Minimize count of divisions by D to obtain at least K equal array elements

Given an array A[ ] of size N and two integers K and D, the task is to calculate the minimum possible number of operations required to obtain at least K equal array elements. Each operation involves replacing an element A[i] by A[i] / D. This operation can be performed any number of times.

Examples: 

Input: N = 5, A[ ] = {1, 2, 3, 4, 5}, K = 3, D = 2 
Output: 2 
Explanation: 
Step 1: Replace A[3] by A[3] / D, i.e. (4 / 2) = 2. Hence, the array modifies to {1, 2, 3, 2, 5} 
Step 2: Replace A[4] by A[4] / D, i.e. (5 / 2) = 2. Hence, the array modifies to {1, 2, 3, 2, 2} 
Hence, the modified array has K(= 3) equal elements. 
Hence, the minimum number of operations required is 2.

Input: N = 4, A[ ] = {1, 2, 3, 6}, K = 2, D = 3 
Output: 1 
Explanation:
Replacing A[3] by A[3] / D, i.e. (6 / 3) = 2. Hence, the array modifies to {1, 2, 3, 2}. 
Hence, the modified array has K(= 2) equal elements. 
Hence, the minimum number of operations required is 1. 

Naive Approach: 
The simplest approach to solve the problem is to generate every possible subset of the given array and perform the given operation on all elements of this subset. The number of operations required for each subset will be equal to the size of the subset. For each subset, count the number of equal elements and check if count is equal to K. If so, compare the then count with minimum moves obtained so far and update accordingly. Finally, print the minimum moves.

Time Complexity: O(2^N *N) 
Auxiliary Space: O(N)

Efficient Approach: 
Follow the steps below to solve the problem: 

• Initialize a 2D vector V, in which, size of a row V[i] denotes the number of array elements that have been reduced to A[i] and every element of the row denotes a count of divisions by D performed on an array element to obtain the number i.
• Traverse the array. For each element A[i], keep dividing it by D until it reduces to 0.
• For each intermediate value of A[i] obtained in the above step, insert the number of divisions required into V[A[i]].
• Once, the above steps are completed for the entire array, iterate over the array V[ ].
• For each V[i], check if the size of V[i] ? K (at most K).
• If V[i] ? K, it denotes that at least K elements in the array have been reduced to i. Sort V[i] and add the first K values, i.e. the smallest K moves required to make K elements equal in the array.
• Compare the sum of K moves with the minimum moves required and update accordingly.
• Once the traversal of the array V[] is completed, print the minimum count of moves obtained finally.

Below is the implementation of the above approach:

// C++ Program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to return minimum
// number of moves required
int getMinimumMoves(int n, int k, int d,
                    vector<int> a)
{
    int MAX = 100000;

    // Stores the number of moves
    // required to obtain respective
    // values from the given array
    vector<int> v[MAX];

    // Traverse the array
    for (int i = 0; i < n; i++) {
        int cnt = 0;

        // Insert 0 into V[a[i]] as
        // it is the initial state
        v[a[i]].push_back(0);

        while (a[i] > 0) {
            a[i] /= d;
            cnt++;

            // Insert the moves required
            // to obtain current a[i]
            v[a[i]].push_back(cnt);
        }
    }

    int ans = INT_MAX;

    // Traverse v[] to obtain
    // minimum count of moves
    for (int i = 0; i < MAX; i++) {

        // Check if there are at least
        // K equal elements for v[i]
        if (v[i].size() >= k) {

            int move = 0;

            sort(v[i].begin(), v[i].end());

            // Add the sum of minimum K moves
            for (int j = 0; j < k; j++) {

                move += v[i][j];
            }

            // Update answer
            ans = min(ans, move);
        }
    }

    // Return the final answer
    return ans;
}

// Driver Code
int main()
{
    int N = 5, K = 3, D = 2;
    vector<int> A = { 1, 2, 3, 4, 5 };

    cout << getMinimumMoves(N, K, D, A);

    return 0;
}

// Java program to implement
// the above approach
import java.util.*;

class GFG{

// Function to return minimum
// number of moves required
@SuppressWarnings("unchecked")
static int getMinimumMoves(int n, int k, 
                           int d, int[] a)
{
    int MAX = 100000;

    // Stores the number of moves
    // required to obtain respective
    // values from the given array
    Vector<Integer> []v = new Vector[MAX];
    for(int i = 0; i < v.length; i++)
        v[i] = new Vector<Integer>();
        
    // Traverse the array
    for(int i = 0; i < n; i++) 
    {
        int cnt = 0;

        // Insert 0 into V[a[i]] as
        // it is the initial state
        v[a[i]].add(0);

        while (a[i] > 0)
        {
            a[i] /= d;
            cnt++;

            // Insert the moves required
            // to obtain current a[i]
            v[a[i]].add(cnt);
        }
    }

    int ans = Integer.MAX_VALUE;

    // Traverse v[] to obtain
    // minimum count of moves
    for(int i = 0; i < MAX; i++)
    {
        
        // Check if there are at least
        // K equal elements for v[i]
        if (v[i].size() >= k)
        {
            int move = 0;

