knapsack.cpp

/*
Knapsack Problems
Mathematical Foundation: Choose items to maximize value within weight constraint
0/1 Knapsack: dp[i][w] = max(dp[i-1][w], dp[i-1][w-weight[i]] + value[i])
Unbounded: Can use item multiple times
Time: O(n*W), Space: O(W) with optimization
*/

#include <bits/stdc++.h>
using namespace std;

// 0/1 Knapsack - 2D DP (Classic Pattern)
// Related LeetCode Problems:
// 416. Partition Equal Subset Sum
// https://leetcode.com/problems/partition-equal-subset-sum/
// 494. Target Sum
// https://leetcode.com/problems/target-sum/
// 474. Ones and Zeroes
// https://leetcode.com/problems/ones-and-zeroes/
// 1049. Last Stone Weight II
// https://leetcode.com/problems/last-stone-weight-ii/
int knapsack01_2D(vector<int>& weights, vector<int>& values, int W) {
    int n = weights.size();
    vector<vector<int>> dp(n + 1, vector<int>(W + 1, 0));
    
    for (int i = 1; i <= n; i++) {
        for (int w = 1; w <= W; w++) {
            if (weights[i - 1] <= w) {
                dp[i][w] = max(dp[i - 1][w], 
                              dp[i - 1][w - weights[i - 1]] + values[i - 1]);
            } else {
                dp[i][w] = dp[i - 1][w];
            }
        }
    }
    return dp[n][W];
}

// 0/1 Knapsack - Space Optimized
// Related LeetCode Problems:
// 416. Partition Equal Subset Sum
// https://leetcode.com/problems/partition-equal-subset-sum/
// 494. Target Sum
// https://leetcode.com/problems/target-sum/
// 1049. Last Stone Weight II
// https://leetcode.com/problems/last-stone-weight-ii/
// 698. Partition to K Equal Sum Subsets
// https://leetcode.com/problems/partition-to-k-equal-sum-subsets/
int knapsack01(vector<int>& weights, vector<int>& values, int W) {
    vector<int> dp(W + 1, 0);
    
    for (int i = 0; i < weights.size(); i++) {
        for (int w = W; w >= weights[i]; w--) {
            dp[w] = max(dp[w], dp[w - weights[i]] + values[i]);
        }
    }
    return dp[W];
}

// Unbounded Knapsack
// Related LeetCode Problems:
// 322. Coin Change
// https://leetcode.com/problems/coin-change/
// 518. Coin Change 2
// https://leetcode.com/problems/coin-change-2/
// 377. Combination Sum IV
// https://leetcode.com/problems/combination-sum-iv/
// 279. Perfect Squares
// https://leetcode.com/problems/perfect-squares/
// 983. Minimum Cost For Tickets
// https://leetcode.com/problems/minimum-cost-for-tickets/
int unboundedKnapsack(vector<int>& weights, vector<int>& values, int W) {
    vector<int> dp(W + 1, 0);
    
    for (int w = 1; w <= W; w++) {
        for (int i = 0; i < weights.size(); i++) {
            if (weights[i] <= w) {
                dp[w] = max(dp[w], dp[w - weights[i]] + values[i]);
            }
        }
    }
    return dp[W];
}

// Coin Change (Minimum Coins)
// LeetCode: 322. Coin Change
// https://leetcode.com/problems/coin-change/
// Related:
// 518. Coin Change 2
// https://leetcode.com/problems/coin-change-2/
// 377. Combination Sum IV
// https://leetcode.com/problems/combination-sum-iv/
// 279. Perfect Squares
// https://leetcode.com/problems/perfect-squares/
// 983. Minimum Cost For Tickets
// https://leetcode.com/problems/minimum-cost-for-tickets/
// 1449. Form Largest Integer With Digits That Add up to Target
// https://leetcode.com/problems/form-largest-integer-with-digits-that-add-up-to-target/
int coinChange(vector<int>& coins, int amount) {
    vector<int> dp(amount + 1, amount + 1);
    dp[0] = 0;
    
    for (int i = 1; i <= amount; i++) {
        for (int coin : coins) {
            if (i >= coin) {
                dp[i] = min(dp[i], dp[i - coin] + 1);
            }
        }
    }
    return dp[amount] > amount ? -1 : dp[amount];
}

// Coin Change II (Number of Ways)
// LeetCode: 518. Coin Change 2
// https://leetcode.com/problems/coin-change-2/
// Related:
// 322. Coin Change
// https://leetcode.com/problems/coin-change/
// 377. Combination Sum IV
// https://leetcode.com/problems/combination-sum-iv/
// 39. Combination Sum
// https://leetcode.com/problems/combination-sum/
// 40. Combination Sum II
// https://leetcode.com/problems/combination-sum-ii/
// 216. Combination Sum III
// https://leetcode.com/problems/combination-sum-iii/
int change(int amount, vector<int>& coins) {
    vector<int> dp(amount + 1, 0);
    dp[0] = 1;
    
    for (int coin : coins) {
        for (int i = coin; i <= amount; i++) {
            dp[i] += dp[i - coin];
        }
    }
    return dp[amount];
}

