Range query for count of set bits

Given an array of positive integer and q query which contains two integers, L & R. Task is to find the number of set bits for a given range.

Prerequisite : Bitwise Hacks

Examples :

Input :  Arr[] = { 1, 5, 6, 10, 9, 4 }
         Query : 2 
         L  &  R
         1     5
         2     4
Output : 9
         6

Input : Arr[] = { 1, 10, 5, 2, 8, 11, 15 }
        Query : 2
        L  &  R
        2     4
        1     5
Output :  4
          9

Simple solution to this problem is to run a loop from L to R and count number of set bits in a Range. This solution take O(nlog(s)) ( where s is bits size ) for each query.

Efficient solution is based on the fact that if we store count of all set bits of numbers in an array “BitCounts”, then we answer each query in O(1) time. So, start traversing the elements of array and count set bits for each element and store in array. Now, find cumulative sum of this array. This array will help in answering queries.

BitCount[] that will store the count of set bits
in a number. 

Run a Loop from 0 to 31 "for 32 bits size integer "       
-> mark elements with i'th bit set     

Run an inner Loop from 0 to size of Array "Arr"  
     -> Check whether the current bit is set or not 
     -> if it's set then mark it.
          long  temp = arr[j] >> i;
             if (temp %2 != 0)
                 BitCount[j] += 1

Below is implementation of above idea.

C++

// C++ program to Range query for
// Count number of set bits
#include <bits/stdc++.h>

using namespace std;
 
// 2-D array that will stored the count
// of bits set in element of array

int BitCount[10000] = { 0 };
 
// Function store the set bit
// count in BitCount Array

void fillSetBitsMatrix(int arr[], int n)
{
 
    // traverse over all bits

    for (int i = 0; i < 32; i++) {
 
        // mark elements with i'th bit set

        for (int j = 0; j < n; j++) {
 
            // Check whether the current bit is

            // set or not if it's set then mark it.

            long temp = arr[j] >> i;

            if (temp % 2 != 0)

                BitCount[j] += 1;

        }

    }
 
    // store cumulative sum of bits

    for (int i = 1; i < n; i++)

        BitCount[i] += BitCount[i - 1];
}
 
// Function to process queries

void Query(int Q[][2], int q)
{

    for (int i = 0; i < q; i++)

        cout << (BitCount[Q[i][1]] - 

                 BitCount[Q[i][0] - 1]) << endl;
}
 
// Driver Code

int main()
{

    int Arr[] = { 1, 5, 6, 10, 9, 4, 67 };

    int n = sizeof(Arr) / sizeof(Arr[0]);
 
    fillSetBitsMatrix(Arr, n);
 
    int q = 2;

    int Q[2][2] = { { 1, 5 }, { 2, 6 } };
 
    Query(Q, q);
 
    return 0;
}

Java

// Java program to Range query for
// Count number of set bits

import java.io.*;
 
class GFG {
 
    // 2-D array that will stored the count

    // of bits set in element of array

    static int BitCount[] = new int[10000];

    // Function store the set bit

    // count in BitCount Array

    static void fillSetBitsMatrix(int arr[], int n)

    {

        // traverse over all bits

        for (int i = 0; i < 32; i++) {

            // mark elements with i'th bit set

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

                // Check whether the current 

                // bit is set or not if it's 

                // set then mark it.

                long temp = arr[j] >> i;

                if (temp % 2 != 0)

                    BitCount[j] += 1;

            }

        }

        // store cumulative sum of bits

        for (int i = 1; i < n; i++)

            BitCount[i] += BitCount[i - 1];

    }

    // Function to process queries

    static void Query(int Q[][], int q)

    {

        for (int i = 0; i < q; i++)

            System.out.println( (BitCount[Q[i][1]]

                        - BitCount[Q[i][0] - 1]));

    }

    // Driver Code

    public static void main (String[] args)

    {

        int Arr[] = { 1, 5, 6, 10, 9, 4, 67 };

        int n = Arr.length;

        fillSetBitsMatrix(Arr, n);

        int q = 2;

        int Q[][] = { { 1, 5 }, { 2, 6 } };

        Query(Q, q);

    }
}
 
// This code is contributed by anuj_67.

