Maximum Subarray sum of Prime length

Given an array arr[] of size N, the task is to find the maximum subarray sum that can be obtained such that the length of the subarray should be prime.

Examples :

Input: arr[] = {2, -1, 3, -2, 1, -1} 
Output: 4 
The subarray {2, -1, 3} of size = 3  (prime number)

input: arr[] = {-2, -3, 4, -1, -2, 1, 5, -3}
Output: 7
The subarray {4, -1, -2, 1, 5} of size = 5 (prime number)

Naive Approach:  The idea is as follows:

Generate all possible subarrays and from them find the ones with prime length. Find the maximum sum among them.

Follow the given steps to solve the problem:

• Generate all possible subarrays of all lengths using nested for-loops.
• Find the sum of each prime length subarray.
• The numbers which are primes can be precomputed by Sieve algorithm 
• Now for each prime length, calculate the sum and take the maximum of it    

Below is the implementation of the above approach:

C++14

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to implement sieve of eratosthenes
void sieve(int N, vector<bool>& prime)
{
    prime[1] = false;
    prime[0] = false;
    for (int i = 2; i * i <= N; i++) {
        if (prime[i]) {
            for (int p = i * i; p <= N; p += i) {
                prime[p] = false;
            }
        }
    }
}

// Function to find the sum
// of prime length subarrays
int primeLenSum(int a[], int N)
{
    vector<bool> prime(N + 1, true);
    sieve(N, prime);
    int ans = INT_MIN;

    // Traversing to find max sum
    // of prime subarray
    for (int i = 0; i < N; i++) {
        for (int j = i + 1; j < N; j++) {
            if (prime[j - i]) {
                int sum = 0;
                for (int k = i; k <= j; k++)
                    sum += a[k];
                ans = max(ans, sum);
            }
        }
    }

    return ans;
}

// Driver code
int main()
{
    int arr[] = { 2, -1, 3, -2, 1, -1 };
    int N = sizeof(arr) / sizeof(arr[0]);

    // Function call
    cout << primeLenSum(arr, N) << "\n";
    return 0;
}

Java

// Java program for the above approach

import java.io.*;

class GFG {

    // Function to implement sieve of eratosthenes
    static boolean[] sieve(int N, boolean[] prime)
    {
        prime[1] = false;
        prime[0] = false;
        for (int i = 2; i * i <= N; i++) {
            if (prime[i]) {
                for (int p = i * i; p <= N; p += i) {
                    prime[p] = false;
                }
            }
        }
        return prime;
    }

    // Function to find the sum of prime length subarrays
    static int primeLenSum(int[] a, int N)
    {
        boolean[] prime = new boolean[N + 1];
        for (int i = 0; i < N; i++) {
            prime[i] = true;
        }
        prime = sieve(N, prime);
        int ans = Integer.MIN_VALUE;

        // Traversing to find max sum of prime subarray
        for (int i = 0; i < N; i++) {
            for (int j = i + 1; j < N; j++) {
                if (prime[j - i]) {
                    int sum = 0;
                    for (int k = i; k <= j; k++) {
                        sum += a[k];
                    }
                    ans = Math.max(ans, sum);
                }
            }
        }
        return ans;
    }

    public static void main(String[] args)
    {
        int[] arr = { 2, -1, 3, -2, 1, -1 };
        int N = arr.length;

        // Function call
        System.out.print(primeLenSum(arr, N));
    }
}

// This code is contributed by lokeshmvs21.

Python3

import sys

def sieve(N, prime):
    prime[1] = False
    prime[0] = False
    for i in range(2,N+1):
        if i*i<=N:
            if (prime[i]):
                for p in range(i*i,N+1):
                    prime[p] = False
                    p+=i

def primeLenSum(a,N):
    prime=[True for i in range(N+1)]
    sieve(N, prime)
    ans = -sys.maxsize - 1

    for i in range(N):
        for j in range(i+1,N):
            if (prime[j - i]):
                sum = 0
                for k in range(i,j+1):
                    sum += a[k]
                ans = max(ans, sum);

    return ans

if __name__ == '__main__':
    arr = [ 2, -1, 3, -2, 1, -1 ]
    N=len(arr)
    # Function call
    print( primeLenSum(arr, N))

    # This code is contributed by vikkycirus.

C#

// C# program for the above approach
using System;
public class GFG {

  // Function to implement sieve of eratosthenes
  static bool[] sieve(int N, bool[] prime)
  {
    prime[1] = false;
    prime[0] = false;
    for (int i = 2; i * i <= N; i++) {
      if (prime[i]) {
        for (int p = i * i; p <= N; p += i) {
          prime[p] = false;
        }
      }
    }
    return prime;
  }

