Queries for counts of multiples in an array
Last Updated :
24 Jun, 2024
Given an array of positive integers and many queries for divisibility. In every query, we are given an integer k ( > 0), we need to count all elements in the array which are perfectly divisible by 'k'.
Example:
Input:
2 4 9 15 21 20
k = 2
k = 3
k = 5
Output:
3
3
2
Explanation:
Multiples of '2' in array are:- {2, 4, 20}
Multiples of '3' in array are:- {9, 15, 21}
Multiples of '5' in array are:- {15, 20}
Simple Approach is to traverse over every value of 'k' in whole array and count total multiples by checking modulus of every element of array i.e., for every element of i (0 < i < n), check whether arr[i] % k == 0 or not. If it's perfectly divisible of k, then increment count. Time complexity of this approach is O(n * k) which is not efficient for large number of queries of k.
Efficient approach is to use the concept of Sieve of Eratosthenes. Let's define the maximum value in array[] is 'Max'. Since multiples of all numbers in array[] will always be less than Max, therefore we will iterate up-to 'Max' only.
Now for every value(say 'q') iterate q, 2q, 3q, ... t.k(tk <= MAX) because all these numbers are multiples of 'q' .Meanwhile store the count of all these number for every value of q(1, 2, ... MAX) in ans[] array. After that we can answer every query in O(1) time.
Implementation:
C++
// C++ program to calculate all multiples
// of integer 'k' in array[]
#include <bits/stdc++.h>
using namespace std;
// ans is global pointer so that both countSieve()
// and countMultiples() can access it.
int* ans = NULL;
// Function to pre-calculate all multiples of
// array elements
void countSieve(int arr[], int n)
{
int MAX = *max_element(arr, arr + n);
int cnt[MAX + 1];
// ans is global pointer so that query function
// can access it.
ans = new int[MAX + 1];
// Initialize both arrays as 0.
memset(cnt, 0, sizeof(cnt));
memset(ans, 0, (MAX + 1) * sizeof(int));
// Store the arr[] elements as index
// in cnt[] array
for (int i = 0; i < n; ++i)
++cnt[arr[i]];
// Iterate over all multiples as 'i'
// and keep the count of array[] ( In
// cnt[] array) elements in ans[] array
for (int i = 1; i <= MAX; ++i)
for (int j = i; j <= MAX; j += i)
ans[i] += cnt[j];
return;
}
int countMultiples(int k)
{
// return pre-calculated result
return ans[k];
}
// Driver code
int main()
{
int arr[] = { 2, 4, 9, 15, 21, 20 };
int n = sizeof(arr) / sizeof(arr[0]);
// pre-calculate all multiples
countSieve(arr, n);
int k = 2;
cout << countMultiples(k) << "\n";
k = 3;
cout << countMultiples(k) << "\n";
k = 5;
cout << countMultiples(k) << "\n";
return 0;
}
// Java program to calculate all multiples
// of integer 'k' in array[]
class CountMultiples {
// ans is global array so that both
// countSieve() and countMultiples()
// can access it.
static int ans[];
// Function to pre-calculate all
// multiples of array elements
static void countSieve(int arr[], int n)
{
int MAX = arr[0];
for (int i = 1; i < n; i++)
MAX = Math.max(arr[i], MAX);
int cnt[] = new int[MAX + 1];
// ans is global array so that
// query function can access it.
ans = new int[MAX + 1];
// Store the arr[] elements as
// index in cnt[] array
for (int i = 0; i < n; ++i)
++cnt[arr[i]];
// Iterate over all multiples as 'i'
// and keep the count of array[]
// (In cnt[] array) elements in ans[]
// array
for (int i = 1; i <= MAX; ++i)
for (int j = i; j <= MAX; j += i)
ans[i] += cnt[j];
return;
}
static int countMultiples(int k)
{
// return pre-calculated result
return ans[k];
}
// Driver code
public static void main(String args[])
{
int arr[] = { 2, 4, 9, 15, 21, 20 };
int n = 6;
// pre-calculate all multiples
countSieve(arr, n);
int k = 2;
System.out.println(countMultiples(k));
k = 3;
System.out.println(countMultiples(k));
k = 5;
System.out.println(countMultiples(k));
}
}
/*This code is contributed by Danish Kaleem */
# Python3 program to calculate all multiples
# of integer 'k' in array[]
# ans is global array so that both countSieve()
# and countMultiples() can access it.
ans = []
# Function to pre-calculate all multiples
# of array elements
# Here, the arguments are as follows
# a: given array
# n: length of given array
def countSieve(arr, n):
MAX=max(arr)
# Accessing the global array in the function
global ans
# Initializing "ans" array with zeros
ans = [0]*(MAX + 1)
# Initializing "cnt" array with zeros
cnt = [0]*(MAX + 1)
#Store the arr[] elements as index in cnt[] array
for i in range(n):
cnt[arr[i]] += 1
# Iterate over all multiples as 'i'
# and keep the count of array[] ( In
# cnt[] array) elements in ans[] array
for i in range(1, MAX+1):
for j in range(i, MAX+1, i):
ans[i] += cnt[j]
def countMultiples(k):
# Return pre-calculated result
return(ans[k])
# Driver code
if __name__ == "__main__":
arr = [2, 4, 9 ,15, 21, 20]
n=len(arr)
# Pre-calculate all multiples
countSieve(arr, n)
k=2
print(countMultiples(2))
k=3
print(countMultiples(3))
k=5
print(countMultiples(5))
# This code is contributed by Pratik Somwanshi
// C# program to calculate all multiples
// of integer 'k' in array[]
using System;
class GFG {
// ans is global array so that both
// countSieve() and countMultiples()
// can access it.
