Given an array arr[] of N non-zero positive integers and an integer K, the task is to find the XOR of the K largest prime and composite numbers.
Examples:
Input: arr[] = {4, 2, 12, 13, 5, 19}, K = 3
Output:
Prime XOR = 27
Composite XOR = 8
5, 13 and 19 are the three maximum primes
from the given array and 5 ^ 13 ^ 19 = 27.
There are only 2 composites in the array i.e. 4 and 12.
And 4 ^ 12 = 8
Input: arr[] = {1, 2, 3, 4, 5, 6, 7}, K = 1
Output:
Prime XOR = 7
Composite XOR = 6
Approach: Using Sieve of Eratosthenes generate a boolean vector upto the size of the maximum element from the array which can be used to check whether a number is prime or not.
Now traverse the array and insert all the numbers which are prime in a max heap maxHeapPrime and all the composite numbers in max heap maxHeapNonPrime.
Now, pop out the top K elements from both the max heaps and take the xor of these elements.
Below is the implementation of the above approach:
// C++ implementation of the approach #include <bits/stdc++.h> using namespace std;
// Function for Sieve of Eratosthenes vector< bool > SieveOfEratosthenes( int max_val)
{ // Create a boolean vector "prime[0..n]". A
// value in prime[i] will finally be false
// if i is Not a prime, else true.
vector< bool > prime(max_val + 1, true );
// Set 0 and 1 as non-primes as
// they don't need to be
// counted as prime numbers
prime[0] = false ;
prime[1] = false ;
for ( int p = 2; p * p <= max_val; p++) {
// If prime[p] is not changed, then
// it is a prime
if (prime[p] == true ) {
// Update all multiples of p
for ( int i = p * 2; i <= max_val; i += p)
prime[i] = false ;
}
}
return prime;
} // Function that calculates the xor // of k smallest and k // largest prime numbers in an array void kMaxXOR( int arr[], int n, int k)
{ // Find maximum value in the array
int max_val = *max_element(arr, arr + n);
// Use sieve to find all prime numbers
// less than or equal to max_val
vector< bool > prime = SieveOfEratosthenes(max_val);
// Min Heaps to store the max K prime
// and composite numbers
priority_queue< int , vector< int >, greater< int > >
minHeapPrime, minHeapNonPrime;
for ( int i = 0; i < n; i++) {
// If current element is prime
if (prime[arr[i]]) {
// Min heap will only store k elements
if (minHeapPrime.size() < k)
minHeapPrime.push(arr[i]);
// If the size of min heap is K and the
// top element is smaller than the current
// element than it needs to be replaced
// by the current element as only
// max k elements are required
else if (minHeapPrime.top() < arr[i]) {
minHeapPrime.pop();
minHeapPrime.push(arr[i]);
}
}
// If current element is composite
else if (arr[i] != 1) {
// Heap will only store k elements
if (minHeapNonPrime.size() < k)
minHeapNonPrime.push(arr[i]);
// If the size of min heap is K and the
// top element is smaller than the current
// element than it needs to be replaced
// by the current element as only
// max k elements are required
else if (minHeapNonPrime.top() < arr[i]) {
minHeapNonPrime.pop();
minHeapNonPrime.push(arr[i]);
}
}
}
long long int primeXOR = 0, nonPrimeXor = 0;
while (k--) {
// Calculate the xor
if (minHeapPrime.size() > 0) {
primeXOR ^= minHeapPrime.top();
minHeapPrime.pop();
}
if (minHeapNonPrime.size() > 0) {
nonPrimeXor ^= minHeapNonPrime.top();
minHeapNonPrime.pop();
}
}
cout << "Prime XOR = " << primeXOR << "\n" ;
cout << "Composite XOR = " << nonPrimeXor << "\n" ;
} // Driver code int main()
{ int arr[] = { 4, 2, 12, 13, 5, 19 };
int n = sizeof (arr) / sizeof (arr[0]);
int k = 3;
kMaxXOR(arr, n, k);
return 0;
} |
// Java implementation of the approach import java.util.*;
class GFG
{ // Function for Sieve of Eratosthenes
static boolean [] SieveOfEratosThenes( int max_val)
{
// Create a boolean vector "prime[0..n]". A
// value in prime[i] will finally be false
// if i is Not a prime, else true.
