Minimize difference between the largest and smallest array elements by K replacements

Given an array A[] consisting of N integers, the task is to find the minimum difference between the largest and the smallest element in the given array after replacing K elements.

Examples:

Input: A[] = {-1, 3, -1, 8, 5, 4}, K = 3
Output: 2
Explanation:Replace A[0] and A[2] by 3 and 4 respectively. Replace A[3] by 5. Modified array A[] is {3, 3, 4, 5, 5, 4}. Therefore, required output = (5-3) = 2.

Input: A[] = {10, 10, 3, 4, 10}, K = 2
Output: 0

Sorting Approach: The idea is to sort the given array. 

• Check for all K + 1 possibilities of 
  • removing X ( 0 ? X ? K ) elements from the start of the array, and 
  • removing K – X elements from the end of the array  
• Then calculate the minimum difference possible. 
• Finally, print the minimum difference obtained.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find minimum difference
// between largest and smallest element
// after K replacements
int minDiff(int A[], int K, int n)
{

    // Sort array in ascending order
    sort(A, A + n);

    if (n <= K)
        return 0;

    // Minimum difference
    int mindiff = A[n - 1] - A[0];

    if (K == 0)
        return mindiff;

    // Check for all K + 1 possibilities
    for (int i = 0, j = n - 1 - K; j < n;) {
        mindiff = min(mindiff, A[j] - A[i]);

        i++;
        j++;
    }

    // Return answer
    return mindiff;
}

// Driver Code
int main()
{

    // Given array
    int A[] = { -1, 3, -1, 8, 5, 4 };
    int K = 3;

    // Length of array
    int n = sizeof(A) / sizeof(A[0]);

    // Prints the minimum possible difference
    cout << minDiff(A, K, n);

    return 0;
}

// This code is contributed by 29AjayKumar

Java

// Java program for the above approach

import java.util.*;

class GFG {

    // Function to find minimum difference
    // between largest and smallest element
    // after K replacements
    static int minDiff(int[] A, int K)
    {
        // Sort array in ascending order
        Arrays.sort(A);

        // Length of array
        int n = A.length;

        if (n <= K)
            return 0;

        // Minimum difference
        int mindiff = A[n - 1] - A[0];
        if (K == 0)
            return mindiff;

        // Check for all K + 1 possibilities
        for (int i = 0, j = n - 1 - K; j < n;) {
            mindiff = Math.min(mindiff, A[j] - A[i]);

            i++;
            j++;
        }

        // Return answer
        return mindiff;
    }

    // Driver Code
    public static void main(String[] args)
    {
        // Given array
        int A[] = { -1, 3, -1, 8, 5, 4 };
        int K = 3;

        // Prints the minimum possible difference
        System.out.println(minDiff(A, K));
    }
}

Python3

# Python3 program for the above approach

# Function to find minimum difference
# between largest and smallest element
# after K replacements


def minDiff(A, K):

    # Sort array in ascending order
    A.sort()

    # Length of array
    n = len(A)
    if (n <= K):
        return 0

    # Minimum difference
    mindiff = A[n - 1] - A[0]
    if (K == 0):
        return mindiff

    # Check for all K + 1 possibilities
    i = 0
    for j in range(n - 1 - K, n):
        mindiff = min(mindiff, A[j] - A[i])

        i += 1
        j += 1

    # Return answer
    return mindiff


# Driver Code
if __name__ == '__main__':

    # Given array
    A = [-1, 3, -1, 8, 5, 4]
    K = 3

    # Prints the minimum possible difference
    print(minDiff(A, K))

    # This code is contributed by 29AjayKumar

C#

// C# program for the above approach
using System;

class GFG {

    // Function to find minimum difference
    // between largest and smallest element
    // after K replacements
    static int minDiff(int[] A, int K)
    {

        // Sort array in ascending order
        Array.Sort(A);

        // Length of array
        int n = A.Length;

        if (n <= K)
            return 0;