// Partition Equal Subset Sum
// LeetCode: 416. Partition Equal Subset Sum
// https://leetcode.com/problems/partition-equal-subset-sum/
// Related:
// 494. Target Sum
// https://leetcode.com/problems/target-sum/
// 1049. Last Stone Weight II
// https://leetcode.com/problems/last-stone-weight-ii/
// 698. Partition to K Equal Sum Subsets
// https://leetcode.com/problems/partition-to-k-equal-sum-subsets/
// 473. Matchsticks to Square
// https://leetcode.com/problems/matchsticks-to-square/
bool canPartition(vector<int>& nums) {
    int sum = accumulate(nums.begin(), nums.end(), 0);
    if (sum % 2) return false;
    int target = sum / 2;
    
    vector<bool> dp(target + 1, false);
    dp[0] = true;
    
    for (int num : nums) {
        for (int j = target; j >= num; j--) {
            dp[j] = dp[j] || dp[j - num];
        }
    }
    return dp[target];
}

// Target Sum
// LeetCode: 494. Target Sum
// https://leetcode.com/problems/target-sum/
// Related:
// 416. Partition Equal Subset Sum
// https://leetcode.com/problems/partition-equal-subset-sum/
// 1049. Last Stone Weight II
// https://leetcode.com/problems/last-stone-weight-ii/
// 698. Partition to K Equal Sum Subsets
// https://leetcode.com/problems/partition-to-k-equal-sum-subsets/
// 2035. Partition Array Into Two Arrays to Minimize Sum Difference
// https://leetcode.com/problems/partition-array-into-two-arrays-to-minimize-sum-difference/
int findTargetSumWays(vector<int>& nums, int target) {
    int sum = accumulate(nums.begin(), nums.end(), 0);
    if (sum < abs(target) || (sum + target) % 2) return 0;
    
    int subset_sum = (sum + target) / 2;
    vector<int> dp(subset_sum + 1, 0);
    dp[0] = 1;
    
    for (int num : nums) {
        for (int j = subset_sum; j >= num; j--) {
            dp[j] += dp[j - num];
        }
    }
    return dp[subset_sum];
}

// Ones and Zeroes
// LeetCode: 474. Ones and Zeroes
// https://leetcode.com/problems/ones-and-zeroes/
// Related:
// 416. Partition Equal Subset Sum
// https://leetcode.com/problems/partition-equal-subset-sum/
// 494. Target Sum
// https://leetcode.com/problems/target-sum/
// 1049. Last Stone Weight II
// https://leetcode.com/problems/last-stone-weight-ii/
// 600. Non-negative Integers without Consecutive Ones
// https://leetcode.com/problems/non-negative-integers-without-consecutive-ones/
int findMaxForm(vector<string>& strs, int m, int n) {
    vector<vector<int>> dp(m + 1, vector<int>(n + 1, 0));
    
    for (string& s : strs) {
        int zeros = count(s.begin(), s.end(), '0');
        int ones = s.size() - zeros;
        
        for (int i = m; i >= zeros; i--) {
            for (int j = n; j >= ones; j--) {
                dp[i][j] = max(dp[i][j], dp[i - zeros][j - ones] + 1);
            }
        }
    }
    return dp[m][n];
}

// Last Stone Weight II
// LeetCode: 1049. Last Stone Weight II
// https://leetcode.com/problems/last-stone-weight-ii/
// Related:
// 416. Partition Equal Subset Sum
// https://leetcode.com/problems/partition-equal-subset-sum/
// 494. Target Sum
// https://leetcode.com/problems/target-sum/
// 1046. Last Stone Weight
// https://leetcode.com/problems/last-stone-weight/
// 2035. Partition Array Into Two Arrays to Minimize Sum Difference
// https://leetcode.com/problems/partition-array-into-two-arrays-to-minimize-sum-difference/
int lastStoneWeightII(vector<int>& stones) {
    int sum = accumulate(stones.begin(), stones.end(), 0);
    int target = sum / 2;
    vector<bool> dp(target + 1, false);
    dp[0] = true;
    
    for (int stone : stones) {
        for (int j = target; j >= stone; j--) {
            dp[j] = dp[j] || dp[j - stone];
        }
    }
    
    for (int i = target; i >= 0; i--) {
        if (dp[i]) return sum - 2 * i;
    }
    return 0;
}

// Perfect Squares
// LeetCode: 279. Perfect Squares
// https://leetcode.com/problems/perfect-squares/
// Related:
// 322. Coin Change
// https://leetcode.com/problems/coin-change/
// 264. Ugly Number II
// https://leetcode.com/problems/ugly-number-ii/
// 204. Count Primes
// https://leetcode.com/problems/count-primes/
// 343. Integer Break
// https://leetcode.com/problems/integer-break/
// 650. 2 Keys Keyboard
// https://leetcode.com/problems/2-keys-keyboard/
int numSquares(int n) {
    vector<int> dp(n + 1, n);
    dp[0] = 0;
    
    for (int i = 1; i <= n; i++) {
        for (int j = 1; j * j <= i; j++) {
            dp[i] = min(dp[i], dp[i - j * j] + 1);
        }
    }
    return dp[n];
}

// Combination Sum IV
// LeetCode: 377. Combination Sum IV
// https://leetcode.com/problems/combination-sum-iv/
// Related:
// 39. Combination Sum
// https://leetcode.com/problems/combination-sum/
// 40. Combination Sum II
// https://leetcode.com/problems/combination-sum-ii/
// 216. Combination Sum III
// https://leetcode.com/problems/combination-sum-iii/
// 518. Coin Change 2
// https://leetcode.com/problems/coin-change-2/
// 70. Climbing Stairs
// https://leetcode.com/problems/climbing-stairs/
int combinationSum4(vector<int>& nums, int target) {
    vector<unsigned int> dp(target + 1, 0);
    dp[0] = 1;
    
    for (int i = 1; i <= target; i++) {
        for (int num : nums) {
            if (i >= num) {
                dp[i] += dp[i - num];
            }
        }
    }
    return dp[target];
}