Minimum XOR Value Pair

Given an array of integers. Find the pair in an array that has a minimum XOR value. 

Examples :

Input : arr[] =  {9, 5, 3}
Output : 6
        All pair with xor value (9 ^ 5) => 12, 
        (5 ^ 3) => 6, (9 ^ 3) => 10.
        Minimum XOR value is 6

Input : arr[] = {1, 2, 3, 4, 5}
Output : 1 

A Simple Solution is to generate all pairs of the given array and compute XOR their values. Finally, return minimum XOR value. This solution takes O(n²) time. 

Implementation:

C++

// C++ program to find minimum XOR value in an array.
#include <bits/stdc++.h>
using namespace std;

// Returns minimum xor value of pair in arr[0..n-1]
int minXOR(int arr[], int n)
{
    int min_xor = INT_MAX; // Initialize result

    // Generate all pair of given array
    for (int i = 0; i < n; i++)
        for (int j = i + 1; j < n; j++)

            // update minimum xor value if required
            min_xor = min(min_xor, arr[i] ^ arr[j]);

    return min_xor;
}

// Driver program
int main()
{
    int arr[] = { 9, 5, 3 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << minXOR(arr, n) << endl;
    return 0;
}

Java

// Java program to find minimum XOR value in an array.
class GFG {

    // Returns minimum xor value of pair in arr[0..n-1]
    static int minXOR(int arr[], int n)
    {
        int min_xor = Integer.MAX_VALUE; // Initialize result

        // Generate all pair of given array
        for (int i = 0; i < n; i++)
            for (int j = i + 1; j < n; j++)

                // update minimum xor value if required
                min_xor = Math.min(min_xor, arr[i] ^ arr[j]);

        return min_xor;
    }

    // Driver program
    public static void main(String args[])
    {
        int arr[] = { 9, 5, 3 };
        int n = arr.length;
        System.out.println(minXOR(arr, n));
    }
}
// This code is contributed by Sumit Ghosh

Python3

# Python program to find minimum
# XOR value in an array.

# Function to find minimum XOR pair
def minXOR(arr, n):

    # Sort given array
    arr.sort();

    min_xor = 999999
    val = 0

    # calculate min xor of
    # consecutive pairs
    for i in range (0, n-1):
        for j in range (i+1, n-1):

            # update minimum xor value
            # if required
            val = arr[i] ^ arr[j]
            min_xor = min(min_xor, val)
    return min_xor

# Driver program
arr = [ 9, 5, 3 ]
n = len(arr)

print(minXOR(arr, n))

# This code is contributed by Sam007.

C#

// C# program to find minimum
// XOR value in an array.
using System;

class GFG {

    // Returns minimum xor value of
    // pair in arr[0..n-1]
    static int minXOR(int[] arr, int n)
    {
         // Initialize result
        int min_xor = int.MaxValue;

        // Generate all pair of given array
        for (int i = 0; i < n; i++)
            for (int j = i + 1; j < n; j++)

            // update minimum xor value if required
            min_xor = Math.Min(min_xor, arr[i] ^ arr[j]);

        return min_xor;
    }

    // Driver program
    public static void Main()
    {
        int[] arr = { 9, 5, 3 };
        int n = arr.Length;
        Console.WriteLine(minXOR(arr, n));
    }
}

// This code is contributed by Sam007

PHP

<?php
// PHP program to find minimum
// XOR value in an array.

// Returns minimum xor value
// of pair in arr[0..n-1]
function minXOR($arr, $n)
{
    // Initialize result
    $min_xor = PHP_INT_MAX;

    // Generate all pair of given array
    for ( $i = 0; $i < $n; $i++)
        for ( $j = $i + 1; $j < $n; $j++)

            // update minimum xor
            // value if required
            $min_xor = min($min_xor, $arr[$i] ^ $arr[$j]);

    return $min_xor;
}

    // Driver Code
    $arr = array(9, 5, 3);
    $n = count($arr);
    echo minXOR($arr, $n);

// This code is contributed by anuj_67.
?>

Javascript

<script>

// Javascript program to find
// minimum XOR value in an array.

