.\n";
+
+    // Dynamic programming
+    res = editDistanceDP(s, t);
+    cout << "Changing " << s << " to " << t << " requires a minimum of " << res << " edits.\n";
+
+    // Space-optimized dynamic programming
+    res = editDistanceDPComp(s, t);
+    cout << "Changing " << s << " to " << t << " requires a minimum of " << res << " edits.\n";
+
+    return 0;
+}

en/codes/cpp/chapter_dynamic_programming/knapsack.cpp

@@ -0,0 +1,109 @@
+#include <algorithm>
+#include <iostream>
+#include <vector>
+
+using namespace std;
+
+/* 0-1 Knapsack: Brute force search */
+int knapsackDFS(vector<int> &wgt, vector<int> &val, int i, int c) {
+    // If all items have been chosen or the knapsack has no remaining capacity, return value 0
+    if (i == 0 || c == 0) {
+        return 0;
+    }
+    // If exceeding the knapsack capacity, can only choose not to put it in the knapsack
+    if (wgt[i - 1] > c) {
+        return knapsackDFS(wgt, val, i - 1, c);
+    }
+    // Calculate the maximum value of not putting in and putting in item i
+    int no = knapsackDFS(wgt, val, i - 1, c);
+    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];
+    // Return the greater value of the two options
+    return max(no, yes);
+}
+
+/* 0-1 Knapsack: Memoized search */
+int knapsackDFSMem(vector<int> &wgt, vector<int> &val, vector<vector<int>> &mem, int i, int c) {
+    // If all items have been chosen or the knapsack has no remaining capacity, return value 0
+    if (i == 0 || c == 0) {
+        return 0;
+    }
+    // If there is a record, return it
+    if (mem[i][c] != -1) {
+        return mem[i][c];
+    }
+    // If exceeding the knapsack capacity, can only choose not to put it in the knapsack
+    if (wgt[i - 1] > c) {
+        return knapsackDFSMem(wgt, val, mem, i - 1, c);
+    }
+    // Calculate the maximum value of not putting in and putting in item i
+    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);
+    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];
+    // Record and return the greater value of the two options
+    mem[i][c] = max(no, yes);
+    return mem[i][c];
+}
+
+/* 0-1 Knapsack: Dynamic programming */
+int knapsackDP(vector<int> &wgt, vector<int> &val, int cap) {
+    int n = wgt.size();
+    // Initialize dp table
+    vector<vector<int>> dp(n + 1, vector<int>(cap + 1, 0));
+    // State transition
+    for (int i = 1; i <= n; i++) {
+        for (int c = 1; c <= cap; c++) {
+            if (wgt[i - 1] > c) {
+                // If exceeding the knapsack capacity, do not choose item i
+                dp[i][c] = dp[i - 1][c];
+            } else {
+                // The greater value between not choosing and choosing item i
+                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);
+            }
+        }
+    }
+    return dp[n][cap];
+}
+
+/* 0-1 Knapsack: Space-optimized dynamic programming */
+int knapsackDPComp(vector<int> &wgt, vector<int> &val, int cap) {
+    int n = wgt.size();
+    // Initialize dp table
+    vector<int> dp(cap + 1, 0);
+    // State transition
+    for (int i = 1; i <= n; i++) {
+        // Traverse in reverse order
+        for (int c = cap; c >= 1; c--) {
+            if (wgt[i - 1] <= c) {
+                // The greater value between not choosing and choosing item i
+                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
+            }
+        }
+    }
+    return dp[cap];
+}
+
+/* Driver Code */
+int main() {
+    vector<int> wgt = {10, 20, 30, 40, 50};
+    vector<int> val = {50, 120, 150, 210, 240};
+    int cap = 50;
+    int n = wgt.size();
+
+    // Brute force search
+    int res = knapsackDFS(wgt, val, n, cap);
+    cout << "The maximum value within the bag capacity is " << res << endl;
+
+    // Memoized search
+    vector<vector<int>> mem(n + 1, vector<int>(cap + 1, -1));
+    res = knapsackDFSMem(wgt, val, mem, n, cap);
+    cout << "The maximum value within the bag capacity is " << res << endl;
+
+    // Dynamic programming
+    res = knapsackDP(wgt, val, cap);
+    cout << "The maximum value within the bag capacity is " << res << endl;
+
+    // Space-optimized dynamic programming
+    res = knapsackDPComp(wgt, val, cap);
+    cout << "The maximum value within the bag capacity is " << res << endl;
+
+    return 0;
+}