boolean [] prime = new boolean [max_val + 1 ];
Arrays.fill(prime, true );
// Set 0 and 1 as non-primes as
// they don't need to be
// counted as prime numbers
prime[ 0 ] = false ;
prime[ 1 ] = false ;
for ( int p = 2 ; p * p <= max_val; p++)
{
// If prime[p] is not changed, then
// it is a prime
if (prime[p])
{
// Update all multiples of p
for ( int i = p * 2 ; i <= max_val; i += p)
prime[i] = false ;
}
}
return prime;
}
// Function that calculates the xor
// of k smallest and k
// largest prime numbers in an array
static void kMinXOR(Integer[] arr, int n, int k)
{
// Find maximum value in the array
int max_val = Collections.max(Arrays.asList(arr));
// Use sieve to find all prime numbers
// less than or equal to max_val
boolean [] prime = SieveOfEratosThenes(max_val);
// Min Heaps to store the max K prime
// and composite numbers
PriorityQueue<Integer> minHeapPrime = new PriorityQueue<>();
PriorityQueue<Integer> minHeapNonPrime = new PriorityQueue<>();
for ( int i = 0 ; i < n; i++)
{
// If current element is prime
if (prime[arr[i]])
{
// Min heap will only store k elements
if (minHeapPrime.size() < k)
minHeapPrime.add(arr[i]);
// If the size of min heap is K and the
// top element is smaller than the current
// element than it needs to be replaced
// by the current element as only
// max k elements are required
else if (minHeapPrime.peek() < arr[i])
{
minHeapPrime.poll();
minHeapPrime.add(arr[i]);
}
}
// If current element is composite
else if (arr[i] != - 1 )
{
// Heap will only store k elements
if (minHeapNonPrime.size() < k)
minHeapNonPrime.add(arr[i]);
// If the size of min heap is K and the
// top element is smaller than the current
// element than it needs to be replaced
// by the current element as only
// max k elements are required
else if (minHeapNonPrime.peek() < arr[i])
{
minHeapNonPrime.poll();
minHeapNonPrime.add(arr[i]);
}
}
}
long primeXOR = 0 , nonPrimeXor = 0 ;
while (k-- > 0 )
{
// Calculate the xor
if (minHeapPrime.size() > 0 )
{
primeXOR ^= minHeapPrime.peek();
minHeapPrime.poll();
}
if (minHeapNonPrime.size() > 0 )
{
nonPrimeXor ^= minHeapNonPrime.peek();
minHeapNonPrime.poll();
}
}
System.out.println( "Prime XOR = " + primeXOR);
System.out.println( "Composite XOR = " + nonPrimeXor);
}
// Driver Code
public static void main(String[] args)
{
Integer[] arr = { 4 , 2 , 12 , 13 , 5 , 19 };
int n = arr.length;
int k = 3 ;
kMinXOR(arr, n, k);