        // Minimum difference
        int mindiff = A[n - 1] - A[0];

        if (K == 0)
            return mindiff;

        // Check for all K + 1 possibilities
        for (int i = 0, j = n - 1 - K; j < n;) {
            mindiff = Math.Min(mindiff, A[j] - A[i]);

            i++;
            j++;
        }

        // Return answer
        return mindiff;
    }

    // Driver Code
    public static void Main(String[] args)
    {

        // Given array
        int[] A = { -1, 3, -1, 8, 5, 4 };
        int K = 3;

        // Prints the minimum possible difference
        Console.WriteLine(minDiff(A, K));
    }
}

// This code is contributed by shikhasingrajput

Javascript

<script>
    // Javascript program for the above approach

    // Function to find minimum difference
    // between largest and smallest element
    // after K replacements
    function minDiff(A, K)
    {

        // Sort array in ascending order
        A.sort(function(a, b){return a - b});

        // Length of array
        let n = A.length;

        if (n <= K)
            return 0;

        // Minimum difference
        let mindiff = A[n - 1] - A[0];

        if (K == 0)
            return mindiff;

        // Check for all K + 1 possibilities
        for(let i = 0, j = n - 1 - K; j < n;)
        {
            mindiff = Math.min(mindiff, A[j] - A[i]);

            i++;
            j++;
        }

        // Return answer
        return mindiff;
    }

    // Given array
    let A = [ -1, 3, -1, 8, 5, 4 ];
    let K = 3;

    // Prints the minimum possible difference
    document.write(minDiff(A, K));

    // This code is contributed by mukesh07.
</script>

Output

2

Time Complexity: O(NlogN)
Auxiliary Space: O(1)

Heap-based Approach: This approach is similar to the above approach, but we will find the K minimum and K maximum array elements using Min Heap and Max Heap respectively.

Follow the steps below to solve the problem: 

1. Initialize two PriorityQueues, min-heap and max-heap.
2. Traverse the given array and add all elements one by one into the Heaps. If the size of the Heap exceeds K, in any of the heaps, remove the element present at the top of that Queue.
3. Store the K maximum and the minimum elements in two separate arrays, maxList and minList, and initialize a variable, say minDiff to store the minimum difference.
4. Iterate over the arrays and for every index, say i, update minDiff as minDiff = min(minDiff, maxList[i]-minList[ K – i ]) and print final value of minDiff as the required answer.

Below is the implementation of the above approach: 

C++

// C++ program for above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find minimum difference
// between the largest and smallest
// element after K replacements
int minDiff(int A[], int K, int N)
{
    if (N <= K + 1)
        return 0;

    // Create a MaxHeap
    priority_queue<int, vector<int>, greater<int> > maxHeap;

    // Create a MinHeap
    priority_queue<int> minHeap;

    // Update maxHeap and MinHeap with highest
    // and smallest K elements respectively
    for (int i = 0; i < N; i++) {

        // Insert current element
        // into the MaxHeap
        maxHeap.push(A[i]);

        // If maxHeap size exceeds K + 1
        if (maxHeap.size() > K + 1)

            // Remove top element
            maxHeap.pop();

        // Insert current element
        // into the MaxHeap
        minHeap.push(A[i]);

        // If maxHeap size exceeds K + 1
        if (minHeap.size() > K + 1)

            // Remove top element
            minHeap.pop();
    }

    // Store all max element from maxHeap
    vector<int> maxList;
    while (maxHeap.size() > 0) {
        maxList.push_back(maxHeap.top());
        maxHeap.pop();
    }

    // Store all min element from minHeap
    vector<int> minList;
    while (minHeap.size() > 0) {
        minList.push_back(minHeap.top());
        minHeap.pop();
    }

    int mindiff = INT_MAX;

    // Generating all K + 1 possibilities
    for (int i = 0; i <= K; i++) {
        mindiff = min(mindiff, maxList[i] - minList[K - i]);
    }

