CSES Solutions - Money Sums

You have N coins with certain values. Your task is to find all money sums you can create using these coins.

Examples:

Input: N = 4, coins[] = {4, 2, 5, 2}
Output:
9
2 4 5 6 7 8 9 11 13
Explanation:

• To create sum = 2, we can use the coin with value = 2.
• To create sum = 4, we can use both the coins with value = 2.
• To create sum = 5, we can use the coin with value = 5.
• To create sum = 6, we can use coin with value = 2 and coin with value = 4.
• To create sum = 7, we can use coin with value = 2 and coin with value = 5.
• To create sum = 8, we can use both the coins with value = 2 and another coin with value = 4.
• To create sum = 9, we can use coin with value = 4 and coin with value = 5.
• To create sum = 11, we can use coin with value = 2, coin with value = 4 and coin with value = 5.
• To create sum = 13, we can use both the coins with value = 2, coin with value = 4 and coin with value = 5.

Input: N = 10, coins[] = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
Output:
10
1 2 3 4 5 6 7 8 9 10

Approach: To solve the problem, follow the below idea:

The problem can be solved using Dynamic Programming. We'll define dp[][], such that dp[i][j] = true if it is possible to make a sum of j with i coins and dp[i][j] = false if it is impossible to make a sum of j with i coins. Iterate through all the coins i and possible sums j:

• If dp[i - 1][j] = true, then dp[i][j] = true because if it is possible to make a sum of j with (i-1) coins, then we can make the same sum with i coins as well.
• Else if dp[i - 1][j - coins[i]] = true, then dp[i][j] = true because if it is possible to make a sum of (j - coins[i]) with i - 1 coins, then we can make sum j by using the ith coin.

For all values of j, dp[N][j] will store whether it is possible to create sum j or not.

Step-by-step algorithm:

• Maintain a boolean dp[][] array such that dp[i][j] = true if it is possible to make sum j using first i coins.
• Initialize a variable sum to store the sum of all coins.
• Iterate over each coin from i = 1 to N
  • Iterate over each sum j = 0 to j <= sum,
    • If dp[i - 1][j] = true, dp[i][j] = true
    • Else if dp[i - 1][j - coins[i]] = true,
    • Else dp[i][j] = false
• Store the sums j from 1 to sum which are possible to construct, that is dp[N][j] = true
• Print the number of possible sums along with all the possible sums.

Below is the implementation of the algorithm:

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

// Function to print all the possible sums
void solve(vector<ll>& coins, ll N)
{
    // Find the maximum value which we can make using all
    // the coins
    ll sum = accumulate(coins.begin(), coins.end(), 0LL);

    // dp[][] array such that dp[i][j] = true if it is
    // possible to construct sum j using i coins
    vector<vector<bool> > dp(N + 1,
                             vector<bool>(sum + 1, false));
    dp[0][0] = true;

    // Iterate over all the coins
    for (int i = 1; i <= N; i++) {
        // Iterate over all the sums
        for (int j = 0; j <= sum; j++) {
            // If it is possible to construct sum j using
            // (i-1) coins, then mark dp[i][j] = true
            dp[i][j] = dp[i - 1][j];
            // If it is possible to construct sum (j -
            // coins[i]) using (i - 1) coins, then mark
            // dp[i][j] = true
            if (j >= coins[i - 1]
                && dp[i - 1][j - coins[i - 1]]) {
                dp[i][j] = true;
            }
        }
    }

    // vector to store all the sums which are possible to
    // construct using all the coins
    vector<int> possibleSums;

    // Store all the possible sums from 1 to sum which are
    // possible to construct
    for (int j = 1; j <= sum; j++) {
        if (dp[N][j]) {
            possibleSums.push_back(j);
        }
    }