// Returns minimum xor value of pair in arr[0..n-1]
function minXOR(arr, n)
{
    // Initialize result
    let min_xor = Number.MAX_VALUE;

    // Generate all pair of given array
    for (let i = 0; i < n; i++)
        for (let j = i + 1; j < n; j++)

            // update minimum xor value if required
            min_xor = Math.min(min_xor, arr[i] ^ arr[j]);

    return min_xor;
}

// Driver program
    let arr = [ 9, 5, 3 ];
    let n = arr.length;
    document.write(minXOR(arr, n));

</script>

Output

6

Space Complexity: O(1) 

An Efficient solution can solve this problem in O(nlogn) time. 

Algorithm: 

1. Sort the given array
2. Traverse and check XOR for every consecutive pair

Below is the implementation of above approach:

C++

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

// Function to find minimum XOR pair
int minXOR(int arr[], int n)
{
    // Sort given array
    sort(arr, arr + n);

    int minXor = INT_MAX;
    int val = 0;

    // calculate min xor of consecutive pairs
    for (int i = 0; i < n - 1; i++) {
        val = arr[i] ^ arr[i + 1];
        minXor = min(minXor, val);
    }

    return minXor;
}

// Driver program
int main()
{
    int arr[] = { 9, 5, 3 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << minXOR(arr, n) << endl;

    return 0;
}

Java

import java.util.Arrays;
class GFG {

    // Function to find minimum XOR pair
    static int minXOR(int arr[], int n)
    {
        // Sort given array
        Arrays.parallelSort(arr);

        int minXor = Integer.MAX_VALUE;
        int val = 0;

        // calculate min xor of consecutive pairs
        for (int i = 0; i < n - 1; i++) {
            val = arr[i] ^ arr[i + 1];
            minXor = Math.min(minXor, val);
        }

        return minXor;
    }

    // Driver program
    public static void main(String args[])
    {
        int arr[] = { 9, 5, 3 };
        int n = arr.length;
        System.out.println(minXOR(arr, n));
    }
}

// This code is contributed by Sumit Ghosh

Python3

import sys

# Function to find minimum XOR pair
def minXOR(arr, n):

    # Sort given array
    arr.sort()

    minXor =  int(sys.float_info.max)
    val = 0

    # calculate min xor of consecutive pairs
    for i in range(0,n-1):
        val = arr[i] ^ arr[i + 1];
        minXor = min(minXor, val);

    return minXor

# Driver program
arr = [9, 5, 3]
n = len(arr)
print(minXOR(arr, n))

# This code is contributed by Sam007.

C#

// C# program to find minimum
// XOR value in an array.
using System;

class GFG {

    // Function to find minimum XOR pair
    static int minXOR(int[] arr, int n)
    {
        // Sort given array
        Array.Sort(arr);

        int minXor = int.MaxValue;
        int val = 0;

        // calculate min xor of consecutive pairs
        for (int i = 0; i < n - 1; i++) {
            val = arr[i] ^ arr[i + 1];
            minXor = Math.Min(minXor, val);
        }

        return minXor;
    }

    // Driver program
    public static void Main()
    {
        int[] arr = { 9, 5, 3 };
        int n = arr.Length;
        Console.WriteLine(minXOR(arr, n));
    }
}

// This code is contributed by Sam007

PHP

<?php
// Function to find minimum XOR pair
function minXOR($arr, $n)
{
    // Sort given array
    sort($arr);

    $minXor = PHP_INT_MAX;
    $val = 0;

    // calculate min xor
    // of consecutive pairs
    for ($i = 0; $i < $n - 1; $i++)
    {
        $val = $arr[$i] ^ $arr[$i + 1];
        $minXor = min($minXor, $val);
    }

    return $minXor;
}

// Driver Code
$arr = array(9, 5, 3);
$n = count($arr);
echo minXOR($arr, $n);

