Minimum elements to be removed to make pairwise Bitwise XOR 0 or 1

Given an array X[] of length N. Then your task is to output the minimum number of elements needed to remove so that the Bitwise XOR of any pair is either 0 or 1.

Examples:

Input: N = 5, X[] = {1, 2, 3, 5, 2}
Output: 2
Explanation: If we remove 1 and 5 from X[] then the remaining elements are {2, 3, 2}. It can be verified that any pair from the remaining X[] will give XOR either 0 or 1.

Input: N = 7, X[] = {1, 4, 6, 12, 2, 6, 7}
Output: 4
Explanation: It can be verified that the minimum number of removed elements must be 4.

Approach: Implement the idea below to solve the problem

The problem is based on the observation, we can solve it using the observation of Bitwise concept. For any two numbers A and B, A^B = 1 if and only if A/2 == B/2, So we just checked the maximum numbers that has same value when divided by 2 and subtracted from the length of array.

Steps were taken to solve the problem:

• Initialize a variable let’s say max to store the maximum element.
• Run a loop for finding maximum element and update max variable.
  • Initialize an integer array Y[] of size [(max / 2) + 1].
  • Update max to -1.
• Run a loop for i = 0 to i < N and follow below mentioned steps under the scope of loop:
  • Y[X[i]/2]++
  • If (Y[X[i]/2] > max), then max = Y[X[i]/2]
• Output N-max as answer.

Below is the implementation of the above idea:

C++

// C++ code for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function for printing minimum number of
// elements needed to remove
void Minimum_deletion(int N, vector<int>& X)
{
    // Variable to store max element
    int max = -1;

    // Loop for finding max element
    for (int i = 0; i < N; i++) {
        if (X[i] > max)
            max = X[i];
    }

    // Initialize a vector Y
    vector<int> Y((max / 2) + 1, 0);
    max = -1;

    // Implement the idea discussed
    // in the Approach section
    for (int i = 0; i < N; i++) {
        Y[X[i] / 2]++;
        if (Y[X[i] / 2] > max)
            max = Y[X[i] / 2];
    }

    // Printing the minimum number of
    // elements needed to remove
    cout << N - max << endl;
}

// Driver code
int main()
{
    // Inputs
    int N = 4;
    vector<int> X = { 1, 2, 2, 5 };

    // Function call
    Minimum_deletion(N, X);

    return 0;
}

// This code is contributed by Abhinav Mahajan (abhinav_m22)

Java

// Java code for the above approach:
import java.io.*;

class GFG {
    public static void main(String[] args)
    {
        // Inputs
        int N = 4;
        int[] X = { 1, 2, 2, 5 };

        // Function call
        Minimum_deletion(N, X);
    }

    // Method for printing minimum number of
    // elements needed to remove
    public static void Minimum_deletion(int N, int X[])
    {

        // Variable to store max element
        int max = -1;

        // Loop for finding max element
        for (int i = 0; i < N; i++) {
            if (X[i] > max)
                max = X[i];
        }

        // Initialize an array Y[]
        int[] Y = new int[(max / 2) + 1];
        max = -1;

        // Implement the idea discussed
        // in Approach section
        for (int i = 0; i < N; i++) {
            Y[X[i] / 2]++;
            if (Y[X[i] / 2] > max)
                max = Y[X[i] / 2];
        }

        // Printing the minimum number of
        // elements needed to remove
        System.out.println(N - max);
    }
}

Python3

def minimum_deletion(N, X):
    # Variable to store max element
    maximum = -1

    # Loop for finding the max element
    for i in range(N):
        if X[i] > maximum:
            maximum = X[i]

    # Initialize a list Y
    Y = [0] * ((maximum // 2) + 1)
    maximum = -1

    # Implement the idea discussed
    # in the Approach section
    for i in range(N):
        Y[X[i] // 2] += 1
        if Y[X[i] // 2] > maximum:
            maximum = Y[X[i] // 2]

    # Printing the minimum number of
    # elements needed to remove
    print(N - maximum)

# Driver code
if __name__ == "__main__":
    # Inputs
    N = 4
    X = [1, 2, 2, 5]

    # Function call
    minimum_deletion(N, X)

C#

using System;
using System.Collections.Generic;

class Geek
{
    // Function for printing minimum number of the
    // elements needed to remove
    static void MinimumDeletion(int N, List<int> X)
    {
        int max = -1;
        // Loop for finding the max element
        foreach (int num in X)
        {
            if (num > max)
                max = num;
        }
        // Initialize a list Y
        List<int> Y = new List<int>();
        for (int i = 0; i <= max / 2; i++)
        {
            Y.Add(0);
        }
        max = -1;
        // Implement the idea discussed in the Approach section
        foreach (int num in X)
        {
            Y[num / 2]++;
            if (Y[num / 2] > max)
                max = Y[num / 2];
        }
        Console.WriteLine(N - max);
    }
    // Driver code
    static void Main(string[] args)
    {
        // Inputs
        int N = 4;
        List<int> X = new List<int> { 1, 2, 2, 5 };
        MinimumDeletion(N, X);
    }
}

Javascript

Javascript// Javascript code for the above approach


// Function for printing minimum number of
// elements needed to remove
function Minimum_deletion(N, X) {
    // Variable to store max element
    let max = -1;

    // Loop for finding max element
    for (let i = 0; i < N; i++) {
        if (X[i] > max)
            max = X[i];
    }

    // Initialize a vector Y
    let Y = new Array(Math.trunc(max / 2) + 1).fill(0);
    max = -1;

    // Implement the idea discussed
    // in the Approach section
    for (let i = 0; i < N; i++) {
        Y[X[i] / 2]++;
        if (Y[X[i] / 2] > max)
            max = Y[X[i] / 2];
    }

    // Printing the minimum number of
    // elements needed to remove
    console.log(N - max);
}

// Driver code
// Inputs
let N = 4;
let X = [1, 2, 2, 5];

// Function call
Minimum_deletion(N, X);

// This code is contributed by ragul21

Output

2

Time Complexity: O(N)
Auxiliary Space: O(N), As another array Y[] is used.