Find minimum number of coins that make a given value

Given a value V, if we want to make a change for V cents, and we have an infinite supply of each of C = { C1, C2, .., Cm} valued coins, what is the minimum number of coins to make the change? If it’s not possible to make a change, print -1.

Examples:

Input: coins[] = {25, 10, 5}, V = 30
Output: Minimum 2 coins required We can use one coin of 25 cents and one of 5 cents 

Input: coins[] = {9, 6, 5, 1}, V = 11
Output: Minimum 2 coins required We can use one coin of 6 cents and 1 coin of 5 cents

This problem is a variation of the problem discussed Coin Change Problem. Here instead of finding the total number of possible solutions, we need to find the solution with the minimum number of coins.

The minimum number of coins for a value V can be computed using the below recursive formula. 

If V == 0, then 0 coins required.
If V > 0
   minCoins(coins[0..m-1], V) = min {1 + minCoins(V-coin[i])} 
                               where i varies from 0 to m-1 
                               and coin[i] <= V

Below is a recursive solution based on the above recursive formula. 

C++

// A Naive recursive C++ program to find minimum of coins
// to make a given change V
#include<bits/stdc++.h>
using namespace std;

// m is size of coins array (number of different coins)
int minCoins(int coins[], int m, int V)
{
   // base case
   if (V == 0) return 0;

   // Initialize result
   int res = INT_MAX;

   // Try every coin that has smaller value than V
   for (int i=0; i<m; i++)
   {
     if (coins[i] <= V)
     {
         int sub_res = minCoins(coins, m, V-coins[i]);

         // Check for INT_MAX to avoid overflow and see if
         // result can minimized
         if (sub_res != INT_MAX && sub_res + 1 < res)
            res = sub_res + 1;
     }
   }
   return res;
}

// Driver program to test above function
int main()
{
    int coins[] =  {9, 6, 5, 1};
    int m = sizeof(coins)/sizeof(coins[0]);
    int V = 11;
    cout << "Minimum coins required is "
         << minCoins(coins, m, V);
    return 0;
}

Java

// A Naive recursive JAVA program to find minimum of coins
// to make a given change V
import java.io.*;
public class coin
{
    // m is size of coins array (number of different coins)
    static int minCoins(int coins[], int m, int V)
    {
       // base case
       if (V == 0) return 0;

       // Initialize result
       int res = Integer.MAX_VALUE;

       // Try every coin that has smaller value than V
       for (int i=0; i<m; i++)
       {
         if (coins[i] <= V)
         {
             int sub_res = minCoins(coins, m, V-coins[i]);

             // Check for INT_MAX to avoid overflow and see if
             // result can minimized
             if (sub_res != Integer.MAX_VALUE && sub_res + 1 < res)
                res = sub_res + 1;
         }
       }
       return res;
    }
    public static void main(String args[])
    {
       int coins[] =  {9, 6, 5, 1};
       int m = coins.length;
       int V = 11;
       System.out.println("Minimum coins required is "+ minCoins(coins, m, V) );
    }
}/* This code is contributed by Rajat Mishra */

Python3

# A Naive recursive python program to find minimum of coins
# to make a given change V

import sys

# m is size of coins array (number of different coins)
def minCoins(coins, m, V):

    # base case
    if (V == 0):
        return 0

    # Initialize result
    res = sys.maxsize

    # Try every coin that has smaller value than V
    for i in range(0, m):
        if (coins[i] <= V):
            sub_res = minCoins(coins, m, V-coins[i])

            # Check for INT_MAX to avoid overflow and see if
            # result can minimized
            if (sub_res != sys.maxsize and sub_res + 1 < res):
                res = sub_res + 1

    return res

# Driver program to test above function
coins = [9, 6, 5, 1]
m = len(coins)
V = 11
print("Minimum coins required is",minCoins(coins, m, V))

