Split array to three subarrays such that sum of first and third subarray is equal and maximum

Given an array of N integers, the task is to print the sum of the first subarray by splitting the array into exactly three subarrays such that the sum of the first and third subarray elements are equal and the maximum. 
Note: All the elements must belong to a subarray and the subarrays can also be empty. 

Examples: 

Input: a[] = {1, 3, 1, 1, 4} 
Output:
Split the N numbers to [1, 3, 1], [] and [1, 4] 

Input: a[] = {1, 3, 2, 1, 4} 
Output:
Split the N numbers to [1, 3], [2, 1] and [4] 

METHOD 1
A naive approach is to check for all possible partitions and use the prefix-sum concept to find out the partitions. The partition which gives the maximum sum of the first subarray will be the answer. 



An efficient approach is as follows:  

  • Store the prefix sum and suffix sum of the N numbers.
  • Hash the suffix sum’s index using a unordered_map in C++ or Hash-map in Java.
  • Iterate from the beginning of the array, and check if the prefix sum exists in the suffix array beyond the current index i.
  • If it does, then check for the previous maximum value and update accordingly.

Below is the implementation of the above approach:  

C++

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// C++ program for Split the array into three
// subarrays such that summation of first
// and third subarray is equal and maximum
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the sum of
// the first subarray
int sumFirst(int a[], int n)
{
    unordered_map<int, int> mp;
    int suf = 0;
 
    // calculate the suffix sum
    for (int i = n - 1; i >= 0; i--)
    {
        suf += a[i];
        mp[suf] = i;
    }
 
    int pre = 0;
    int maxi = -1;
 
    // iterate from beginning
    for (int i = 0; i < n; i++)
    {
        // prefix sum
        pre += a[i];
 
        // check if it exists beyond i
        if (mp[pre] > i)
        {
            // if greater then previous
            // then update maximum
            if (pre > maxi)
            {
                maxi = pre;
            }
        }
    }
 
    // First and second subarray empty
    if (maxi == -1)
        return 0;
 
    // partition done
    else
        return maxi;
}
 
// Driver Code
int main()
{
    int a[] = { 1, 3, 2, 1, 4 };
    int n = sizeof(a) / sizeof(a[0]);
   
    // Function call
    cout << sumFirst(a, n);
    return 0;
}

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Java

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// Java program for Split the array into three
// subarrays such that summation of first
// and third subarray is equal and maximum
import java.util.HashMap;
import java.util.Map;
 
class GfG {
 
    // Function to return the sum
    // of the first subarray
    static int sumFirst(int a[], int n)
    {
        HashMap<Integer, Integer> mp = new HashMap<>();
        int suf = 0;
 
        // calculate the suffix sum
        for (int i = n - 1; i >= 0; i--)
        {
            suf += a[i];
            mp.put(suf, i);
        }
 
        int pre = 0, maxi = -1;
 
        // iterate from beginning
        for (int i = 0; i < n; i++)
        {
            // prefix sum
            pre += a[i];
 
            // check if it exists beyond i
            if (mp.containsKey(pre) && mp.get(pre) > i)
            {
                // if greater then previous
                // then update maximum
                if (pre > maxi)
                {
                    maxi = pre;
                }
            }
        }
 
        // First and second subarray empty
        if (maxi == -1)
            return 0;
 
        // partition done
        else
            return maxi;
    }
 
    // Driver code
    public static void main(String[] args)
    {
 
        int a[] = { 1, 3, 2, 1, 4 };
        int n = a.length;
       
        // Function call
        System.out.println(sumFirst(a, n));
    }
}
 
// This code is contributed by Rituraj Jain

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Python3

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# Python 3 program for Split the array into three
# subarrays such that summation of first
# and third subarray is equal and maximum
 