// This code is contributed by Smitha.
?>

Javascript

<script>

// Function to find minimum XOR pair
function minXOR(arr, n)
{
    // Sort given array
    arr.sort();

    let minXor = Number.MAX_VALUE;
    let val = 0;

    // calculate min xor of consecutive pairs
    for (let i = 0; i < n - 1; i++) {
        val = arr[i] ^ arr[i + 1];
        minXor = Math.min(minXor, val);
    }

    return minXor;
}

// Driver program
    let arr = [ 9, 5, 3 ];
    let n = arr.length;
    document.write(minXOR(arr, n));

</script>

Output

6

Time Complexity: O(N*logN) 
Space Complexity: O(1) 

A further more Efficient solution can solve the above problem in O(n) time under the assumption that integers take fixed number of bits to store. The idea is to use Trie Data Structure.

Algorithm:

1. Create an empty trie. Every node of trie contains two children for 0 and 1 bits.
2. Initialize min_xor = INT_MAX, insert arr[0] into trie
3. Traversal all array element one-by-one starting from second.
   1. First find minimum setbet difference value in trie 
      • do xor of current element with minimum setbit diff that value 
   2. update min_xor value if required
   3. insert current array element in trie 
4. return min_xor

Below is the implementation of above algorithm.

C++

// C++ program to find minimum XOR value in an array.
#include <bits/stdc++.h>
using namespace std;
#define INT_SIZE 32

// A Trie Node
struct TrieNode {
    int value; // used in leaf node
    TrieNode* Child[2];
};

// Utility function to create a new Trie node
TrieNode* getNode()
{
    TrieNode* newNode = new TrieNode;
    newNode->value = 0;
    newNode->Child[0] = newNode->Child[1] = NULL;
    return newNode;
}

// utility function insert new key in trie
void insert(TrieNode* root, int key)
{
    TrieNode* temp = root;

    // start from the most significant bit, insert all
    // bit of key one-by-one into trie
    for (int i = INT_SIZE - 1; i >= 0; i--) {
        // Find current bit in given prefix
        bool current_bit = (key & (1 << i));

        // Add a new Node into trie
        if (temp->Child[current_bit] == NULL)
            temp->Child[current_bit] = getNode();

        temp = temp->Child[current_bit];
    }

    // store value at leafNode
    temp->value = key;
}

// Returns minimum XOR value of an integer inserted
// in Trie and given key.
int minXORUtil(TrieNode* root, int key)
{
    TrieNode* temp = root;

    for (int i = INT_SIZE - 1; i >= 0; i--) {
        // Find current bit in given prefix
        bool current_bit = (key & (1 << i));

        // Traversal Trie, look for prefix that has
        // same bit
        if (temp->Child[current_bit] != NULL)
            temp = temp->Child[current_bit];

        // if there is no same bit.then looking for
        // opposite bit
        else if (temp->Child[1 - current_bit] != NULL)
            temp = temp->Child[1 - current_bit];
    }

    // return xor value of minimum bit difference value
    // so we get minimum xor value
    return key ^ temp->value;
}

// Returns minimum xor value of pair in arr[0..n-1]
int minXOR(int arr[], int n)
{
    int min_xor = INT_MAX; // Initialize result

    // create a True and insert first element in it
    TrieNode* root = getNode();
    insert(root, arr[0]);

    // Traverse all array element and find minimum xor
    // for every element
    for (int i = 1; i < n; i++) {
        // Find minimum XOR value of current element with
        // previous elements inserted in Trie
        min_xor = min(min_xor, minXORUtil(root, arr[i]));

        // insert current array value into Trie
        insert(root, arr[i]);
    }
    return min_xor;
}

// Driver code
int main()
{
    int arr[] = { 9, 5, 3 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << minXOR(arr, n) << endl;
    return 0;
}

Java

// Java program to find minimum XOR value in an array.
class GFG {
    static final int INT_SIZE = 32;

    // A Trie Node
    static class TrieNode {
        int value; // used in leaf node
        TrieNode[] Child = new TrieNode[2];

        public TrieNode()
        {
            value = 0;
            Child[0] = null;
            Child[1] = null;
        }
    }
    static TrieNode root;

