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Minimum count of elements that sums to a given number
  • Difficulty Level : Hard
  • Last Updated : 11 May, 2021

Given infinte number of elements of form 10^n and 25*100^n ( n >= 0 ). The task is to find the minimum count of elements chosen such that there sum is equal to K
Examples:
 

Input : K = 48 
Output :
elements chosen are: (1 + 1 + 1 + 10 + 10 + 25)
Input : 69 
Output :
elements chosen are: (1 + 1 + 1 + 1 + 10 + 10 + 10 + 10 + 25) 
 

 

Approach: 
There are infinite number of the following elements : 
1, 10, 25, 100, 1000, 2500, 10000, 100000, 250000 … and so on.
Greedy Approach won’t work here. For K = 66, by Greedy Approach minimum count will be 9 and chosen elements are 25 + 25 + 10 + 1 + 1 + 1 + 1 + 1 + 1 = 66. But its optimum answer is 6 when these elements are chosen: 25 + 10 + 10 + 10 + 10 + 1 = 66. So, Dynamic Programming will work here. But simple DP cannot be applied because K can go upto 10^9 . 
Dynamic Programming approach: 
 

  • Precompute the minimum no. of elements chosen that constitutes sum upto 99 and store it in memo array.
  • Also, sums upto 99 can only be formed by the combinations of 1, 10 and 25.
  • Starting from end of K, iterate over each last 2 digits to find minimum no. of elements chosen that will sum to last two digits.
  • Sum them all to find the minimum count.

 



Illustration of the above approach: 
Let’s take K = 250166
Let min_count = 0, last 2 digits = 66 
add minimum number of elements to min_count that sums to 66 (it is calculated from memo array 
that we have precomputed). 
min_count = min_count + 6, 
Now, min_count = 6, last 2 digits = 01
add minimum number of elements to min_count sums to 1. 
min_count = min_count + 1, 
Now, min_count = 7, last 2 digits = 25
add minimum number of elements to min_count sums to 25. 
min_count = min_count + 1, 
Now, min_count = 8.
So, minimum number of elements chosen that sums to 250166 is 8. 
Optimal chosen Elements are (250000, 100, 25, 10, 10, 10, 10, 1) 
 

Below is the implementation of the above approach. 
 

C++




// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
 
int minCount(int K)
{
    // we will only store min counts
    // of sum upto 100
    int memo[100];
 
    // initialize with INT_MAX
    for (int i = 0; i < 100; i++) {
        memo[i] = INT_MAX;
    }
 
    // memo[0] = 0 as 0 is
    // made from 0 elements
    memo[0] = 0;
 
    // fill memo array with min counts
    // of elements that will constitute
    // sum upto 100
 
    for (int i = 1; i < 100; i++) {
        memo[i] = min(memo[i - 1] + 1, memo[i]);
    }
 
    for (int i = 10; i < 100; i++) {
        memo[i] = min(memo[i - 10] + 1, memo[i]);
    }
 
    for (int i = 25; i < 100; i++) {
        memo[i] = min(memo[i - 25] + 1, memo[i]);
    }
 
    // min_count will store min
    // count of elements chosen
    long min_count = 0;
 
    // starting from end iterate over
    // each 2 digits and add min count
    // of elements to min_count
    while (K > 0) {
        min_count += memo[K % 100];
        K /= 100;
    }
 
    return min_count;
}
 
// Driver code
int main()
{
 
    int K = 69;
 
    cout << minCount(K) << endl;
 
    return 0;
}

Java




// Java implementation of the above approach
 
class GFG
{
     
    static int minCount(int K)
    {
        // we will only store min counts
        // of sum upto 100
        int memo[] = new int[100];
     
        // initialize with INT_MAX
        for (int i = 0; i < 100; i++)
        {
            memo[i] = Integer.MAX_VALUE;
        }
     
        // memo[0] = 0 as 0 is
        // made from 0 elements
        memo[0] = 0;
     
        // fill memo array with min counts
        // of elements that will constitute
        // sum upto 100
     
        for (int i = 1; i < 100; i++)
        {
            memo[i] = Math.min(memo[i - 1] + 1, memo[i]);
        }
     
        for (int i = 10; i < 100; i++)
        {
            memo[i] = Math.min(memo[i - 10] + 1, memo[i]);
        }
     
        for (int i = 25; i < 100; i++)
        {
            memo[i] = Math.min(memo[i - 25] + 1, memo[i]);
        }
     
