# XOR of K largest prime and composite numbers from the given array

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

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 ` `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 minHeapPrime = ``new` `PriorityQueue<>(); ` `        ``PriorityQueue 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 implimentation 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 `

Output:
```Prime XOR = 27
Composite XOR = 8
```

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