# This code is contributed by
# Smitha Dinesh Semwal

C#

// A Naive recursive C# program
// to find minimum of coins
// to make a given change V
using System;
class coin
{

    // m is size of coins array
    // (number of different coins)
    static int minCoins(int []coins, int m, int V)
    {

        // base case
        if (V == 0) return 0;

        // Initialize result
        int res = int.MaxValue;

        // Try every coin that has
        // smaller value than V
        for (int i = 0; i < m; i++)
        {
            if (coins[i] <= V)
            {
                int sub_res = minCoins(coins, m,
                                  V - coins[i]);

                // Check for INT_MAX to
                // avoid overflow and see
                // if result can minimized
                if (sub_res != int.MaxValue &&
                            sub_res + 1 < res)
                    res = sub_res + 1;
            }
        }
        return res;
    }

    // Driver Code
    public static void Main()
    {
        int []coins = {9, 6, 5, 1};
        int m = coins.Length;
        int V = 11;
        Console.Write("Minimum coins required is "+
                             minCoins(coins, m, V));
    }
}

// This code is contributed by nitin mittal.

PHP

<?php
// A Naive recursive PHP
// program to find minimum
// of coins to make a given
// change V

// m is size of coins array
// (number of different coins)
function minCoins($coins,
                  $m, $V)
{

// base case
if ($V == 0) return 0;

// Initialize result
$res = PHP_INT_MAX;

// Try every coin that has
// smaller value than V
for ($i = 0; $i < $m; $i++)
{
    if ($coins[$i] <= $V)
    {
        $sub_res = minCoins($coins, $m,
                            $V - $coins[$i]);

        // Check for INT_MAX to
        // avoid overflow and see
        // if result can minimized
        if ($sub_res != PHP_INT_MAX &&
            $sub_res + 1 < $res)
            $res = $sub_res + 1;
    }
}
return $res;
}

// Driver Code
$coins = array(9, 6, 5, 1);
$m = sizeof($coins);
$V = 11;
echo "Minimum coins required is ",
         minCoins($coins, $m, $V);

// This code is contributed by ajit
?>

Javascript

<script>

// A Naive recursive Javascript program to
// find minimum of coins to make a given
// change V

// m is size of coins array
// (number of different coins)
function minCoins(coins, m, V)
{

    // Base case
    if (V == 0)
        return 0;

    // Initialize result
    let res = Number.MAX_VALUE;

    // Try every coin that has smaller
    // value than V
    for(let i = 0; i < m; i++)
    {
        if (coins[i] <= V)
        {
            let sub_res = minCoins(coins, m,
                               V - coins[i]);

            // Check for INT_MAX to avoid overflow and
            // see if result can minimized
            if (sub_res != Number.MAX_VALUE &&
                sub_res + 1 < res)
                res = sub_res + 1;
        }
    }
    return res;
}

// Driver code
let coins = [ 9, 6, 5, 1 ];
let m = coins.length;
let V = 11;

document.write("Minimum coins required is " +
               minCoins(coins, m, V) );

// This code is contributed by avanitrachhadiya2155

</script>

Output

Minimum coins required is 2

The time complexity of the above solution is exponential and space complexity is way greater than O(n). If we draw the complete recursion tree, we can observe that many subproblems are solved again and again. For example, when we start from V = 11, we can reach 6 by subtracting one 5 times and by subtracting 5 one time. So the subproblem for 6 is called twice. 

Since the same subproblems are called again, this problem has the Overlapping Subproblems property. So the min coins problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of the same subproblems can be avoided by constructing a temporary array table[][] in a bottom-up manner. Below is Dynamic Programming based solution. 

