Solving the Minimum XOR Value Problem Efficiently | Bit Manipulation

In this article, we tackle the problem of finding the pair of integers in an array with the minimum XOR value. Given an array of integers, the goal is to identify two numbers that, when subjected to bitwise XOR, yield the smallest possible result among all possible pairs within the array.

We begin by providing an overview of the problem and its significance in various real-world scenarios, such as autocomplete suggestions and network routing. Next, we delve into two distinct approaches to solve this problem efficiently.

Problem Link:https://www.interviewbit.com/problems/min-xor-value/

Given an integer array A of N integers, find the pair of integers in the array which have minimum XOR value. Report the minimum XOR value.

Input Format:

 First and only argument of input contains an integer array A

Output Format:

 return a single integer denoting minimum XOR value

Constraints:

2 <= N <= 100 000  
0 <= A[i] <= 1 000 000 000

For Examples :

Example Input 1:
    A = [0, 2, 5, 7]
Example Output 1:
    2
Explanation:
    0 xor 2 = 2
Example Input 2:
    A = [0, 4, 7, 9]
Example Output 2:
    3

Explanation:

The problem involves finding a pair of integers in an array such that their bitwise XOR operation results in the minimum possible value among all possible pairs in the array. In other words, you want to identify two numbers from the array whose binary representations have the fewest differing bits.

Here's a brief overview of the problem:

• You are given an array A containing N integers.

• You need to find two integers, A[i] and A[j], from the array A such that A[i] XOR A[j] is minimized.

• The XOR operation calculates the bitwise exclusive OR of two numbers, which results in a new number with bits set where the corresponding bits in the original numbers are different.

The goal is to determine both the minimum XOR value and the pair of integers that produce this minimum XOR.

This problem can be solved efficiently using various approaches, such as sorting the array or using a Tries data structure, as shown in the previous code examples. The final result will be the minimum XOR value found in the array.

Approach 1(Brute-Force):

To find the pair of integers in the array with the minimum XOR value using a brute-force approach with a time complexity of O(N^2), you can compare all possible pairs of integers in the array and keep track of the minimum XOR value. Here's a JS code implementation of this approach:

Solution Code:

function findMinimumXOR(arr) {
    let n = arr.length;
    let minXOR = Number.MAX_SAFE_INTEGER;
    
    for (let i = 0; i < n; i++) {
        for (let j = i + 1; j < n; j++) {
            let xor = arr[i] ^ arr[j];
            minXOR = Math.min(minXOR, xor);
        }
    }
    
    return minXOR;
}

// Example usage:
const arr = [9, 5, 3, 7, 2, 8];
const result = findMinimumXOR(arr);
console.log("Minimum XOR value:", result);

This JavaScript code defines a function findMinimumXOR that iterates through all possible pairs of integers in the array and calculates the XOR value for each pair. It keeps track of the minimum XOR value encountered and returns it. The example usage at the end demonstrates how to use this function with an example array.

Approach 2(Sorting):

You can find the pair of integers in an array with the minimum XOR value in O(N log N) time complexity using sorting. Here's a JavaScript code implementation:

function findMinimumXOR(arr) {
    arr.sort((a, b) => a - b);
    
    let minXOR = Number.MAX_SAFE_INTEGER;
    
    for (let i = 0; i < arr.length - 1; i++) {
        const XOR = arr[i] ^ arr[i + 1];
        minXOR = Math.min(minXOR, XOR);
    }
    
    return minXOR;
}

// Example usage:
const arr = [9, 5, 3, 7, 2, 8];
const result = findMinimumXOR(arr);
console.log("Minimum XOR value:", result);

This code first sorts the input array in ascending order. Then, it iterates through the sorted array and calculates the XOR value for each adjacent pair of integers. It keeps track of the minimum XOR value encountered and returns it. The time complexity of this solution is O(N log N) due to the sorting step.