Find maximum XOR of given integer in a stream of integers

You are given a number of queries Q and each query will be of the following types:

1. Query 1 : add(x) This means add x into your data structure.
2. Query 2 : maxXOR(y) This means print the maximum possible XOR of y with all the elements already stored in the data structure.

1 <= x, y <= 10^9 1 <= 10^5 <= Q The data structure begins with only a 0 in it. Example:

Input: (1 10), (1 13), (2 10), (1 9), (1 5), (2 6)
Output: 7 15

Add 10 and 13 to stream.
Find maximum XOR with 10, which is 7
Insert 9 and 5
Find maximum XOR with 6 which is 15.

A good way to solve this problem is to use a Trie. A prerequisite for this post is Trie Insert and Search. Each Trie Node will look following:

struct TrieNode
{
    // We use binary and hence we 
    // need only 2 children
    TrieNode* Children[2]; 
    bool isLeaf;
};

Another thing to handle is that we have to pad the binary equivalent of each input number by a suitable number of zeros to the left before storing them. The maximum possible value of x or y is 10^9 and hence 32 bits will be sufficient. So how does this work? Assume we have to insert 3 and 7 into Trie. The Trie starts out with 0 and after these three insertions can be visualized like this: For simplification, the padding has been done to store each number using 3 bits. Note that in binary: 3 is 011 7 is 111 Now if we have to insert 1 into our Trie, we can note that 1 is 001 and we already have path for 00. So we make a new node for the last set bit and after connecting, we get this: Now if we have to take XOR with 5 which is 101, we note that for the leftmost bit (position 2), we can choose a 0 starting at the root and thus we go to the left. This is the position 2 and we add 2^2 to the answer. For position 1, we have a 0 in 5 and we see that we can choose a 1 from our current node. Thus we go right and add 2^1 to the answer. For position 0, we have a 1 in 5 and we see that we cannot choose a 0 from our current node, thus we go right. The path taken for 5 is shown above. The answer is thus 2^2 + 2^1 = 6. 

CPP

// C++ program to find maximum XOR in
// a stream of integers
#include<bits/stdc++.h>
using namespace std;

struct TrieNode
{
    TrieNode* children[2];
    bool isLeaf;
};

// This checks if the ith position in
// binary of N is a 1 or a 0
bool check(int N, int i)
{
    return (bool)(N & (1<<i));
}

// Create a new Trie node
TrieNode* newNode()
{
    TrieNode* temp = new TrieNode;
    temp->isLeaf = false;
    temp->children[0] = NULL;
    temp->children[1] = NULL;
    return temp;
}

// Inserts x into the Trie
void insert(TrieNode* root, int x)
{
    TrieNode* Crawler = root;

    // padding upto 32 bits
    for (int i = 31; i >= 0; i--)
    {
        int f = check(x, i);
        if (! Crawler->children[f])
            Crawler->children[f] = newNode();
        Crawler = Crawler->children[f];
    }
    Crawler->isLeaf = true;
}

// Finds maximum XOR of x with stream of
// elements so far.
int query2(TrieNode *root, int x)
{
    TrieNode* Crawler = root;

    // Do XOR from root to a leaf path
    int ans = 0;
    for (int i = 31; i >= 0; i--)
    {
        // Find i-th bit in x
        int f = check(x, i);

        // Move to the child whose XOR with f
        // is 1.
        if ((Crawler->children[f ^ 1]))
        {
            ans = ans + (1 << i); // update answer
            Crawler = Crawler->children[f ^ 1];
        }

        // If child with XOR 1 doesn't exist
        else
            Crawler = Crawler->children[f];
    }

    return ans;
}

// Process x (Add x to the stream)
void query1(TrieNode *root, int x)
{
    insert(root, x);
}

// Driver code
int main()
{
    TrieNode* root = newNode();
    query1(root, 10);
    query1(root, 13);
    cout << query2(root, 10) << endl;
    query1(root, 9);
    query1(root, 5);
    cout << query2(root, 6) << endl;
    return 0;
}