    // Return answer
    return mindiff;
}

// Driver Code
int main()
{

    // Given array
    int A[] = { -1, 3, -1, 8, 5, 4 };
    int N = sizeof(A) / sizeof(A[0]);
    int K = 3;

    // Function call
    cout << minDiff(A, K, N);
    return 0;
}

// This code is contributed by Dharanendra L V

Java

// Java program for above approach

import java.lang.*;
import java.util.*;

class GFG {

    // Function to find minimum difference
    // between the largest and smallest
    // element after K replacements
    static int minDiff(int[] A, int K)
    {
        if (A.length <= K + 1)
            return 0;

        // Create a MaxHeap
        PriorityQueue<Integer> maxHeap
            = new PriorityQueue<>();

        // Create a MinHeap
        PriorityQueue<Integer> minHeap
            = new PriorityQueue<>(
                Collections.reverseOrder());

        // Update maxHeap and MinHeap with highest
        // and smallest K elements respectively
        for (int n : A) {

            // Insert current element
            // into the MaxHeap
            maxHeap.add(n);

            // If maxHeap size exceeds K + 1
            if (maxHeap.size() > K + 1)

                // Remove top element
                maxHeap.poll();

            // Insert current element
            // into the MaxHeap
            minHeap.add(n);

            // If maxHeap size exceeds K + 1
            if (minHeap.size() > K + 1)

                // Remove top element
                minHeap.poll();
        }

        // Store all max element from maxHeap
        List<Integer> maxList = new ArrayList<>();
        while (maxHeap.size() > 0)
            maxList.add(maxHeap.poll());

        // Store all min element from minHeap
        List<Integer> minList = new ArrayList<>();
        while (minHeap.size() > 0)
            minList.add(minHeap.poll());

        int mindiff = Integer.MAX_VALUE;

        // Generating all K + 1 possibilities
        for (int i = 0; i <= K; i++) {
            mindiff = Math.min(mindiff,
                               maxList.get(i)
                                   - minList.get(K - i));
        }
        // Return answer
        return mindiff;
    }

    // Driver Code
    public static void main(String[] args)
    {

        // Given array
        int A[] = { -1, 3, -1, 8, 5, 4 };
        int K = 3;

        // Function call
        System.out.println(minDiff(A, K));
    }
}

Python3

# Python3 program for above approach
import sys
import heapq
# Function to find minimum difference
# between the largest and smallest
# element after K replacements


def minDiff(A, K):
    if (len(A) <= K + 1):
        return 0
    # Create a MaxHeap
    maxHeap = []
    heapq.heapify(maxHeap)
    # Create a MinHeap
    minHeap = []
    heapq.heapify(minHeap)
    # Update maxHeap and MinHeap with highest
    # and smallest K elements respectively
    for n in A:
        # Insert current element
        # into the MaxHeap
        heapq.heappush(maxHeap, n)
        # If maxHeap size exceeds K + 1
        if (len(maxHeap) > K + 1):
            # Remove top element
            heapq.heappop(maxHeap)
        # Insert current element
        # into the MaxHeap
        heapq.heappush(minHeap, -n)
        # If maxHeap size exceeds K + 1
        if (len(minHeap) > K + 1):
            # Remove top element
            heapq.heappop(minHeap)
    # Store all max element from maxHeap
    maxList = []
    while (len(maxHeap) > 0):
        maxList.append(heapq.heappop(maxHeap))
    # Store all min element from minHeap
    minList = []
    while (len(minHeap) > 0):
        minList.append(-1 * heapq.heappop(minHeap))
    mindiff = sys.maxsize
    # Generating all K + 1 possibilities
    for i in range(K):
        mindiff = min(mindiff, maxList[i] - minList[K - i])
    # Return answer
    return mindiff


# Drive Code
if __name__ == "__main__":
    # Given array
    A = [-1, 3, -1, 8, 5, 4]
    K = 3
    # Function call
    print(minDiff(A, K))
    # This code is contributed by divyesh072019.

