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# Coin change problem : Greedy algorithm

Today, we will learn a very common problem which can be solved using the greedy algorithm. If you are not very familiar with a greedy algorithm, here is the gist: At every step of the algorithm, you take the best available option and hope that everything turns optimal at the end which usually does. The problem at hand is coin change problem, which goes like given coins of denominations 1,5,10,25,100; find out a way to give a customer an amount with the fewest number of coins. For example, if I ask you to return me change for 30, there are more than two ways to do so like

```
Amount: 30
Solutions : 3 X 10  ( 3 coins )
6 X 5   ( 6 coins )
1 X 25 + 5 X 1 ( 6 coins )
1 X 25 + 1 X 5 ( 2 coins )```

The last solution is the optimal one as it gives us a change of amount only with 2 coins, where as all other solutions provide it in more than two coins.

Solution for coin change problem using greedy algorithm is very intuitive and called as cashier’s algorithm. Basic principle is : At every iteration in search of a coin, take the largest coin which can fit into remaining amount we need change for at the instance. At the end you will have optimal solution.

## Coin change problem : Algorithm

1. Sort n denomination coins in increasing order of value.
2. Initialize set of coins as empty. S = {}
3. While amount is not zero:
3.1 Ck is largest coin such that amount > Ck
3.1.1 If there is no such coin return “no viable solution”
3.1.2 Else include the coin in the solution S.
3.1.3 Decrease the remaining amount = amount – Ck

### Coin change problem : implementation

```#include <stdio.h>

int coins[] = { 1,5,10,25,100 };

int findMaxCoin(int amount, int size){
for(int i=0; i<size; i++){
if(amount < coins[i]) return i-1;
}
return -1;
}

int findMinimumCoinsForAmount(int amount, int change[]){

int numOfCoins = sizeof(coins)/sizeof(coins[0]);
int count = 0;
while(amount){
int k = findMaxCoin(amount, numOfCoins);
if(k == -1)
printf("No viable solution");
else{
amount-= coins[k];
change[count++] = coins[k];
}
}
return count;
}

int main(void) {
int change[10]; // This needs to be dynamic
int amount = 34;
int count = findMinimumCoinsForAmount(amount, change);

printf("\n Number of coins for change of %d : %d", amount, count);
printf("\n Coins : ");
for(int i=0; i<count; i++){
printf("%d ", change[i]);
}
return 0;
}
```

What will the time complexity of the implementation? First of all, we are sorting the array of coins of size n, hence complexity with `O(nlogn)`. While loop, the worst case is `O(amount)`. If all we have is the coin with 1-denomination. Overall complexity for coin change problem becomes `O(n log n) + O(amount)`.

Will this algorithm work for all sort of denominations? The answer is no. It will not give any solution if there is no coin with denomination 1. So be careful while applying this algorithm.

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