en/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"
+
+/* Climbing stairs with minimum cost: Dynamic programming */
+int minCostClimbingStairsDP(vector<int> &cost) {
+    int n = cost.size() - 1;
+    if (n == 1 || n == 2)
+        return cost[n];
+    // Initialize dp table, used to store subproblem solutions
+    vector<int> dp(n + 1);
+    // Initial state: preset the smallest subproblem solution
+    dp[1] = cost[1];
+    dp[2] = cost[2];
+    // State transition: gradually solve larger subproblems from smaller ones
+    for (int i = 3; i <= n; i++) {
+        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];
+    }
+    return dp[n];
+}
+
+/* Climbing stairs with minimum cost: Space-optimized dynamic programming */
+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 << "Input the cost list for stairs";
+    printVector(cost);
+
+    int res = minCostClimbingStairsDP(cost);
+    cout << "Minimum cost to climb the stairs " << res << endl;
+
+    res = minCostClimbingStairsDPComp(cost);
+    cout << "Minimum cost to climb the stairs " << res << endl;
+
+    return 0;
+}

en/codes/cpp/chapter_dynamic_programming/min_path_sum.cpp

@@ -0,0 +1,116 @@
+/**
+ * File: min_path_sum.cpp
+ * Created Time: 2023-07-10
+ * Author: krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* Minimum path sum: Brute force search */
+int minPathSumDFS(vector<vector<int>> &grid, int i, int j) {
+    // If it's the top-left cell, terminate the search
+    if (i == 0 && j == 0) {
+        return grid[0][0];
+    }
+    // If the row or column index is out of bounds, return a +∞ cost
+    if (i < 0 || j < 0) {
+        return INT_MAX;
+    }
+    // Calculate the minimum path cost from the top-left to (i-1, j) and (i, j-1)
+    int up = minPathSumDFS(grid, i - 1, j);
+    int left = minPathSumDFS(grid, i, j - 1);
+    // Return the minimum path cost from the top-left to (i, j)
+    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
+}
+
+/* Minimum path sum: Memoized search */
+int minPathSumDFSMem(vector<vector<int>> &grid, vector<vector<int>> &mem, int i, int j) {
+    // If it's the top-left cell, terminate the search
+    if (i == 0 && j == 0) {
+        return grid[0][0];
+    }
+    // If the row or column index is out of bounds, return a +∞ cost
+    if (i < 0 || j < 0) {
+        return INT_MAX;
+    }
+    // If there is a record, return it
+    if (mem[i][j] != -1) {
+        return mem[i][j];
+    }
+    // The minimum path cost from the left and top cells
+    int up = minPathSumDFSMem(grid, mem, i - 1, j);
+    int left = minPathSumDFSMem(grid, mem, i, j - 1);
+    // Record and return the minimum path cost from the top-left to (i, j)
+    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
+    return mem[i][j];
+}
+
+/* Minimum path sum: Dynamic programming */
+int minPathSumDP(vector<vector<int>> &grid) {
+    int n = grid.size(), m = grid[0].size();
+    // Initialize dp table
+    vector<vector<int>> dp(n, vector<int>(m));
+    dp[0][0] = grid[0][0];
+    // State transition: first row
+    for (int j = 1; j < m; j++) {
+        dp[0][j] = dp[0][j - 1] + grid[0][j];
+    }
+    // State transition: first column
+    for (int i = 1; i < n; i++) {
+        dp[i][0] = dp[i - 1][0] + grid[i][0];
+    }
+    // State transition: the rest of the rows and columns
+    for (int i = 1; i < n; i++) {
+        for (int j = 1; j < m; j++) {
+            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
+        }
+    }
+    return dp[n - 1][m - 1];
+}
+
+/* Minimum path sum: Space-optimized dynamic programming */
+int minPathSumDPComp(vector<vector<int>> &grid) {
+    int n = grid.size(), m = grid[0].size();
+    // Initialize dp table
+    vector<int> dp(m);
+    // State transition: first row
+    dp[0] = grid[0][0];
+    for (int j = 1; j < m; j++) {
+        dp[j] = dp[j - 1] + grid[0][j];
+    }
+    // State transition: the rest of the rows
+    for (int i = 1; i < n; i++) {
+        // State transition: first column
+        dp[0] = dp[0] + grid[i][0];
+        // State transition: the rest of the columns
+        for (int j = 1; j < m; j++) {
+            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];
+        }
+    }
+    return dp[m - 1];
+}
+
+/* Driver Code */
+int main() {
+    vector<vector<int>> grid = {{1, 3, 1, 5}, {2, 2, 4, 2}, {5, 3, 2, 1}, {4, 3, 5, 2}};
+    int n = grid.size(), m = grid[0].size();
+
+    // Brute force search
+    int res = minPathSumDFS(grid, n - 1, m - 1);
+    cout << "The minimum path sum from the top left corner to the bottom right corner is " << res << endl;
+
+    // Memoized search
+    vector<vector<int>> mem(n, vector<int>(m, -1));
+    res = minPathSumDFSMem(grid, mem, n - 1, m - 1);
+    cout << "The minimum path sum from the top left corner to the bottom right corner is " << res << endl;
+
+    // Dynamic programming
+    res = minPathSumDP(grid);
+    cout << "The minimum path sum from the top left corner to the bottom right corner is " << res << endl;
+
+    // Space-optimized dynamic programming
+    res = minPathSumDPComp(grid);
+    cout << "The minimum path sum from the top left corner to the bottom right corner is " << res << endl;
+
+    return 0;
+}