  // Function to find the sum of prime length subarrays
  static int primeLenSum(int[] a, int N)
  {
    bool[] prime = new bool[N + 1];
    for (int i = 0; i < N; i++) {
      prime[i] = true;
    }
    prime = sieve(N, prime);
    int ans = Int32.MinValue;

    // Traversing to find max sum of prime subarray
    for (int i = 0; i < N; i++) {
      for (int j = i + 1; j < N; j++) {
        if (prime[j - i]) {
          int sum = 0;
          for (int k = i; k <= j; k++) {
            sum += a[k];
          }
          ans = Math.Max(ans, sum);
        }
      }
    }
    return ans;
  }

  static public void Main()
  {

    // Code
    int[] arr = { 2, -1, 3, -2, 1, -1 };
    int N = arr.Length;

    // Function call
    Console.Write(primeLenSum(arr, N));
  }
}

// This code is contributed by lokeshmvs21.

Javascript

// Javascript program for the above approach

// Function to implement sieve of eratosthenes
function sieve(N, prime) {
  prime[1] = false;
  prime[0] = false;
  for (let i = 2; i * i <= N; i++) {
    if (prime[i]) {
      for (let p = i * i; p <= N; p += i) {
        prime[p] = false;
      }
    }
  }
}

// Function to find the sum
// of prime length subarrays
function primeLenSum(a, N) {
  let prime = new Array(N + 1).fill(true);
  sieve(N, prime);
  let ans = Number.MIN_SAFE_INTEGER;

  // Traversing to find max sum
  // of prime subarray
  for (let i = 0; i < N; i++) {
    for (let j = i + 1; j < N; j++) {
      if (prime[j - i]) {
        let sum = 0;
        for (let k = i; k <= j; k++) sum += a[k];
        ans = Math.max(ans, sum);
      }
    }
  }

  return ans;
}

// Driver code
let arr = [2, -1, 3, -2, 1, -1];
let N = arr.length;

// Function call
console.log(primeLenSum(arr, N));

// This code is contributed by akashish__

Output

4

Time complexity: O(N³)
Auxiliary Space: O(N) 

Efficient Approach: To solve the problem follow the below idea:

Use Kadane’s algorithm, and update the answer only if the length of the subarray is prime.

 Follow the given steps to solve the problem:

• Initialize max_so_far = INT_MIN  (since sum can be negative), and max_ending_here = 0 (to keep track of the current sum )
• Loop to iterate each element of the array:
  • max_ending_here is equal to max_ending_here + arr[i]
  • If max_so_far is less than max_ending_here then update max_so_far
  • If max_ending_here is less than 0 then set max_ending_here = 0
  • Return max_so_far
• Now for calculating the subarray sum of prime length we have to keep track of the subarray size and have to check whether the size is prime or not 

Below is the implementation of the above idea :

C++14

// C++ code for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to implement sieve of eratosthenes
void sieve(int n, vector<bool>& prime)
{
    prime[1] = false;
    prime[0] = false;
    for (int i = 2; i * i <= n; i++) {
        if (prime[i]) {
            for (int p = i * i; p <= n; p += i) {
                prime[p] = false;
            }
        }
    }
}

// Function to find the maximum subarray sum
// having prime length
int primeLenSum(int a[], int N)
{
    vector<bool> prime(N + 1, true);
    sieve(N, prime);
    int max_ending_here = 0;
    int max_for_primes = 0, sz = 0;

    // Finding prime length subarray
    // with maximum sum
    for (int i = 0; i < N; i++) {
        max_ending_here += a[i];
        sz = sz + 1;
        if (max_ending_here < 0) {
            max_ending_here = 0;
            sz = 0;
        }

        if (max_ending_here > max_for_primes && prime[sz])
            max_for_primes = max_ending_here;
    }
    return max_for_primes;
}

// Driver code
int main()
{
    int arr[] = { 2, -1, 3, -2, 1, -1 };
    int N = sizeof(arr) / sizeof(arr[0]);

    // Function call
    cout << primeLenSum(arr, N);
    return 0;
}

Java

// Java code for the above approach

import java.io.*;
import java.util.*;

class GFG {

    // Function to implement sieve of eratosthenes
    static boolean[] sieve(int n, boolean[] prime)
    {
        prime[1] = false;
        prime[0] = false;
        for (int i = 2; i * i <= n; i++) {
            if (prime[i]) {
                for (int p = i * i; p <= n; p += i) {
                    prime[p] = false;
                }
            }
        }
        return prime;
    }