    // Print the number of possible sums
    cout << possibleSums.size() << endl;
    // Print all the possible sums
    for (int i = 0; i < possibleSums.size(); i++)
        cout << possibleSums[i] << " ";
    cout << endl;
}

int main()
{
    // Sample Input
    ll N = 4;

    vector<ll> coins = { 4, 2, 5, 2 };
    solve(coins, N);
}

import java.util.ArrayList;
import java.util.List;

public class Main {    
    public static void main(String[] args) {
        int N = 4;
        int[] coins = {4, 2, 5, 2};
        gfg(N, coins);
    }    
    public static void gfg(int N, int[] coins) {
        // Calculate the sum of the all coins
        int sum = 0;
        for (int coin : coins) {
            sum += coin;
        }     
        boolean[][] dp = new boolean[N + 1][sum + 1];       
        // Base case: When no coins are selected and sum = 0 is possible
        dp[0][0] = true;      
        for (int i = 1; i <= N; i++) {
            // Iterate over each sum j from the 0 to sum
            for (int j = 0; j <= sum; j++) {
                if (dp[i - 1][j]) {
                    dp[i][j] = true;
                } else if (j - coins[i - 1] >= 0 && dp[i - 1][j - coins[i - 1]]) {
                    dp[i][j] = true;
                }
            }
        }  
        // Initialize a list to store the possible sums
        List<Integer> possibleSums = new ArrayList<>();       
        for (int j = 1; j <= sum; j++) {
            if (dp[N][j]) {
                possibleSums.add(j);
            }
        }        
        // Print the number of the possible sums
        System.out.println(possibleSums.size());        
        // Print the possible sums separated by the space
        for (int i = 0; i < possibleSums.size(); i++) {
            System.out.print(possibleSums.get(i) + " ");
        }
    }
}

function GFG(N, coins) {
    // Calculate the sum of all coins
    let sum = coins.reduce((acc, curr) => acc + curr, 0);
    let dp = new Array(N + 1).fill(false).map(() => new Array(sum + 1).fill(false));
    dp[0][0] = true;
    // Iterate over each coin from i = 1 to N
    for (let i = 1; i <= N; i++) {
        // Iterate over each sum j from 0 to sum
        for (let j = 0; j <= sum; j++) {
            if (dp[i - 1][j]) {
                dp[i][j] = true;
            }
            else if (j - coins[i - 1] >= 0 && dp[i - 1][j - coins[i - 1]]) {
                dp[i][j] = true;
            }
        }
    }
    // Initialize an array to the store the possible sums
    let possibleSums = [];
    for (let j = 1; j <= sum; j++) {
        if (dp[N][j]) {
            possibleSums.push(j);
        }
    }
    // Print the number of the possible sums
    console.log(possibleSums.length);
    console.log(possibleSums.join(' '));
}
// Example usage:
const N = 4;
const coins = [4, 2, 5, 2];
GFG(N, coins);

from typing import List

# Function to print all the possible sums
def solve(coins: List[int], N: int):
    # Find the maximum value which we can make using all the coins
    sum_coins = sum(coins)

    # dp[][] array such that dp[i][j] = true if it is possible to construct sum j using i coins
    dp = [[False for _ in range(sum_coins + 1)] for _ in range(N + 1)]
    dp[0][0] = True

    # Iterate over all the coins
    for i in range(1, N + 1):
        # Iterate over all the sums
        for j in range(sum_coins + 1):
            # If it is possible to construct sum j using (i-1) coins, then mark dp[i][j] = true
            dp[i][j] = dp[i - 1][j]
            # If it is possible to construct sum (j - coins[i]) using (i - 1) coins, then mark dp[i][j] = true
            if j >= coins[i - 1] and dp[i - 1][j - coins[i - 1]]:
                dp[i][j] = True

    # list to store all the sums which are possible to construct using all the coins
    possible_sums = []

    # Store all the possible sums from 1 to sum which are possible to construct
    for j in range(1, sum_coins + 1):
        if dp[N][j]:
            possible_sums.append(j)

    # Print the number of possible sums
    print(len(possible_sums))
    # Print all the possible sums
    print(*possible_sums)

# Sample Input
N = 4
coins = [4, 2, 5, 2]
solve(coins, N)

9
2 4 5 6 7 8 9 11 13 

Time Complexity: O(N * X), where N is the number of coins and X is the sum of values of all coins
Auxiliary Space: O(N * X)