        // min_count will store min
        // count of elements chosen
        int min_count = 0;
     
        // starting from end iterate over
        // each 2 digits and add min count
        // of elements to min_count
        while (K > 0)
        {
            min_count += memo[K % 100];
            K /= 100;
        }
     
        return min_count;
    }
     
    // Driver code
    public static void main (String[] args)
    {
         
                int K = 69;
     
                System.out.println(minCount(K));
    }
}
 
// This code is contributed by AnkitRai01

Python3




# Python3 implementation of the above approach
 
def minCount(K):
     
    # we will only store min counts
    # of sum upto 100
    memo=[10**9 for i in range(100)]
 
    # memo[0] = 0 as 0 is
    # made from 0 elements
    memo[0] = 0
 
    # fill memo array with min counts
    # of elements that will constitute
    # sum upto 100
 
    for i in range(1,100):
        memo[i] = min(memo[i - 1] + 1, memo[i])
 
    for i in range(10,100):
        memo[i] = min(memo[i - 10] + 1, memo[i])
 
    for i in range(25,100):
        memo[i] = min(memo[i - 25] + 1, memo[i])
 
    # min_count will store min
    # count of elements chosen
    min_count = 0
 
    # starting from end iterate over
    # each 2 digits and add min count
    # of elements to min_count
    while (K > 0):
        min_count += memo[K % 100]
        K //= 100
 
    return min_count
 
# Driver code
 
K = 69
 
print(minCount(K))
 
# This code is contributed by mohit kumar 29

C#




// C# implementation of the above approach
using System;
 
class GFG
{
         
    static int minCount(int K)
    {
        // we will only store min counts
        // of sum upto 100
        int []memo = new int[100];
     
        // initialize with INT_MAX
        for (int i = 0; i < 100; i++)
        {
            memo[i] = int.MaxValue;
        }
     
        // memo[0] = 0 as 0 is
        // made from 0 elements
        memo[0] = 0;
     
        // fill memo array with min counts
        // of elements that will constitute
        // sum upto 100
     
        for (int i = 1; i < 100; i++)
        {
            memo[i] = Math.Min(memo[i - 1] + 1, memo[i]);
        }
     
        for (int i = 10; i < 100; i++)
        {
            memo[i] = Math.Min(memo[i - 10] + 1, memo[i]);
        }
     
        for (int i = 25; i < 100; i++)
        {
            memo[i] = Math.Min(memo[i - 25] + 1, memo[i]);
        }
     
        // min_count will store min
        // count of elements chosen
        int min_count = 0;
     
        // starting from end iterate over
        // each 2 digits and add min count
        // of elements to min_count
        while (K > 0)
        {
            min_count += memo[K % 100];
            K /= 100;
        }
     
        return min_count;
    }
     
    // Driver code
    static public void Main ()
    {
         
        int K = 69;
        Console.WriteLine(minCount(K));
    }
}
 
// This code is contributed by ajit

Javascript




<script>
    // Javascript implementation of the above approach
     
    function minCount(K)
    {
        // we will only store min counts
        // of sum upto 100
        let memo = new Array(100);
       
        // initialize with INT_MAX
        for (let i = 0; i < 100; i++)
        {
            memo[i] = Number.MAX_VALUE;
        }
       
        // memo[0] = 0 as 0 is
        // made from 0 elements
        memo[0] = 0;
       
        // fill memo array with min counts
        // of elements that will constitute
        // sum upto 100
       
        for (let i = 1; i < 100; i++)
        {
            memo[i] = Math.min(memo[i - 1] + 1, memo[i]);
        }
       
        for (let i = 10; i < 100; i++)
        {
            memo[i] = Math.min(memo[i - 10] + 1, memo[i]);
        }
       
        for (let i = 25; i < 100; i++)
        {
            memo[i] = Math.min(memo[i - 25] + 1, memo[i]);
        }
       
        // min_count will store min
        // count of elements chosen
        let min_count = 0;
       
        // starting from end iterate over
        // each 2 digits and add min count
        // of elements to min_count
        while (K > 0)
        {
            min_count += memo[K % 100];
            K = parseInt(K / 100, 10);
        }
       
        return min_count;
    }
     
    let K = 69;
      document.write(minCount(K));
 
</script>
Output: 
9

 

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