    // Function to find the maximum subarray sum having
    // prime length
    static int primeLenSum(int[] a, int N)
    {
        boolean[] prime = new boolean[N + 1];
        Arrays.fill(prime, true);
        prime = sieve(N, prime);
        int max_ending_here = 0;
        int max_for_primes = 0, sz = 0;

        // Finding prime length subarray with maximum sum
        for (int i = 0; i < N; i++) {
            max_ending_here += a[i];
            sz = sz + 1;
            if (max_ending_here < 0) {
                max_ending_here = 0;
                sz = 0;
            }
            if (max_ending_here > max_for_primes
                && prime[sz]) {
                max_for_primes = max_ending_here;
            }
        }
        return max_for_primes;
    }

    public static void main(String[] args)
    {
        int[] arr = { 2, -1, 3, -2, 1, -1 };
        int N = arr.length;

        // Function call
        System.out.print(primeLenSum(arr, N));
    }
}

// This code is contributed by lokeshmvs21.

Python3

# Function to implement sieve of eratosthenes
def sieve(n, prime):
    prime[1] = False
    prime[0] = False
    for i in range(2, int(n**(1/2))+1):
        if prime[i]:
            for p in range(i*i, n+1, i):
                prime[p] = False

# Function to find the maximum subarray sum
# having prime length
def primeLenSum(a, N):
    prime = [True] * (N + 1)
    sieve(N, prime)
    max_ending_here = 0
    max_for_primes = 0
    sz = 0

    # Finding prime length subarray
    # with maximum sum
    for i in range(N):
        max_ending_here += a[i]
        sz = sz + 1
        if max_ending_here < 0:
            max_ending_here = 0
            sz = 0
        if max_ending_here > max_for_primes and prime[sz]:
            max_for_primes = max_ending_here
    return max_for_primes

# Driver code
arr = [2, -1, 3, -2, 1, -1]
N = len(arr)

# Function call
print(primeLenSum(arr, N))

# This code is contributed by akashish__

C#

// C# code to implement the approach

using System;
using System.Collections.Generic;

public class Gfg {

    // Function to implement sieve of eratosthenes
    static void sieve(int n, int[] prime)
    {
        prime[1] = 0;
        prime[0] = 0;
        for (int i = 2; i * i <= n; i++) {
            if (prime[i]==1) {
                for (int p = i * i; p <= n; p += i) {
                    prime[p] = 0;
                }
            }
        }
    }

    // Function to find the maximum subarray sum
    // having prime length
    static int primeLenSum(int []a, int N)
    {
        int[] prime = new int[N+1];
        for(int i=0; i<N+1; i++)
            prime[i]=1;

        sieve(N, prime);
        int max_ending_here = 0;
        int max_for_primes = 0, sz = 0;

        // Finding prime length subarray
        // with maximum sum
        for (int i = 0; i < N; i++) {
            max_ending_here += a[i];
            sz = sz + 1;
            if (max_ending_here < 0) {
                max_ending_here = 0;
                sz = 0;
            }

            if (max_ending_here > max_for_primes && prime[sz]==1)
                max_for_primes = max_ending_here;
        }
        return max_for_primes;
    }

    // Driver code
    public static void Main(string[] args)
    {
        int[] arr = { 2, -1, 3, -2, 1, -1 };
        int N = arr.Length;

        // Function call
        Console.WriteLine(primeLenSum(arr, N));
    }
}

Javascript

// Function to implement sieve of eratosthenes
function sieve(n, prime) {
  prime[1] = false;
  prime[0] = false;
  for (let i = 2; i * i <= n; i++) {
    if (prime[i]) {
      for (let p = i * i; p <= n; p += i) {
        prime[p] = false;
      }
    }
  }
}

// Function to find the maximum subarray sum
// having prime length
function primeLenSum(a, N) {
  const prime = Array(N + 1).fill(true);
  sieve(N, prime);
  let max_ending_here = 0;
  let max_for_primes = 0,
    sz = 0;

  // Finding prime length subarray
  // with maximum sum
  for (let i = 0; i < N; i++) {
    max_ending_here += a[i];
    sz = sz + 1;
    if (max_ending_here < 0) {
      max_ending_here = 0;
      sz = 0;
    }

    if (max_ending_here > max_for_primes && prime[sz])
      max_for_primes = max_ending_here;
  }
  return max_for_primes;
}

// Driver code

const arr = [2, -1, 3, -2, 1, -1];
const N = arr.length;

// Function call
console.log(primeLenSum(arr, N));

// This code is contributed by akashish__

Output

4

Time Complexity: O(N * log(logN))
Auxiliary Space: O(N)