Bitwise Operations in Subarrays: Query and Result

Given a positive integer array A[] of size N and Q queries. Given a 2D integer array Queries[][] of length Q where each query has four integers l1, r1, l2, r2, the task is to calculate (Al1 & Al1+1 & Al1+2 …& Ar1) | (Al2 & Al2+1 & Al2+2….& Ar2) and return an integer array answer of length Q where answer[i] represents the answer for ith query.

Examples:

Input: N = 5, Q = 2, A[] = {2, 3, 7, 11, 15}, Queries[][] = {{2, 4, 3, 5}, {1, 5, 2, 3}}
Output: {3, 3}
Explanation:

• for query 2 4 3 5: answer = (3&7&11) | (7&11&15) = (3) | (3) = 3
• for query 1 5 2 3: answer = (2&3&7&11&15) | (3&7) = (2) | (3) = 3

Input: N = 6, Q = 2, A[] = {3, 7, 13, 12, 14, 15}, Queries[][] = {{1, 4, 2, 5}, {1, 2, 3, 6}}
Output: {4 15}
Explanation:

• for query 1 4 2 5: answer = (3&7&13&12) | (7&13&12&14) = (0) | (4) = 4
• for query 1 2 3 6: answer = (3&7) | (13&12&14&15) = (3) | (12) = 15

Approach: This can be solved with the following idea:

The intuition behind this solution is to utilize bitwise operations efficiently. By pre-calculating the counts of each bit up to a certain index, the code can quickly determine if a particular bit position is set in a given subarray. The final answer is calculated by performing bitwise OR operations for the bits that satisfy the conditions specified in the query.

Below are the steps involved:

• pre is initialized as an empty list to store the prefix counts for each bit.
  • bit is initialized as a list of 30 zeroes, which will be used to count the occurrences of each bit in the array A.
• The initial state of bit is appended to pre as the first element.
• The code then iterates through each element e in array A and for each bit position (from 0 to 29), it checks if the bit is set.
• If it is set, it increments the count of that bit position in the bit array.
• After each iteration through the elements of A, a copy of the current state of the bit array is appended to the pre array. This allows us to keep track of the cumulative counts of each bit up to a certain index in A.
• The code initializes an empty list res to store the results for each query.
• It initializes ans as 0, which will be the answer for each query.
• For each query, it calculates the lengths d1 and d2 based on the ranges provided in the query.
• It then iterates through each bit position from 0 to 29 and checks if either of the subarrays (defined by the query ranges) has all the bits set for that position. If it finds such a position, it adds 2^i (i.e., 1 << i) to the answer ans.
• Finally, it appends the answer ans for the current query to the result list res.

Below is the implementation of the code:

C++

// C++ code for the above approach:
#include <bits/stdc++.h>
#include <iostream>
using namespace std;

// Function to find minimum query
vector<int> minQuery(int N, int Q, vector<int>& A,
                     vector<vector<int> >& Queries)
{
    // Intialize a 2D vector
    vector<vector<int> > v(N + 1, vector<int>(31, 0));

    // Iterate in given array
    for (int i = 1; i <= N; i++) {

        // Iterate for 31 bits
        for (int j = 0; j < 31; j++) {

            v[i][j] = v[i - 1][j] + ((A[i - 1] >> j) & 1);
        }
    }

    vector<int> ans(Q);
    for (int i = 0; i < Q; i++) {
        int a1 = 0;
        int a2 = 0;

        // Iterate for queries
        int l1 = Queries[i][0], r1 = Queries[i][1],
            l2 = Queries[i][2], r2 = Queries[i][3];

        for (int j = 0; j < 31; j++) {
            if (v[r1][j] - v[l1 - 1][j] == (r1 - l1 + 1))
                a1 += (1 << j);
            if (v[r2][j] - v[l2 - 1][j] == (r2 - l2 + 1))
                a2 += (1 << j);
        }

        // Do OR of a1 and a2
        ans[i] = (a1 | a2);
    }

    return ans;
}

// Driver code
int main()
{

    int N = 5;
    int Q = 2;
    vector<int> A = { 2, 3, 7, 11, 15 };
    vector<vector<int> > Queries
        = { { 2, 4, 3, 5 }, { 1, 5, 2, 3 } };

    // Function call
    vector<int> ans = minQuery(N, Q, A, Queries);

    for (auto a : ans) {
        cout << a << " ";
    }
    return 0;
}

Java

import java.util.ArrayList;
import java.util.List;

public class MinimumQuery {

    // Function to find minimum query
    static List<Integer> minQuery(int N, int Q, List<Integer> A, List<List<Integer>> Queries) {
        // Initialize a 2D array
        int[][] v = new int[N + 1][31];

        // Iterate in the given array
        for (int i = 1; i <= N; i++) {
            // Iterate for 31 bits
            for (int j = 0; j < 31; j++) {
                v[i][j] = v[i - 1][j] + ((A.get(i - 1) >> j) & 1);
            }
        }

        List<Integer> ans = new ArrayList<>();
        for (int i = 0; i < Q; i++) {
            int a1 = 0;
            int a2 = 0;

            // Iterate for queries
            int l1 = Queries.get(i).get(0), r1 = Queries.get(i).get(1),
                l2 = Queries.get(i).get(2), r2 = Queries.get(i).get(3);

            for (int j = 0; j < 31; j++) {
                if (v[r1][j] - v[l1 - 1][j] == (r1 - l1 + 1))
                    a1 += (1 << j);
                if (v[r2][j] - v[l2 - 1][j] == (r2 - l2 + 1))
                    a2 += (1 << j);
            }