    // utility function insert new key in trie
    static void insert(int key)
    {
        TrieNode temp = root;

        // start from the most significant bit, insert all
        // bit of key one-by-one into trie
        for (int i = INT_SIZE - 1; i >= 0; i--) {
            // Find current bit in given prefix
            int current_bit = (key & (1 << i)) >= 1 ? 1 : 0;

            // Add a new Node into trie
            if (temp != null && temp.Child[current_bit] == null)
                temp.Child[current_bit] = new TrieNode();

            temp = temp.Child[current_bit];
        }

        // store value at leafNode
        temp.value = key;
    }

    // Returns minimum XOR value of an integer inserted
    // in Trie and given key.
    static int minXORUtil(int key)
    {
        TrieNode temp = root;

        for (int i = INT_SIZE - 1; i >= 0; i--) {
            // Find current bit in given prefix
            int current_bit = (key & (1 << i)) >= 1 ? 1 : 0;

            // Traversal Trie, look for prefix that has
            // same bit
            if (temp.Child[current_bit] != null)
                temp = temp.Child[current_bit];

            // if there is no same bit.then looking for
            // opposite bit
            else if (temp.Child[1 - current_bit] != null)
                temp = temp.Child[1 - current_bit];
        }

        // return xor value of minimum bit difference value
        // so we get minimum xor value
        return key ^ temp.value;
    }

    // Returns minimum xor value of pair in arr[0..n-1]
    static int minXOR(int arr[], int n)
    {
        int min_xor = Integer.MAX_VALUE; // Initialize result

        // create a True and insert first element in it
        root = new TrieNode();
        insert(arr[0]);

        // Traverse all array element and find minimum xor
        // for every element
        for (int i = 1; i < n; i++) {
            // Find minimum XOR value of current element with
            // previous elements inserted in Trie
            min_xor = Math.min(min_xor, minXORUtil(arr[i]));

            // insert current array value into Trie
            insert(arr[i]);
        }
        return min_xor;
    }

    // Driver code
    public static void main(String args[])
    {
        int arr[] = { 9, 5, 3 };
        int n = arr.length;
        System.out.println(minXOR(arr, n));
    }
}
// This code is contributed by Sumit Ghosh

Python

# class for the basic Trie Node
class TrieNode:
    def __init__(self):

        # Child array with 0 and 1
        self.child = [None]*2

        # meant for the lead Node
        self.value = None

class Trie:

    def __init__(self):
        # initialise the root Node
        self.root = self.getNode()

    def getNode(self):
        # get a new Trie Node
        return TrieNode()

    # inserts a new element
    def insert(self,key):
        temp = self.root

        # 32 bit valued binary digit
        for i in range(31,-1,-1):

            # finding the bit at ith position
            curr = (key>>i)&(1)

            # if the child is None create one
            if(temp.child[curr] is None):
                temp.child[curr] = self.getNode()
            temp = temp.child[curr]

        # add value to the leaf node
        temp.value = key

    # traverse the trie and xor with the most similar element
    def xorUtil(self,key):
        temp = self.root

        # 32 bit valued binary digit
        for i in range(31,-1,-1):

            # finding the bit at ith position
            curr = (key>>i)&1

            # traverse for the same bit
            if(temp.child[curr] is not None):
                temp = temp.child[curr]

            # traverse if the same bit is not set in trie
              elif (temp.child[1-curr] is not None):
                temp = temp.child[1-curr]

        # return with the xor of the value
        return temp.value^key


def minXor(arr):

        # set m to a large number
        m = 2**30

        # initialize Trie
        trie = Trie()

        # insert the first element
        trie.insert(arr[0])

        # for each element in the array
        for i in range(1,len(arr)):

            # find the minimum xor value
            m = min(m,trie.xorUtil(arr[i]))

            # insert the new element
            trie.insert(arr[i])
        return m

# Driver Code
if __name__=="__main__":
    sample = [9,5,3]
    print(minXor(sample))

#code contributed by Shushant Kumar

C#

// Include namespace system
using System;