}
} // This code is contributed by // sanjeev2552 |
# Python implementation of above approach import heapq
# Function for Sieve of Eratosthenes def SieveOfEratosthenes(max_val: int ) - > list :
# Create a boolean vector "prime[0..n]". A
# value in prime[i] will finally be false
# if i is Not a prime, else true.
prime = [ True ] * (max_val + 1 )
# Set 0 and 1 as non-primes as
# they don't need to be
# counted as prime numbers
prime[ 0 ] = False
prime[ 1 ] = False
p = 2
while p * p < = max_val:
# If prime[p] is not changed, then
# it is a prime
if prime[p]:
# Update all multiples of p
for i in range (p * 2 , max_val + 1 , p):
prime[i] = False
p + = 1
return prime
# Function that calculates the xor # of k smallest and k # largest prime numbers in an array def kMaxXOR(arr: list , n: int , k: int ):
# Find maximum value in the array
max_val = max (arr)
# Use sieve to find all prime numbers
# less than or equal to max_val
prime = SieveOfEratosthenes(max_val)
# Min Heaps to store the max K prime
# and composite numbers
minHeapPrime, minHeapNonPrime = [], []
heapq.heapify(minHeapPrime)
heapq.heapify(minHeapNonPrime)
for i in range (n):
# If current element is prime
if prime[arr[i]]:
# Min heap will only store k elements
if len (minHeapPrime) < k:
heapq.heappush(minHeapPrime, arr[i])
# If the size of min heap is K and the
# top element is smaller than the current
# element than it needs to be replaced
# by the current element as only
# max k elements are required
elif heapq.nsmallest( 1 , minHeapPrime)[ 0 ] < arr[i]:
heapq.heappop(minHeapPrime)
heapq.heappush(minHeapPrime, arr[i])
# If current element is composite
elif arr[i] ! = 1 :
# Heap will only store k elements
if len (minHeapNonPrime) < k:
heapq.heappush(minHeapNonPrime, arr[i])
# If the size of min heap is K and the
# top element is smaller than the current
# element than it needs to be replaced
# by the current element as only
# max k elements are required
elif heapq.nsmallest( 1 , minHeapNonPrime)[ 0 ] < arr[i]:
heapq.heappop(minHeapNonPrime)
heapq.heappush(minHeapNonPrime, arr[i])
primeXOR = 0
nonPrimeXor = 0
while k > 0 :
# Calculate the xor
if len (minHeapPrime) > 0 :
primeXOR ^ = heapq.nsmallest( 1 , minHeapPrime)[ 0 ]
heapq.heappop(minHeapPrime)
if len (minHeapNonPrime) > 0 :
nonPrimeXor ^ = heapq.nsmallest( 1 , minHeapNonPrime)[ 0 ]
heapq.heappop(minHeapNonPrime)
k - = 1
print ( "Prime XOR =" , primeXOR)
print ( "Composite XOR =" , nonPrimeXor)
# Driver Code if __name__ = = "__main__" :
arr = [ 4 , 2 , 12 , 13 , 5 , 19 ]
n = len (arr)
k = 3
kMaxXOR(arr, n, k)
# This code is contributed by # sanjeev2552 |
// C# implementation of the approach using System;
using System.Collections.Generic;
class GFG {
// Function for Sieve of Eratosthenes
static bool [] SieveOfEratosThenes( int max_val)
{
// Create a boolean vector "prime[0..n]". A
// value in prime[i] will finally be false
// if i is Not a prime, else true.
bool [] prime = new bool [max_val + 1];
for ( int i = 0; i < max_val + 1; i++)
{
prime[i] = true ;
}
// Set 0 and 1 as non-primes as
// they don't need to be
// counted as prime numbers
prime[0] = false ;
prime[1] = false ;
for ( int p = 2; p * p <= max_val; p++)
{
// If prime[p] is not changed, then
// it is a prime
if (prime[p])
{
// Update all multiples of p
for ( int i = p * 2; i <= max_val; i += p)
prime[i] = false ;
}
}
return prime;
}
// Function that calculates the xor
// of k smallest and k
// largest prime numbers in an array
static void kMinXOR( int [] arr, int n, int k)
{
// Find maximum value in the array
int max_val = Int32.MinValue;
for ( int i = 0; i < arr.Length; i++)
{
max_val = Math.Max(max_val,arr[i]);
}
// Use sieve to find all prime numbers
// less than or equal to max_val
bool [] prime = SieveOfEratosThenes(max_val);
// Min Heaps to store the max K prime
// and composite numbers
List< int > minHeapPrime = new List< int >();
List< int > minHeapNonPrime = new List< int >();
for ( int i = 0; i < n; i++)
{
// If current element is prime
if (prime[arr[i]])
{
// Min heap will only store k elements