C++

// A Dynamic Programming based C++ program to find minimum
// of coins to make a given change V
#include <bits/stdc++.h>
using namespace std;

// m is size of coins array (number of different coins)
int minCoins(int coins[], int m, int V)
{
    // table[i] will be storing the minimum number of coins
    // required for i value.  So table[V] will have result
    int table[V + 1];

    // Base case (If given value V is 0)
    table[0] = 0;

    // Initialize all table values as Infinite
    for (int i = 1; i <= V; i++)
        table[i] = INT_MAX;

    // Compute minimum coins required for all
    // values from 1 to V
    for (int i = 1; i <= V; i++) {
        // Go through all coins smaller than i
        for (int j = 0; j < m; j++)
            if (coins[j] <= i) {
                int sub_res = table[i - coins[j]];
                if (sub_res != INT_MAX
                    && sub_res + 1 < table[i])
                    table[i] = sub_res + 1;
            }
    }

    if (table[V] == INT_MAX)
        return -1;

    return table[V];
}

// Driver program to test above function
int main()
{
    int coins[] = { 9, 6, 5, 1 };
    int m = sizeof(coins) / sizeof(coins[0]);
    int V = 11;
    cout << "Minimum coins required is "
         << minCoins(coins, m, V);
    return 0;
}

Java

// A Dynamic Programming based Java
// program to find minimum of coins
// to make a given change V
import java.io.*;

class GFG
{
    // m is size of coins array
    // (number of different coins)
    static int minCoins(int coins[], int m, int V)
    {
        // table[i] will be storing
        // the minimum number of coins
        // required for i value. So
        // table[V] will have result
        int table[] = new int[V + 1];

        // Base case (If given value V is 0)
        table[0] = 0;

        // Initialize all table values as Infinite
        for (int i = 1; i <= V; i++)
        table[i] = Integer.MAX_VALUE;

        // Compute minimum coins required for all
        // values from 1 to V
        for (int i = 1; i <= V; i++)
        {
            // Go through all coins smaller than i
            for (int j = 0; j < m; j++)
            if (coins[j] <= i)
            {
                int sub_res = table[i - coins[j]];
                if (sub_res != Integer.MAX_VALUE
                       && sub_res + 1 < table[i])
                       table[i] = sub_res + 1;


            }

        }

          if(table[V]==Integer.MAX_VALUE)
            return -1;

        return table[V];

    }

    // Driver program
    public static void main (String[] args)
    {
        int coins[] = {9, 6, 5, 1};
        int m = coins.length;
        int V = 11;
        System.out.println ( "Minimum coins required is "
                            + minCoins(coins, m, V));
    }
}

//This Code is contributed by vt_m.

Python3

# A Dynamic Programming based Python3 program to
# find minimum of coins to make a given change V
import sys

# m is size of coins array (number of
# different coins)
def minCoins(coins, m, V):

    # table[i] will be storing the minimum
    # number of coins required for i value.
    # So table[V] will have result
    table = [0 for i in range(V + 1)]

    # Base case (If given value V is 0)
    table[0] = 0

    # Initialize all table values as Infinite
    for i in range(1, V + 1):
        table[i] = sys.maxsize

    # Compute minimum coins required
    # for all values from 1 to V
    for i in range(1, V + 1):

        # Go through all coins smaller than i
        for j in range(m):
            if (coins[j] <= i):
                sub_res = table[i - coins[j]]
                if (sub_res != sys.maxsize and
                    sub_res + 1 < table[i]):
                    table[i] = sub_res + 1

    if table[V] == sys.maxsize:
        return -1

    return table[V]

# Driver Code
if __name__ == "__main__":

    coins = [9, 6, 5, 1]
    m = len(coins)
    V = 11
    print("Minimum coins required is ",
                 minCoins(coins, m, V))

# This code is contributed by ita_c

C#

// A Dynamic Programming based
// Java program to find minimum
// of coins to make a given
// change V
using System;

class GFG
{

// m is size of coins array
// (number of different coins)
static int minCoins(int []coins,
                    int m, int V)
{
    // table[i] will be storing
    // the minimum number of coins
    // required for i value. So
    // table[V] will have result
    int []table = new int[V + 1];