C#

// C# program for above approach
using System;
using System.Collections.Generic;

class GFG {

    // Function to find minimum difference
    // between the largest and smallest
    // element after K replacements
    static int minDiff(int[] A, int K)
    {
        if (A.Length <= K + 1)
            return 0;

        // Create a MaxHeap
        List<int> maxHeap = new List<int>();

        // Create a MinHeap
        List<int> minHeap = new List<int>();

        // Update maxHeap and MinHeap with highest
        // and smallest K elements respectively
        foreach(int n in A)
        {

            // Insert current element
            // into the MaxHeap
            maxHeap.Add(n);
            maxHeap.Sort();

            // If maxHeap size exceeds K + 1
            if (maxHeap.Count > K + 1)

                // Remove top element
                maxHeap.RemoveAt(0);

            // Insert current element
            // into the MaxHeap
            minHeap.Add(n);
            minHeap.Sort();
            minHeap.Reverse();

            // If maxHeap size exceeds K + 1
            if (minHeap.Count > K + 1)

                // Remove top element
                minHeap.RemoveAt(0);
        }

        // Store all max element from maxHeap
        List<int> maxList = new List<int>();

        while (maxHeap.Count > 0) {
            maxList.Add(maxHeap[0]);
            maxHeap.RemoveAt(0);
        }

        // Store all min element from minHeap
        List<int> minList = new List<int>();

        while (minHeap.Count > 0) {
            minList.Add(minHeap[0]);
            minHeap.RemoveAt(0);
        }

        int mindiff = Int32.MaxValue;

        // Generating all K + 1 possibilities
        for (int i = 0; i < K; i++) {
            mindiff = Math.Min(mindiff,
                               maxList[i] - minList[K - i]);
        }

        // Return answer
        return mindiff;
    }

    // Driver code
    static void Main()
    {

        // Given array
        int[] A = { -1, 3, -1, 8, 5, 4 };
        int K = 3;

        // Function call
        Console.WriteLine(minDiff(A, K));
    }
}

// This code is contributed by divyeshrabadiya07

Javascript

<script>

// Javascript program for above approach

// Function to find minimum difference
// between the largest and smallest
// element after K replacements
function minDiff(A, K, N)
{
    if (N <= K + 1)
        return 0;

    // Create a MaxHeap
    var maxHeap = [];

    // Create a MinHeap
    var minHeap = [];

    // Update maxHeap and MinHeap with highest
    // and smallest K elements respectively
    for (var i = 0; i < N; i++)
    {

        // Insert current element
        // into the MaxHeap
        maxHeap.push(A[i]);
        maxHeap.sort((a,b)=>b-a);

        // If maxHeap size exceeds K + 1
        if (maxHeap.length > K + 1)

            // Remove top element
            maxHeap.pop();

        // Insert current element
        // into the MaxHeap
        minHeap.push(A[i]);
        minHeap.sort((a,b)=>a-b);

        // If maxHeap size exceeds K + 1
        if (minHeap.length > K + 1)

            // Remove top element
            minHeap.pop();
    }

    // Store all max element from maxHeap
    var maxList = [];
    while (maxHeap.length > 0)
    {
        maxList.push(maxHeap[maxHeap.length-1]);
        maxHeap.pop();
    }

    // Store all min element from minHeap
    var minList = [];
    while (minHeap.length > 0)
    {
        minList.push(minHeap[minHeap.length-1]);
        minHeap.pop();
    }

    var mindiff = 1000000000;

    // Generating all K + 1 possibilities
    for (var i = 0; i <= K; i++)
    {
        mindiff = Math.min(mindiff, maxList[i] - minList[K - i]);
    }

    // Return answer
    return mindiff;
}

// Driver Code
// Given array
var A = [-1, 3, -1, 8, 5, 4];
var N = A.length;
var K = 3;

// Function call
document.write(minDiff(A, K, N));

// This code is contributed by noob2000.
</script>

Output

2

Time Complexity: O(N*log N) where N is the size of the given array.
Auxiliary Space: O(N)