Python3

# Python3 program to Range query for
# Count number of set bits
 
# 2-D array that will stored the count
# of bits set in element of array

BitCount = [0] * 10000
 
# Function store the set bit
# count in BitCount Array

def fillSetBitsmatrix(arr: list, n: int):

    global BitCount
 
    # traverse over all bits

    for i in range(32):
 
        # mark elements with i'th bit set

        for j in range(n):
 
            # Check whether the current bit is

            # set or not if it's set then mark it.

            temp = arr[j] >> i

            if temp % 2 != 0:

                BitCount[j] += 1
 
    # store cumulative sum of bits

    for i in range(1, n):

        BitCount[i] += BitCount[i - 1]
 
# Function to process queries

def Query(Q: list, q: int):

    for i in range(q):

        print(BitCount[Q[i][1]] - BitCount[Q[i][0] - 1])
 
# Driver Code

if __name__ == "__main__":
 
    Arr = [1, 5, 6, 10, 9, 4, 67]

    n = len(Arr)
 
    fillSetBitsmatrix(Arr, n)
 
    q = 2

    Q = [(1, 5), (2, 6)]

    Query(Q, q)
 
# This code is contributed by
# sanjeev2552

// C# program to Range query for
// Count number of set bits

using System;
 
class GFG {
 
    // 2-D array that will stored the count

    // of bits set in element of array

    static int []BitCount = new int[10000];

    // Function store the set bit

    // count in BitCount Array

    static void fillSetBitsMatrix(int []arr, int n)

    {

        // traverse over all bits

        for (int i = 0; i < 32; i++) {

            // mark elements with i'th bit set

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

                // Check whether the current 

                // bit is set or not if it's 

                // set then mark it.

                long temp = arr[j] >> i;

                if (temp % 2 != 0)

                    BitCount[j] += 1;

            }

        }

        // store cumulative sum of bits

        for (int i = 1; i < n; i++)

            BitCount[i] += BitCount[i - 1];

    }

    // Function to process queries

    static void Query(int [,]Q, int q)

    {

        for (int i = 0; i < q; i++)

            Console.WriteLine( (BitCount[Q[i,1]]

                        - BitCount[Q[i,0] - 1]));

    }

    // Driver Code

    public static void Main ()

    {

        int []Arr = { 1, 5, 6, 10, 9, 4, 67 };

        int n = Arr.Length;

        fillSetBitsMatrix(Arr, n);

        int q = 2;

        int [,]Q = { { 1, 5 }, { 2, 6 } };

        Query(Q, q);

    }
}
 
// This code is contributed by anuj_67.

Javascript

<script>
 
// Javascript program to Range query for
// Count number of set bits
 
// 2-D array that will stored the count
// of bits set in element of array

var BitCount = Array.from({length: 10000}, (_, i) => 0);
 
// Function store the set bit
// count in BitCount Array

function fillSetBitsMatrix(arr, n)
{
 
    // traverse over all bits

    for(i = 0; i < 32; i++)

    {

        // mark elements with i'th bit set

        for(j = 0; j < n; j++) 

        {

            // Check whether the current 

            // bit is set or not if it's 

            // set then mark it.

            var temp = arr[j] >> i;

            if (temp % 2 != 0)

                BitCount[j] += 1;

        }

    }
 
    // store cumulative sum of bits

    for(i = 1; i < n; i++)

        BitCount[i] += BitCount[i - 1];
}
 
// Function to process queries

function Query(Q, q)
{

    for(i = 0; i < q; i++)

        document.write((BitCount[Q[i][1]] - 

                        BitCount[Q[i][0] - 1]) + "<br>");
}
 
// Driver Code

var Arr = [ 1, 5, 6, 10, 9, 4, 67 ];

var n = Arr.length;
 
fillSetBitsMatrix(Arr, n);
 
var q = 2;

var Q = [ [ 1, 5 ], [ 2, 6 ] ];
 
Query(Q, q);
 
// This code is contributed by Rajput-Ji 
 
</script>

Output

9
10

Time Complexity : O(1) for each query.
Auxiliary Space: O(k) where k=10000

Article Tags :

Arrays

Bit Magic

DSA

array-range-queries