Java

// Java program to find maximum XOR in
// a stream of integers
import java.util.ArrayList;
import java.util.List;

class TrieNode {
    TrieNode[] children = new TrieNode[2];
    boolean isLeaf;

    TrieNode()
    {
        this.isLeaf = false;
        children[0] = null;
        children[1] = null;
    }
}

public class Main {
    // This checks if the ith position in
    // binary of N is a 1 or a 0
    static boolean check(int N, int i)
    {
        return (N & (1 << i)) != 0;
    }

    // Create a new Trie node
    static TrieNode newNode() { return new TrieNode(); }

    // Inserts x into the Trie
    static void insert(TrieNode root, int x)
    {
        TrieNode Crawler = root;

        // padding upto 32 bits
        for (int i = 31; i >= 0; i--) {
            int f = check(x, i) ? 1 : 0;
            if (Crawler.children[f] == null) {
                Crawler.children[f] = newNode();
            }
            Crawler = Crawler.children[f];
        }
        Crawler.isLeaf = true;
    }

    // Finds maximum XOR of x with stream of
    // elements so far.
    static int query2(TrieNode root, int x)
    {

        TrieNode Crawler = root;

        // Do XOR from root to a leaf path
        int ans = 0;
        for (int i = 31; i >= 0; i--) {
            // Find i-th bit in x
            int f = check(x, i) ? 1 : 0;

            // Move to the child whose XOR with f
            // is 1.
            if (Crawler.children[f ^ 1] != null) {
                // update answer
                ans += (1 << i);
                Crawler = Crawler.children[f ^ 1];
            }
            // If child with XOR 1 doesn't exist
            else {
                Crawler = Crawler.children[f];
            }
        }
        return ans;
    }

    // Process x (Add x to the stream)
    static void query1(TrieNode root, int x)
    {
        insert(root, x);
    }

    // Driver code
    public static void main(String[] args)
    {
        TrieNode root = newNode();
        query1(root, 10);
        query1(root, 13);
        System.out.println(query2(root, 10));
        query1(root, 9);
        query1(root, 5);
        System.out.println(query2(root, 6));
    }
}

// This code is contributed by Aman Kumar

Python3

# Define TrieNode class
class TrieNode:
    def __init__(self):
        self.children = [None, None]
        self.isLeaf = False

# Check if ith bit of N is 1 or 0
def check(N, i):
    return (N & (1 << i)) != 0

# Create a new TrieNode
def newNode():
    return TrieNode()

# Insert x into Trie
def insert(root, x):
    crawler = root

    # Padding up to 32 bits
    for i in range(31, -1, -1):
        f = 1 if check(x, i) else 0
        if crawler.children[f] is None:
            crawler.children[f] = newNode()
        crawler = crawler.children[f]
    crawler.isLeaf = True
    return root

# Find maximum XOR of x with stream of elements so far
def query2(root, x):
    crawler = root

    # Do XOR from root to a leaf path
    ans = 0
    for i in range(31, -1, -1):
        # Find ith bit in x
        f = 1 if check(x, i) else 0

        # Move to the child whose XOR with f is 1
        if crawler.children[f ^ 1] is not None:
            # Update answer
            ans += 1 << i
            crawler = crawler.children[f ^ 1]
        # If child with XOR 1 doesn't exist
        else:
            crawler = crawler.children[f]
    return ans

# Process x (add x to the stream)
def query1(root, x):
    return insert(root, x)


# Driver code
root = TrieNode()
root = query1(root, 10)
root = query1(root, 13)
print(query2(root, 10))
root = query1(root, 9)
root = query1(root, 5)
print(query2(root, 6))

C#

// C# program to find maximum XOR in
// a stream of integers
using System;

public class TrieNode {
public TrieNode[] children = new TrieNode[2];
public bool isLeaf;

public TrieNode()
{
    this.isLeaf = false;
    children[0] = null;
    children[1] = null;
}
}

public class Program {
// This checks if the ith position in
// binary of N is a 1 or a 0
static bool Check(int N, int i)
{
return (N & (1 << i)) != 0;
}

