 # Minimum Coin Change Problem

How to solve Minimum Coin Change Problem using bottom up dp?

You can refer this video. It clearly explain each and every step of this question.

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For bottom up solution, you can start filling the values from 0 and then maintaining a dp table fill the values for all the values upto n.

``````for (i = 1; i < n+1; i++)
{
for (j = 0; j < m; j++)
{
// Count of solutions including S[j]
x = (i-S[j] >= 0)? table[i - S[j]][j]: 0;

// Count of solutions excluding S[j]
y = (j >= 1)? table[i][j-1]: 0;

// total count
table[i][j] = min(x,y);
}
}
``````

For more explanation check the GeeksForGeeks Solution. Please note Geeks For Geeks Solution is for total no of ways, for minimum coins just replace x+y with min(x,y).

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I want to the minimum numbers of coins to fulfill amount .

table[i][j] = min(x,y), will give you the minimum amount to fulfil amount i using coins upto value S[j].
So table[n][m-1] will give you minimum amount to fulfill n using coins upto S[m-1] i.e. all the coins.

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In this question there is a little twist in initialization part then regular question i.e. we need to fill the 2nd row of matrix as well.
Here is my code using bottom up approach.

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

int coin_change(int coin[], int sum, int n){

``````int i,j;
int t[n+1][sum+1];

for(i=0;i<n+1;i++){
for(j=0;j<sum+1;j++){
if(j==0)
t[i][j] = 0;
else
t[i][j] = INT_MAX-1;
}
}
for(j=1;j<sum+1;j++){
if(j%coin==0)
t[i][j] = j/coin;
else
t[i][j] = INT_MAX-1;
}

for(i=2;i<n+1;i++){
for(j=1;j<sum+1;j++){
if(coin[i-1]<=j)
t[i][j] = min(1+t[i][j-coin[i-1]], t[i-1][j]);
else
t[i][j] = t[i-1][j];
}
}
return t[n][sum];
``````

}

int main(){
int n,i,sum;
cin>>n;
int coin[n];
for(i=0;i<n;i++){
cin>>coin[i];
}
cin>>sum;
int k=coin_change(coin, sum, n);
cout<<k<<endl;

``````return 0;
``````

}