    // Base case (If given
    // value V is 0)
    table[0] = 0;

    // Initialize all table
    // values as Infinite
    for (int i = 1; i <= V; i++)
    table[i] = int.MaxValue;

    // Compute minimum coins
    // required for all
    // values from 1 to V
    for (int i = 1; i <= V; i++)
    {
        // Go through all coins
        // smaller than i
        for (int j = 0; j < m; j++)
        if (coins[j] <= i)
        {
            int sub_res = table[i - coins[j]];
            if (sub_res != int.MaxValue &&
                sub_res + 1 < table[i])
                table[i] = sub_res + 1;
        }
    }

    return table[V];

}

// Driver Code
static public void Main ()
{
    int []coins = {9, 6, 5, 1};
    int m = coins.Length;
    int V = 11;
    Console.WriteLine("Minimum coins required is " +
                             minCoins(coins, m, V));
}
}

// This code is contributed
// by akt_mit

PHP

<?php
// A Dynamic Programming based
// PHP program to find minimum
// of coins to make a given
// change V.

// m is size of coins
// array (number of different coins)
function minCoins($coins, $m, $V)
{
    // table[i] will be storing the
    // minimum number of coins
    // required for i value. So
    // table[V] will have result
    $table[$V + 1] = array();

    // Base case (If given
    // value V is 0)
    $table[0] = 0;

    // Initialize all table
    // values as Infinite
    for ($i = 1; $i <= $V; $i++)
        $table[$i] = PHP_INT_MAX;

    // Compute minimum coins
    // required for all
    // values from 1 to V
    for ($i = 1; $i <= $V; $i++)
    {
        // Go through all coins
        // smaller than i
        for ($j = 0; $j < $m; $j++)
        if ($coins[$j] <= $i)
        {
            $sub_res = $table[$i - $coins[$j]];
            if ($sub_res != PHP_INT_MAX &&
                $sub_res + 1 < $table[$i])
                $table[$i] = $sub_res + 1;
        }
    }

      if ($table[$V] == PHP_INT_MAX)
        return -1;
    return $table[$V];
}

// Driver Code
$coins = array(9, 6, 5, 1);
$m = sizeof($coins);
$V = 11;
echo "Minimum coins required is ",
    minCoins($coins, $m, $V);

// This code is contributed by ajit
?>

Javascript

<script>
// A Dynamic Programming based Javascript
// program to find minimum of coins
// to make a given change V

    // m is size of coins array
    // (number of different coins)
    function minCoins(coins,m,v)
    {

        // table[i] will be storing
        // the minimum number of coins
        // required for i value. So
        // table[V] will have result
        let table = new Array(V+1);
        // Initialize first table value as zero
        table[0] = 0;

        // Initialize all table values as Infinite except for the first one
        for (let i = 1; i <= V; i++)
        {
            table[i] = Number.MAX_VALUE;
        }
        // Compute minimum coins required for all
        // values from 1 to V
        for (let i = 1; i <= V; i++)
        {
            // Go through all coins smaller than i
            for (let j = 0; j < m; j++)
            if (coins[j] <= i)
            {
                let sub_res = table[i - coins[j]];
                if (sub_res != Number.MAX_VALUE
                       && sub_res + 1 < table[i])
                    table[i] = sub_res + 1;
            }
        }

        if(table[V] == Number.MAX_VALUE)
            return -1;

        return table[V];
    }

    // Driver program
    let coins = [9, 6, 5, 1];
    let m = coins.length;
    let V = 11;
    document.write( "Minimum coins required is " + minCoins(coins, m, V))

    //  This code is contributed by rag2127
</script>

Output

Minimum coins required is 2

Time complexity: O(m*V).
Auxiliary space: O(V) because using extra space for array table

Thanks to Goku for suggesting the above solution in a comment here and thanks to Vignesh Mohan for suggesting this problem and initial solution.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above