Queries for counts of multiples in an array

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

// 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

# 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#

// 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

PHP

<?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
?>

Javascript

<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>

Output

3
3
2

Time complexity: O(M*log(M)) where M is the maximum value in array elements. 
Auxiliary space: O(MAX)