Maximum Bitwise XOR of node values of an Acyclic Graph made up of N given vertices using M edges

Given N nodes valued by [1, N], an array arr[] consisting of N positive integers such that the i^th node ( 1-based indexing ) has the value arr[i] and an integer M, the task is to find the maximum Bitwise XOR of node values of an acyclic graph formed by M edges.

Examples:

Input: arr[]= {1, 2, 3, 4}, M = 2
Output: 7
Explanation:
Acyclic graphs having M(= 2) edges can be formed by vertices as:

1. {1, 2, 3}: The value of the Bitwise XOR of vertices is 1^2^3 = 0.
2. {2, 3, 4}: The value of the Bitwise XOR of vertices is 2^3^4 = 5.
3. {1, 2, 4}: The value of the Bitwise XOR of vertices is 1^2^4 = 7.
4. {1, 4, 3}: The value of the Bitwise XOR of vertices is 1^4^3 = 6.

Therefore, the maximum Bitwise XOR among all possible acyclic graphs is 7. 

Input: arr[] = {2, 4, 8, 16}, M = 2
Output: 28

Approach: The given problem can be solved by using the fact that an acyclic graph having M edges must have (M + 1) vertices. Therefore, the task is reduced to finding the maximum Bitwise XOR of a subset of the array arr[] having (M + 1) vertices. Follow the steps below to solve the problem:

• Initialize a variable, say maxAns as 0 that stores the maximum Bitwise XOR of an acyclic graph having M edges.
• Generate all possible subsets of the array arr[] and for each subset find the Bitwise XOR of the elements of the subset and update the value of maxAns to the maximum of maxAns and Bitwise XOR.
• After completing the above steps, print the value of maxAns as the result.

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 the maximum Bitwise
// XOR of any subset of the array of size K
int maximumXOR(int arr[], int n, int K)
{
    // Number of node must K + 1 for
    // K edges
    K++;

    // Stores the maximum Bitwise XOR
    int maxXor = INT_MIN;

    // Generate all subsets of the array
    for (int i = 0; i < (1 << n); i++) {

        // __builtin_popcount() returns
        // the number of sets bits in
        // an integer
        if (__builtin_popcount(i) == K) {

            // Initialize current xor as 0
            int cur_xor = 0;

            for (int j = 0; j < n; j++) {

                // If jth bit is set in i
                // then include jth element
                // in the current xor
                if (i & (1 << j))
                    cur_xor = cur_xor ^ arr[j];
            }

            // Update the maximum Bitwise
            // XOR obtained so far
            maxXor = max(maxXor, cur_xor);
        }
    }

    // Return the maximum XOR
    return maxXor;
}

// Driver Code
int main()
{
    int arr[] = { 1, 2, 3, 4 };
    int N = sizeof(arr) / sizeof(int);
    int M = 2;
    cout << maximumXOR(arr, N, M);

    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.lang.*;
import java.util.*;

class GFG{

// Function to find the maximum Bitwise
// XOR of any subset of the array of size K
static int maximumXOR(int arr[], int n, int K)
{

    // Number of node must K + 1 for
    // K edges
    K++;

    // Stores the maximum Bitwise XOR
    int maxXor = Integer.MIN_VALUE;

    // Generate all subsets of the array
    for(int i = 0; i < (1 << n); i++)
    {

        // Integer.bitCount() returns
        // the number of sets bits in
        // an integer
        if (Integer.bitCount(i) == K)
        {

            // Initialize current xor as 0
            int cur_xor = 0;

            for(int j = 0; j < n; j++)
            {

                // If jth bit is set in i
                // then include jth element
                // in the current xor
                if ((i & (1 << j)) != 0)
                    cur_xor = cur_xor ^ arr[j];
            }

            // Update the maximum Bitwise
            // XOR obtained so far
            maxXor = Math.max(maxXor, cur_xor);
        }
    }