// Create a new Trie node
static TrieNode NewNode() { return new TrieNode(); }

// Inserts x into the Trie
static void insert(TrieNode root, int x)
{
    TrieNode Crawler = root;

    // padding upto 32 bits
    for (int i = 31; i >= 0; i--)
    {
        int f = Check(x, i) ? 1 : 0;
        if (Crawler.children[f] == null)
        {
            Crawler.children[f] = NewNode();
        }
        Crawler = Crawler.children[f];
    }
    Crawler.isLeaf = true;
}

// Finds maximum XOR of x with stream of
// elements so far.
static int query2(TrieNode root, int x)
{
    TrieNode Crawler = root;

    // Do XOR from root to a leaf path
    int ans = 0;
    for (int i = 31; i >= 0; i--)
    {
        // Find i-th bit in x
        int f = Check(x, i) ? 1 : 0;

        // Move to the child whose XOR with f
        // is 1.
        if (Crawler.children[f ^ 1] != null)
        {
            // update answer
            ans += (1 << i);
            Crawler = Crawler.children[f ^ 1];
        }
        // If child with XOR 1 doesn't exist
        else
        {
            Crawler = Crawler.children[f];
        }
    }
    return ans;
}

// Process x (Add x to the stream)
static void Query1(TrieNode root, int x)
{
    insert(root, x);
}

// Driver code
public static void Main()
{
    TrieNode root = NewNode();
    Query1(root, 10);
    Query1(root, 13);
    Console.WriteLine(query2(root, 10));
    Query1(root, 9);
    Query1(root, 5);
    Console.WriteLine(query2(root, 6));
}
}

// This code is contributed by Pushpesh Raj.

Javascript

// Define TrieNode class
class TrieNode {
constructor() {
this.children = [null, null];
this.isLeaf = false;
}
}

// Check if ith bit of N is 1 or 0
function check(N, i) {
return (N & (1 << i)) !== 0;
}

// Create a new TrieNode
function newNode() {
return new TrieNode();
}

// Insert x into Trie
function insert(root, x) {
let crawler = root;

// Padding up to 32 bits
for (let i = 31; i >= 0; i--) {
const f = check(x, i) ? 1 : 0;
if (crawler.children[f] === null) {
crawler.children[f] = newNode();
}
crawler = crawler.children[f];
}
crawler.isLeaf = true;
return root;
}

// Find maximum XOR of x with stream of elements so far
function query2(root, x) {
let crawler = root;

// Do XOR from root to a leaf path
let ans = 0;
for (let i = 31; i >= 0; i--) {
// Find ith bit in x
const f = check(x, i) ? 1 : 0;


// Move to the child whose XOR with f is 1
if (crawler.children[f ^ 1] !== null) {
  // Update answer
  ans += 1 << i;
  crawler = crawler.children[f ^ 1];
}
// If child with XOR 1 doesn't exist
else {
  crawler = crawler.children[f];
}
}
return ans;
}

// Process x (add x to the stream)
function query1(root, x) {
return insert(root, x);
}

// Driver code
let root = new TrieNode();
root = query1(root, 10);
root = query1(root, 13);
document.write(query2(root, 10));
root = query1(root, 9);
root = query1(root, 5);
document.write(query2(root, 6));

7
15

The space taken by the Trie is O(n*log(n)). Each query of type 1 takes O(log(n)) time. Each query of type 2 takes O(log(n)) time too. Here n is the largest query number. Follow up problem: What if we are given three queries instead of two? 1) add(x) This means add x into your data structure (duplicates are allowed). 2) maxXOR(y) This means print the maximum possible XOR of y with all the elements already stored in the data structure. 3) remove(z) This means remove one instance of z from the data structure. What changes in the Trie solution can achieve this?