// C# program to find minimum XOR value in an array.
public class GFG
{
  public const int INT_SIZE = 32;

  // A Trie Node
  public class TrieNode
  {
    public int value;

    // used in leaf node
    public TrieNode[] Child = new TrieNode[2];
    public TrieNode()
    {
      this.value = 0;
      this.Child[0] = null;
      this.Child[1] = null;
    }
  }
  public static TrieNode root;

  // utility function insert new key in trie
  public static void insert(int key)
  {
    var temp = root;

    // start from the most significant bit, insert all
    // bit of key one-by-one into trie
    for (int i = GFG.INT_SIZE - 1; i >= 0; i--)
    {

      // Find current bit in given prefix
      var current_bit = (key & (1 << i)) >= 1 ? 1 : 0;

      // Add a new Node into trie
      if (temp != null && temp.Child[current_bit] == null)
      {
        temp.Child[current_bit] = new TrieNode();
      }
      temp = temp.Child[current_bit];
    }

    // store value at leafNode
    temp.value = key;
  }

  // Returns minimum XOR value of an integer inserted
  // in Trie and given key.
  public static int minXORUtil(int key)
  {
    var temp = root;
    for (int i = GFG.INT_SIZE - 1; i >= 0; i--)
    {

      // Find current bit in given prefix
      var current_bit = (key & (1 << i)) >= 1 ? 1 : 0;

      // Traversal Trie, look for prefix that has
      // same bit
      if (temp.Child[current_bit] != null)
      {
        temp = temp.Child[current_bit];
      }
      else if (temp.Child[1 - current_bit] != null)
      {
        temp = temp.Child[1 - current_bit];
      }
    }

    // return xor value of minimum bit difference value
    // so we get minimum xor value
    return key ^ temp.value;
  }

  // Returns minimum xor value of pair in arr[0..n-1]
  public static int minXOR(int[] arr, int n)
  {
    var min_xor = int.MaxValue;

    // Initialize result
    // create a True and insert first element in it
    root = new TrieNode();
    GFG.insert(arr[0]);

    // Traverse all array element and find minimum xor
    // for every element
    for (int i = 1; i < n; i++)
    {

      // Find minimum XOR value of current element with
      // previous elements inserted in Trie
      min_xor = Math.Min(min_xor,GFG.minXORUtil(arr[i]));

      // insert current array value into Trie
      GFG.insert(arr[i]);
    }
    return min_xor;
  }

  // Driver code
  public static void Main(String[] args)
  {
    int[] arr = {9, 5, 3};
    var n = arr.Length;
    Console.WriteLine(GFG.minXOR(arr, n));
  }
}

// This code is contributed by aadityaburujwale.

Javascript

class TrieNode {
constructor() {
this.child = new Array(2);
this.value = null;
}
}

class Trie {
constructor() {
this.root = this.getNode();
}

getNode() {
return new TrieNode();
}

insert(key) {
let temp = this.root;
for (let i = 31; i >= 0; i--) {
let curr = (key >> i) & 1;
if (!temp.child[curr]) temp.child[curr] = this.getNode();
temp = temp.child[curr];
}
temp.value = key;
}

xorUtil(key) {
let temp = this.root;
for (let i = 31; i >= 0; i--) {
let curr = (key >> i) & 1;
if (temp.child[curr]) temp = temp.child[curr];
else if (temp.child[1 - curr]) temp = temp.child[1 - curr];
}
return temp.value ^ key;
}
}

function minXor(arr) {
let m = 2 ** 30;
let trie = new Trie();
trie.insert(arr[0]);
for (let i = 1; i < arr.length; i++) {
m = Math.min(m, trie.xorUtil(arr[i]));
trie.insert(arr[i]);
}
return m;
}

if (typeof module !== 'undefined') {
module.exports = {
minXor: minXor,
};
}

console.log(minXor([9, 5, 3]));

// This code is contributed by akashish__

Output

6

Time Complexity O(n)

Space Complexity: O(n*INT_SIZE) 

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