en/codes/cpp/chapter_dynamic_programming/unbounded_knapsack.cpp

@@ -0,0 +1,64 @@
+/**
+ * File: unbounded_knapsack.cpp
+ * Created Time: 2023-07-11
+ * Author: krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* Complete knapsack: Dynamic programming */
+int unboundedKnapsackDP(vector<int> &wgt, vector<int> &val, int cap) {
+    int n = wgt.size();
+    // Initialize dp table
+    vector<vector<int>> dp(n + 1, vector<int>(cap + 1, 0));
+    // State transition
+    for (int i = 1; i <= n; i++) {
+        for (int c = 1; c <= cap; c++) {
+            if (wgt[i - 1] > c) {
+                // If exceeding the knapsack capacity, do not choose item i
+                dp[i][c] = dp[i - 1][c];
+            } else {
+                // The greater value between not choosing and choosing item i
+                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);
+            }
+        }
+    }
+    return dp[n][cap];
+}
+
+/* Complete knapsack: Space-optimized dynamic programming */
+int unboundedKnapsackDPComp(vector<int> &wgt, vector<int> &val, int cap) {
+    int n = wgt.size();
+    // Initialize dp table
+    vector<int> dp(cap + 1, 0);
+    // State transition
+    for (int i = 1; i <= n; i++) {
+        for (int c = 1; c <= cap; c++) {
+            if (wgt[i - 1] > c) {
+                // If exceeding the knapsack capacity, do not choose item i
+                dp[c] = dp[c];
+            } else {
+                // The greater value between not choosing and choosing item i
+                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
+            }
+        }
+    }
+    return dp[cap];
+}
+
+/* Driver code */
+int main() {
+    vector<int> wgt = {1, 2, 3};
+    vector<int> val = {5, 11, 15};
+    int cap = 4;
+
+    // Dynamic programming
+    int res = unboundedKnapsackDP(wgt, val, cap);
+    cout << "The maximum value within the bag capacity is " << res << endl;
+
+    // Space-optimized dynamic programming
+    res = unboundedKnapsackDPComp(wgt, val, cap);
+    cout << "The maximum value within the bag capacity is " << res << endl;
+
+    return 0;
+}