DP – Coin Change: Find number of ways of representing n cents

Given an infinite supply of 25 cents, 10 cents, 5 cents, and 1 cent. Find the number of ways of representing n cents. (The order doesn’t matter). This is a coin change problem and will require the implementation of Dynamic Programming.

Ex: n = 10 => {1,1,1,1,1,1,1,1,1,1}
              {1,1,1,1,1,5}
              {5,5}
              {10}

Algorithm:

We are given a set of 4 Coins of type 1 cents, 5 cents, 10 cents, 25 cents. To find the number of ways of making n cents using these 4 cents, we will consider 2 conditions:

1. Try to make n cents by including the ith cent from the set of 4 coins. number_of_ways(n-coin_arr[m], coin_arr m);  here m in the number of coins in given set(here m=4).
2. Try to make n cents by not including the ith cent from the given set of the 4 coins.                                 number_of_ways(n, coin_arr, m-1);

This problem involves the repetition of subproblems, so we will use Dynamic Programming.

Implementation of the above Algorithm

1. Iterative Approach

#include <bits/stdc++.h>
using namespace std;
 
int number_of_ways(int cents, vector<int>coin_arr, int n){
    vector<vector<int> >dp(n+1, vector<int>(cents+1,0));
 
    for(int j=0; j<n; j++)
        dp[j][0] = 1;
     
    for(int i=1; i<=cents; i++){
        for(int j=0; j<n; j++){
             
            if(i-coin_arr[j]>=0) dp[j][i] += dp[j][i-coin_arr[j]];
            if(j>0) dp[j][i] += dp[j-1][i];
 
        }
    }
    return dp[n-1][cents];
}
 
int main()
{
    int cents;
    vector<int> coin_arr{1,5,10,25};
    cin>>cents;
 
    cout<<number_of_ways(cents, coin_arr, coin_arr.size())<<endl;
 
    return 0;
}

2. Recursive Approach

#include <bits/stdc++.h>
using namespace std;
 
int number_of_ways(int cents, vector<int>coin_arr, int n, vector<vector<int> > &dp){
 
    if(cents<0 || n<0)
        return 0;
 
    if(cents == 0)
        return 1;
 
    if(dp[cents][n]==-1)
        dp[cents][n] =  number_of_ways(cents-coin_arr[n], coin_arr, n, dp) +number_of_ways(cents, coin_arr, n-1, dp);
    return dp[cents][n];
}
 
int main()
{
    int cents;
    vector<int> coin_arr{1,5,10,25};
    cin>>cents;
    //creating a 2D array for storing subproblems 
    //of size= dp[cents+1][coin_arr.size()+1]
    vector<vector<int> > dp(cents+1, vector<int>(coin_arr.size()+1,-1));
 
    cout<<number_of_ways(cents, coin_arr, coin_arr.size()-1, dp)<<endl;
 
    return 0;
}

Output

Coins       Output
n = 5        2
n = 26       13
n = 1000     142511

Time Complexity

Since we have used Dynamic Programming here, hence the time complexity of the above program for coin change problem is O(NM), N the total cents to make and M is the number of the coin in a given set.