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 arrayint BitCount[10000] = { 0 };// Function store the set bit// count in BitCount Arrayvoid 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 queriesvoid 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 Codeint 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 bitsimport 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 arrayBitCount = [0] * 10000# Function store the set bit# count in BitCount Arraydef 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 queriesdef Query(Q: list, q: int): for i in range(q): print(BitCount[Q[i][1]] - BitCount[Q[i][0] - 1])# Driver Codeif __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#
// C# program to Range query for// Count number of set bitsusing 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 arrayvar BitCount = Array.from({length: 10000}, (_, i) => 0);// Function store the set bit// count in BitCount Arrayfunction 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 queriesfunction Query(Q, q){ for(i = 0; i < q; i++) document.write((BitCount[Q[i][1]] - BitCount[Q[i][0] - 1]) + "<br>");}// Driver Codevar 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
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