static int[] ans;
// Function to pre-calculate all
// multiples of array elements
static void countSieve(int[] arr, int n)
{
int MAX = arr[0];
for (int i = 1; i < n; i++)
MAX = Math.Max(arr[i], MAX);
int[] cnt = new int[MAX + 1];
// ans is global array so that
// query function can access it.
ans = new int[MAX + 1];
// Store the arr[] elements as
// index in cnt[] array
for (int i = 0; i < n; ++i)
++cnt[arr[i]];
// Iterate over all multiples as
// 'i' and keep the count of
// array[] (In cnt[] array)
// elements in ans[] array
for (int i = 1; i <= MAX; ++i)
for (int j = i; j <= MAX; j += i)
ans[i] += cnt[j];
return;
}
static int countMultiples(int k)
{
// return pre-calculated result
return ans[k];
}
// Driver code
public static void Main()
{
int[] arr = { 2, 4, 9, 15, 21, 20 };
int n = 6;
// pre-calculate all multiples
countSieve(arr, n);
int k = 2;
Console.WriteLine(countMultiples(k));
k = 3;
Console.WriteLine(countMultiples(k));
k = 5;
Console.WriteLine(countMultiples(k));
}
}
// This code is contributed by nitin mittal
<script>
// Javascript program to calculate all multiples
// of integer 'k' in array[]
// ans is global array so that both
// countSieve() and countMultiples()
// can access it.
let ans = [];
// Function to pre-calculate all
// multiples of array elements
function countSieve(arr, n)
{
let MAX = arr[0];
for (let i = 1; i < n; i++)
MAX = Math.max(arr[i], MAX);
let cnt = Array.from({length: MAX + 1}, (_, i) => 0);
// ans is global array so that
// query function can access it.
ans = Array.from({length: MAX + 1}, (_, i) => 0);
// Store the arr[] elements as
// index in cnt[] array
for (let i = 0; i < n; ++i)
++cnt[arr[i]];
// Iterate over all multiples as 'i'
// and keep the count of array[]
// (In cnt[] array) elements in ans[]
// array
for (let i = 1; i <= MAX; ++i)
for (let j = i; j <= MAX; j += i)
ans[i] += cnt[j];
return;
}
function countMultiples(k)
{
// return pre-calculated result
return ans[k];
}
// driver function
let arr = [ 2, 4, 9, 15, 21, 20 ];
let n = 6;
// pre-calculate all multiples
countSieve(arr, n);
let k = 2;
document.write(countMultiples(k) + "<br/>");
k = 3;
document.write(countMultiples(k) + "<br/>");
k = 5;
document.write(countMultiples(k) + "<br/>");
</script>
<?php
// PHP program to calculate
// all multiples of integer
// 'k' in array[]
// ans is global array so
// that both countSieve()
// and countMultiples()
// can access it.
$ans;
// Function to pre-calculate all
// multiples of array elements
function countSieve($arr, $n)
{
global $ans;
$MAX = $arr[0];
for ($i = 1; $i < $n; $i++)
$MAX = max($arr[$i], $MAX);
$cnt = array_fill(0, $MAX + 1, 0);
// ans is global array so that
// query function can access it.
$ans = array_fill(0, $MAX + 1, 0);
// Store the arr[] elements
// as index in cnt[] array
for ($i = 0; $i < $n; ++$i)
++$cnt[$arr[$i]];
// Iterate over all multiples as 'i'
// and keep the count of array[]
// (In cnt[] array) elements in ans[]
// array
for ($i = 1; $i <= $MAX; ++$i)
for ($j = $i; $j <= $MAX; $j += $i)
$ans[$i] += $cnt[$j];
return;
}
function countMultiples($k)
{
global $ans;
// return pre-calculated result
return $ans[$k];
}
// Driver code
$arr = array( 2, 4, 9, 15, 21, 20);
$n = 6;
// pre-calculate
// all multiples
countSieve($arr, $n);
$k = 2;
echo countMultiples($k) . "\n";
$k = 3;
echo countMultiples($k) . "\n";
$k = 5;
echo countMultiples($k) . "\n";
// This code is contributed by mits
?>
Time complexity: O(M*log(M)) where M is the maximum value in array elements.
Auxiliary space: O(MAX)
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