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.

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// c++ program to Range query for
// Count number of set bits
#include <iostream>
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;
}
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// 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.
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// 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.
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Output:
9
10

Time Complexity : O(1) for each query.




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