if (minHeapPrime.Count < k)
{
minHeapPrime.Add(arr[i]);
}
// If the size of min heap is K and the
// top element is smaller than the current
// element than it needs to be replaced
// by the current element as only
// max k elements are required
else if (minHeapPrime[0] < arr[i])
{
minHeapPrime.RemoveAt(0);
minHeapPrime.Add(arr[i]);
}
minHeapPrime.Sort();
}
// If current element is composite
else if (arr[i] != -1)
{
// Heap will only store k elements
if (minHeapNonPrime.Count < k)
minHeapNonPrime.Add(arr[i]);
// If the size of min heap is K and the
// top element is smaller than the current
// element than it needs to be replaced
// by the current element as only
// max k elements are required
else if (minHeapNonPrime[0] < arr[i])
{
minHeapNonPrime.RemoveAt(0);
minHeapNonPrime.Add(arr[i]);
}
minHeapNonPrime.Sort();
}
}
long primeXOR = 0, nonPrimeXor = 0;
while (k-- > 0)
{
// Calculate the xor
if (minHeapPrime.Count > 0)
{
primeXOR ^= minHeapPrime[0];
minHeapPrime.RemoveAt(0);
}
if (minHeapNonPrime.Count > 0)
{
nonPrimeXor ^= minHeapNonPrime[0];
minHeapNonPrime.RemoveAt(0);
}
}
Console.WriteLine( "Prime XOR = " + primeXOR);
Console.WriteLine( "Composite XOR = " + nonPrimeXor);
}
// Driver code
static void Main()
{
int [] arr = { 4, 2, 12, 13, 5, 19 };
int n = arr.Length;
int k = 3;
kMinXOR(arr, n, k);
}
} // This code is contributed by divyesh072019. |
<script> // Javascript implementation of the approach
// Function for Sieve of Eratosthenes
function SieveOfEratosThenes(max_val)
{
// Create a boolean vector "prime[0..n]". A
// value in prime[i] will finally be false
// if i is Not a prime, else true.
let prime = new Array(max_val + 1);
for (let i = 0; i < max_val + 1; i++)
{
prime[i] = true ;
}
// Set 0 and 1 as non-primes as
// they don't need to be
// counted as prime numbers
prime[0] = false ;
prime[1] = false ;
for (let p = 2; p * p <= max_val; p++)
{
// If prime[p] is not changed, then
// it is a prime
if (prime[p])
{
// Update all multiples of p
for (let i = p * 2; i <= max_val; i += p)
prime[i] = false ;
}
}
return prime;
}
// Function that calculates the xor
// of k smallest and k
// largest prime numbers in an array
function kMinXOR(arr, n, k)
{
// Find maximum value in the array
let max_val = Number.MIN_VALUE;
for (let i = 0; i < arr.length; i++)
{
max_val = Math.max(max_val,arr[i]);
}
// Use sieve to find all prime numbers
// less than or equal to max_val
let prime = SieveOfEratosThenes(max_val);
// Min Heaps to store the max K prime
// and composite numbers
let minHeapPrime = [];
let minHeapNonPrime = [];
for (let i = 0; i < n; i++)
{
// If current element is prime
if (prime[arr[i]])
{
// Min heap will only store k elements
if (minHeapPrime.length < k)
{
minHeapPrime.push(arr[i]);
}
// If the size of min heap is K and the
// top element is smaller than the current
// element than it needs to be replaced
// by the current element as only
// max k elements are required
else if (minHeapPrime[0] < arr[i])
{
minHeapPrime.shift();
minHeapPrime.push(arr[i]);
}
minHeapPrime.sort( function (a, b){ return a - b});
}
// If current element is composite
else if (arr[i] != -1)
{
// Heap will only store k elements
if (minHeapNonPrime.length < k)
minHeapNonPrime.push(arr[i]);
// If the size of min heap is K and the
// top element is smaller than the current
// element than it needs to be replaced
// by the current element as only
// max k elements are required
else if (minHeapNonPrime[0] < arr[i])
{
minHeapNonPrime.shift();
minHeapNonPrime.push(arr[i]);
}
minHeapNonPrime.sort( function (a, b){ return a - b});
}
}
let primeXOR = 0, nonPrimeXor = 0;
while (k-- > 0)
{
// Calculate the xor
if (minHeapPrime.length > 0)
{
primeXOR ^= minHeapPrime[0];
minHeapPrime.shift();
}
if (minHeapNonPrime.length > 0)
{
nonPrimeXor ^= minHeapNonPrime[0];
minHeapNonPrime.shift();
}
}
document.write( "Prime XOR = " + primeXOR + "</br>" );
document.write( "Composite XOR = " + nonPrimeXor);
}
let arr = [ 4, 2, 12, 13, 5, 19 ];
let n = arr.length;
let k = 3;
kMinXOR(arr, n, k);
</script> |
Prime XOR = 27 Composite XOR = 8