    // Return the maximum XOR
    return maxXor;
}

// Driver Code
public static void main(String[] args)
{
    int arr[] = { 1, 2, 3, 4 };
    int N = arr.length;
    int M = 2;

    System.out.println(maximumXOR(arr, N, M));
}
}

// This code is contributed by Kingash

Python3

# Python3 program for the above approach

# Function to find the maximum Bitwise
# XOR of any subset of the array of size K
def maximumXOR(arr, n, K):

    # Number of node must K + 1 for
    # K edges
    K += 1

    # Stores the maximum Bitwise XOR
    maxXor = -10**9

    # Generate all subsets of the array
    for i in range(1<<n):

        # __builtin_popcount() returns
        # the number of sets bits in
        # an integer
        if (bin(i).count('1') == K):

            # Initialize current xor as 0
            cur_xor = 0

            for j in range(n):

                # If jth bit is set in i
                # then include jth element
                # in the current xor
                if (i & (1 << j)):
                    cur_xor = cur_xor ^ arr[j]

            # Update the maximum Bitwise
            # XOR obtained so far
            maxXor = max(maxXor, cur_xor)

    # Return the maximum XOR
    return maxXor

# Driver Code
if __name__ == '__main__':
    arr= [1, 2, 3, 4 ]
    N = len(arr)
    M = 2
    print (maximumXOR(arr, N, M))

# This code is contributed by mohit kumar 29.

C#

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

class GFG{

// Function to find the maximum Bitwise
// XOR of any subset of the array of size K
static int maximumXOR(int []arr, int n, int K)
{

    // Number of node must K + 1 for
    // K edges
    K++;

    // Stores the maximum Bitwise XOR
    int maxXor = Int32.MinValue;

    // Generate all subsets of the array
    for(int i = 0; i < (1 << n); i++)
    {

        // Finding number of sets
        // bits in an integer
        if (Convert.ToString(i, 2).Count(c => c == '1') == K)
        {

            // Initialize current xor as 0
            int cur_xor = 0;

            for(int j = 0; j < n; j++)
            {

                // If jth bit is set in i
                // then include jth element
                // in the current xor
                if ((i & (1 << j)) != 0)
                    cur_xor = cur_xor ^ arr[j];
            }

            // Update the maximum Bitwise
            // XOR obtained so far
            maxXor = Math.Max(maxXor, cur_xor);
        }
    }

    // Return the maximum XOR
    return maxXor;
}

// Driver code
static void Main()
{
    int [] arr = { 1, 2, 3, 4 };
    int N = arr.Length;
    int M = 2;

    Console.WriteLine(maximumXOR(arr, N, M));
}
}

// This code is contributed by jana_sayantan.

Javascript

<script>
// Javascript program for the above approach

// Function to find the maximum Bitwise
// XOR of any subset of the array of size K
function maximumXOR(arr, n, K) {
    // Number of node must K + 1 for
    // K edges
    K++;

    // Stores the maximum Bitwise XOR
    let maxXor = Number.MIN_SAFE_INTEGER;

    // Generate all subsets of the array
    for (let i = 0; i < (1 << n); i++) {

        // __builtin_popcount() returns
        // the number of sets bits in
        // an integer
        if ((i).toString(2).split('').
            filter(x => x == '1').length == K) {

            // Initialize current xor as 0
            let cur_xor = 0;

            for (let j = 0; j < n; j++) {

                // If jth bit is set in i
                // then include jth element
                // in the current xor
                if (i & (1 << j))
                    cur_xor = cur_xor ^ arr[j];
            }

            // Update the maximum Bitwise
            // XOR obtained so far
            maxXor = Math.max(maxXor, cur_xor);
        }
    }

    // Return the maximum XOR
    return maxXor;
}

// Driver Code

let arr = [1, 2, 3, 4];
let N = arr.length;
let M = 2;
document.write(maximumXOR(arr, N, M));


// This code is contributed by _saurabh_jaiswal
</script>

Output:

7

Time Complexity: O(N * 2^N) 
Auxiliary Space: O(1)