            Collections.sort(v[i]);

            // Add the sum of minimum K moves
            for(int j = 0; j < k; j++) 
            {
                move += v[i].get(j);
            }

            // Update answer
            ans = Math.min(ans, move);
        }
    }

    // Return the final answer
    return ans;
}

// Driver Code
public static void main(String[] args)
{
    int N = 5, K = 3, D = 2;
    int []A = { 1, 2, 3, 4, 5 };

    System.out.print(getMinimumMoves(N, K, D, A));
}
}

// This code is contributed by Amit Katiyar

# Python3 program to implement 
# the above approach 

# Function to return minimum
# number of moves required
def getMinimumMoves(n, k, d, a):

    MAX = 100000

    # Stores the number of moves
    # required to obtain respective
    # values from the given array
    v = []
    for i in range(MAX):
        v.append([])

    # Traverse the array
    for i in range(n):
        cnt = 0

        # Insert 0 into V[a[i]] as
        # it is the initial state
        v[a[i]] += [0]

        while(a[i] > 0):
            a[i] //= d
            cnt += 1

            # Insert the moves required
            # to obtain current a[i]
            v[a[i]] += [cnt]

    ans = float('inf')

    # Traverse v[] to obtain
    # minimum count of moves
    for i in range(MAX):

        # Check if there are at least
        # K equal elements for v[i]
        if(len(v[i]) >= k):
            move = 0
            v[i].sort()

            # Add the sum of minimum K moves
            for j in range(k):
                move += v[i][j]

            # Update answer
            ans = min(ans, move)

    # Return the final answer
    return ans

# Driver Code
if __name__ == '__main__':

    N = 5
    K = 3
    D = 2
    A = [ 1, 2, 3, 4, 5 ]

    # Function call
    print(getMinimumMoves(N, K, D, A))

# This code is contributed by Shivam Singh

// C# program to implement
// the above approach
using System;
using System.Collections.Generic;

class GFG{

// Function to return minimum
// number of moves required

static int getMinimumMoves(int n, int k, 
                           int d, int[] a)
{
    int MAX = 100000;

    // Stores the number of moves
    // required to obtain respective
    // values from the given array
    List<int> []v = new List<int>[MAX];
    for(int i = 0; i < v.Length; i++)
        v[i] = new List<int>();
        
    // Traverse the array
    for(int i = 0; i < n; i++) 
    {
        int cnt = 0;

        // Insert 0 into V[a[i]] as
        // it is the initial state
        v[a[i]].Add(0);

        while (a[i] > 0)
        {
            a[i] /= d;
            cnt++;

            // Insert the moves required
            // to obtain current a[i]
            v[a[i]].Add(cnt);
        }
    }

    int ans = int.MaxValue;

    // Traverse v[] to obtain
    // minimum count of moves
    for(int i = 0; i < MAX; i++)
    {
        
        // Check if there are at least
        // K equal elements for v[i]
        if (v[i].Count >= k)
        {
            int move = 0;

            v[i].Sort();

            // Add the sum of minimum K moves
            for(int j = 0; j < k; j++) 
            {
                move += v[i][j];
            }

            // Update answer
            ans = Math.Min(ans, move);
        }
    }

    // Return the final answer
    return ans;
}

// Driver Code
public static void Main(String[] args)
{
    int N = 5, K = 3, D = 2;
    int []A = { 1, 2, 3, 4, 5 };

    Console.Write(getMinimumMoves(N, K, D, A));
}
}

// This code is contributed by 29AjayKumar

<script>

// JavaScript Program to implement
// the above approach


// Function to return minimum
// number of moves required
function getMinimumMoves(n, k, d,a)
{
    let MAX = 100000;

    // Stores the number of moves
    // required to obtain respective
    // values from the given array
    let v = new Array(MAX).fill(0).map(()=>new Array());

    // Traverse the array
    for (let i = 0; i < n; i++) {
        let cnt = 0;

        // Insert 0 into V[a[i]] as
        // it is the initial state
        v[a[i]].push(0);

        while (a[i] > 0) {
            a[i] = Math.floor(a[i] / d);
            cnt++;

            // Insert the moves required
            // to obtain current a[i]
            v[a[i]].push(cnt);
        }
    }

    let ans = Number.MAX_VALUE;

    // Traverse v[] to obtain
    // minimum count of moves
    for (let i = 0; i < MAX; i++) {

        // Check if there are at least
        // K equal elements for v[i]
        if (v[i].length >= k) {

            let move = 0;

            v[i].sort((a,b)=>a-b);

            // Add the sum of minimum K moves
            for (let j = 0; j < k; j++) {

                move += v[i][j];
            }

            // Update answer
            ans = Math.min(ans, move);
        }
    }

    // Return the final answer
    return ans;
}

// driver code

let N = 5, K = 3, D = 2;
let A = [ 1, 2, 3, 4, 5 ];

document.write(getMinimumMoves(N, K, D, A),"</br>");

// This code is contributed by shinjanpatra

</script>

2

Time Complexity: O(MlogM), where M is the maximum number taken 
Auxiliary Space: O(M)