Minimum count of elements that sums to a given number

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 : 6
elements chosen are: (1 + 1 + 1 + 10 + 10 + 25)

Input : 69
Output : 9
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++

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// 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;
}

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Java

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// 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

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Python3

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# 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

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C#

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// 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

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Output:

9


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