# Function to return the sum of
# the first subarray
 
 
def sumFirst(a, n):
    mp = {i: 0 for i in range(7)}
    suf = 0
    i = n - 1
 
    # calculate the suffix sum
    while(i >= 0):
        suf += a[i]
        mp[suf] = i
        i -= 1
 
    pre = 0
    maxi = -1
 
    # iterate from beginning
    for i in range(n):
 
        # prefix sum
        pre += a[i]
 
        # check if it exists beyond i
        if (mp[pre] > i):
 
            # if greater then previous
            # then update maximum
            if (pre > maxi):
                maxi = pre
 
    # First and second subarray empty
    if (maxi == -1):
        return 0
 
    # partition done
    else:
        return maxi
 
 
# Driver Code
if __name__ == '__main__':
    a = [1, 3, 2, 1, 4]
    n = len(a)
     
    # Function call
    print(sumFirst(a, n))
 
# This code is contributed by
# Surendra_Gangwar

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C#

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// C# program for Split the array into three
// subarrays such that summation of first
// and third subarray is equal and maximum
using System;
using System.Collections.Generic;
 
class GfG {
 
    // Function to return the sum
    // of the first subarray
    static int sumFirst(int[] a, int n)
    {
        Dictionary<int, int> mp
            = new Dictionary<int, int>();
        int suf = 0;
 
        // calculate the suffix sum
        for (int i = n - 1; i >= 0; i--)
        {
            suf += a[i];
            mp.Add(suf, i);
            if (mp.ContainsKey(suf))
            {
                mp.Remove(suf);
            }
            mp.Add(suf, i);
        }
 
        int pre = 0, maxi = -1;
 
        // iterate from beginning
        for (int i = 0; i < n; i++)
        {
 
            // prefix sum
            pre += a[i];
 
            // check if it exists beyond i
            if (mp.ContainsKey(pre) && mp[pre] > i)
            {
                // if greater then previous
                // then update maximum
                if (pre > maxi)
                {
                    maxi = pre;
                }
            }
        }
 
        // First and second subarray empty
        if (maxi == -1)
            return 0;
 
        // partition done
        else
            return maxi;
    }
 
    // Driver code
    public static void Main(String[] args)
    {
 
        int[] a = { 1, 3, 2, 1, 4 };
        int n = a.Length;
       
        // Function call
        Console.WriteLine(sumFirst(a, n));
    }
}
 
// This code is contributed by Rajput-Ji

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Output

4

METHOD 2

Approach: We will use two pointers concept where one pointer will start from the front and the other from the back. In each iteration, the sum of the first and last subarray is compared and if they are the same then the sum is updated in the answer variable.

Algorithm:

  • Initialize front_pointer to 0 and back_pointer to n-1.
  • Initialize prefixsum to arr[ front_pointer ] and suffixsum to arr[back_pointer].
  • The summations are compared.
    If prefixsum > suffixsum ,back_pointer is decremented by 1 and suffixsum+= arr[ back_pointer ]. 
    If prefixsum < suffixsum, front_pointer is incremented by 1 and prefixsum+= arr[ front_pointer ]
    If they are the same then the sum is updated in the answer variable and both the pointers are moved by one step 
       and both prefixsum and suffixsum are updated accordingly.
  • The above step is continued until the front_pointer is no less than the back_pointer.

Below is the implementation of the above approach:

C++

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// C++ program for Split the array into three
// subarrays such that summation of first
// and third subarray is equal and maximum
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the sum of
// the first subarray
int sumFirst(int a[], int n)
{
    // two pointers are initialized
    // one at the front and the other
    // at the back
    int front_pointer = 0;
    int back_pointer = n - 1;
 
    // prefixsum and suffixsum initialized
    int prefixsum = a[front_pointer];
    int suffixsum = a[back_pointer];
 
    // answer variable initialized to 0
    int answer = 0;
 
    while (front_pointer < back_pointer)
    {
        // if the summation are equal
        if (prefixsum == suffixsum)
        {
            // answer updated
            answer = max(answer, prefixsum);
 
            // both the pointers are moved by step
            front_pointer++;
            back_pointer--;
 
            // prefixsum and suffixsum are updated
            prefixsum += a[front_pointer];
            suffixsum += a[back_pointer];
        }
        else if (prefixsum > suffixsum)
        {
            // if prefixsum is more,then back pointer is
            // moved by one step and suffixsum updated.
            back_pointer--;
            suffixsum += a[back_pointer];
        }
        else
        {
            // if prefixsum is less,then front pointer is
            // moved by one step and prefixsum updated.
            front_pointer++;
            prefixsum += a[front_pointer];
        }
    }
 
    // answer is returned
    return answer;
}
 
// Driver code
int main()
{
    int arr[] = { 1, 3, 2, 1, 4 };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    // Function call
    cout << sumFirst(arr, n);
 
    // This code is contributed by Arif
}

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Output

4

Time Complexity: O(n)
Auxiliary Space: O(1)

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