            // Do OR of a1 and a2
            ans.add(a1 | a2);
        }

        return ans;
    }

    // Driver code
    public static void main(String[] args) {
        int N = 5;
        int Q = 2;
        List<Integer> A = List.of(2, 3, 7, 11, 15);
        List<List<Integer>> Queries = List.of(
            List.of(2, 4, 3, 5),
            List.of(1, 5, 2, 3)
        );

        // Function call
        List<Integer> ans = minQuery(N, Q, A, Queries);

        for (int a : ans) {
            System.out.print(a + " ");
        }
    }
}

Python3

def min_query(N, Q, A, Queries):
    # Initialize a 2D list
    v = [[0] * 31 for _ in range(N + 1)]

    # Iterate through the given array
    for i in range(1, N + 1):
        # Iterate for 31 bits
        for j in range(31):
            v[i][j] = v[i - 1][j] + ((A[i - 1] >> j) & 1)

    ans = []
    for i in range(Q):
        a1 = 0
        a2 = 0

        # Iterate through queries
        l1, r1, l2, r2 = Queries[i]

        for j in range(31):
            if v[r1][j] - v[l1 - 1][j] == (r1 - l1 + 1):
                a1 += (1 << j)
            if v[r2][j] - v[l2 - 1][j] == (r2 - l2 + 1):
                a2 += (1 << j)

        # Do OR of a1 and a2
        ans.append(a1 | a2)

    return ans

# Driver code
if __name__ == "__main__":
    N = 5
    Q = 2
    A = [2, 3, 7, 11, 15]
    Queries = [[2, 4, 3, 5], [1, 5, 2, 3]]

    # Function call
    result = min_query(N, Q, A, Queries)

    # Output the result
    for res in result:
        print(res, end=" ")

C#

using System;
using System.Collections.Generic;

class Program
{
    // Function to find minimum query
    static List<int> MinQuery(int N, int Q, List<int> A, List<List<int>> Queries)
    {
        // Initialize a 2D list
        List<List<int>> v = new List<List<int>>(N + 1);
        for (int i = 0; i <= N; i++)
        {
            v.Add(new List<int>(new int[31]));
        }

        // Iterate in the given array
        for (int i = 1; i <= N; i++)
        {
            // Iterate for 31 bits
            for (int j = 0; j < 31; j++)
            {
                v[i][j] = v[i - 1][j] + ((A[i - 1] >> j) & 1);
            }
        }

        List<int> ans = new List<int>(Q);
        for (int i = 0; i < Q; i++)
        {
            int a1 = 0;
            int a2 = 0;

            // Iterate for queries
            int l1 = Queries[i][0], r1 = Queries[i][1],
                l2 = Queries[i][2], r2 = Queries[i][3];

            for (int j = 0; j < 31; j++)
            {
                if (v[r1][j] - v[l1 - 1][j] == (r1 - l1 + 1))
                    a1 += (1 << j);
                if (v[r2][j] - v[l2 - 1][j] == (r2 - l2 + 1))
                    a2 += (1 << j);
            }

            // Do OR of a1 and a2
            ans.Add(a1 | a2);
        }

        return ans;
    }

    // Driver code
    static void Main()
    {
        int N = 5;
        int Q = 2;
        List<int> A = new List<int> { 2, 3, 7, 11, 15 };
        List<List<int>> Queries = new List<List<int>>
        {
            new List<int> { 2, 4, 3, 5 },
            new List<int> { 1, 5, 2, 3 }
        };

        // Function call
        List<int> result = MinQuery(N, Q, A, Queries);

        // Print the results
        foreach (var a in result)
        {
            Console.Write($"{a} ");
        }


    }
}

Javascript

// Function to find minimum query
function minQuery(N, Q, A, Queries) {
    // Initialize a 2D array
    let v = new Array(N + 1);
    for (let i = 0; i <= N; i++) {
        v[i] = new Array(31).fill(0);
    }

    // Iterate through the given array
    for (let i = 1; i <= N; i++) {
        // Iterate for 31 bits
        for (let j = 0; j < 31; j++) {
            v[i][j] = v[i - 1][j] + ((A[i - 1] >> j) & 1);
        }
    }

    let ans = new Array(Q);
    for (let i = 0; i < Q; i++) {
        let a1 = 0;
        let a2 = 0;

        // Iterate for queries
        let [l1, r1, l2, r2] = Queries[i];

        for (let j = 0; j < 31; j++) {
            if (v[r1][j] - v[l1 - 1][j] === r1 - l1 + 1) {
                a1 += (1 << j);
            }
            if (v[r2][j] - v[l2 - 1][j] === r2 - l2 + 1) {
                a2 += (1 << j);
            }
        }

        // Do OR of a1 and a2
        ans[i] = (a1 | a2);
    }

    return ans;
}

// Driver code
function main() {
    let N = 5;
    let Q = 2;
    let A = [2, 3, 7, 11, 15];
    let Queries = [[2, 4, 3, 5], [1, 5, 2, 3]];

    // Function call
    let result = minQuery(N, Q, A, Queries);

    console.log(result.join(' '));
}

// Run the main function
main();

Output

3 3 

Time Complexity: O(N + Q)
Auxiliary Space: O(N)