Maximum subarray sum with same first and last element formed by removing elements

Given an array arr[] of N integers, the task is to find the maximum sum of subarray having length an at least 2 whose first and last elements are the same after removing any number of array elements. If there exists no such array, then print 0.

Examples:

Input: arr[] = {-1, -3, -2, 4, -1, 3}
Output: 2
Explanation: Choose the sub-array { -1, -3, -2, 4, -1} and remove the elements -3 and -2 which makes the sub-array as { -1, 4, -1} with sum equal to 2 which is the maximum.

Input: arr[] = {-1}
Output: 0

Approach: The given problem can be solved by using the idea is that first find all the subarrays having the first and the last element same and removing all the negative elements between those first and the last element. This idea can be implemented by the idea discussed in this article using an unordered map. Follow the steps below to solve the given problem:

• Initialize a variable, say res as INT_MIN that stores the resultant maximum sum of the subarray.
• Initialize a variable, say currentSum as 0 that stores the running prefix sum of the array.
• Initialize a unordered_map, say memo[] that stores the value of each array element mapped with its prefix sum.
• Iterate over the range [0, N) using the variable i and perform the following tasks:
  • Add the value of the current array element arr[i] to the variable currentSum.
  • If arr[i] is present in the map, and If the value of arr[i] is positive then update the value of res to the maximum of res and (currentSum - M[arr[i]] + arr[i]). Otherwise,  update the value of res to the maximum of res and (currentSum - M[arr[i]] + 2*arr[i]).
  • Otherwise, Insert the value arr[i] mapped with currentSum.
  • If the current value arr[i] is negative, then decrement it from currentSum to exclude the negative element for the possible subarray.
• After completing the above steps, print the value of res as the result.

Below is the implementation of the above approach:

// C++ program for the above approach

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

// Function to find the maximum sum
// of sub-array
int maximumSubarraySum(vector<int>& arr)
{

    // Initialize the variables
    int N = arr.size();
    unordered_map<int, int> memo;
    int res = INT_MIN;
    int currsum = 0, currval = 0;

    // Traverse over the range
    for (int i = 0; i < N; ++i) {

        // Add the current value to the
        // variable currsum for prefix sum
        currval = arr[i];
        currsum = currsum + currval;

        // Calculate the result
        if (memo.count(currval) != 0) {
            if (currval > 0)
                res = max(
                    res,
                    currsum - memo[currval]
                        + currval);
            else
                res = max(
                    res,
                    currsum - memo[currval]
                        + 2 * currval);
        }
        else
            memo[currval] = currsum;
        if (currval < 0)
            currsum = currsum - currval;
    }

    // Return the answer
    return res;
}

// Driver Code
int main()
{
    vector<int> arr = { -1, -3, 4, 0, -1, -2 };
    cout << maximumSubarraySum(arr);

    return 0;
}

// Java program for the above approach
import java.io.*;
import java.util.*;
class GFG 
{
  
    // Function to find the maximum sum
    // of sub-array
    static int maximumSubarraySum(int arr[], int N)
    {

        // Initialize the variables
        HashMap<Integer, Integer> memo = new HashMap<>();
        int res = Integer.MIN_VALUE;
        int currsum = 0, currval = 0;

        // Traverse over the range
        for (int i = 0; i < N; ++i) {

            // Add the current value to the
            // variable currsum for prefix sum
            currval = arr[i];
            currsum = currsum + currval;

            // Calculate the result
            if (memo.containsKey(currval)) {
                if (currval > 0)
                    res = Math.max(
                        res, currsum - memo.get(currval)
                                 + currval);
                else
                    res = Math.max(
                        res, currsum - memo.get(currval)
                                 + 2 * currval);
            }
            else
                memo.put(currval, currsum);
            if (currval < 0)
                currsum = currsum - currval;
        }

        // Return the answer
        return res;
    }
  
    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { -1, -3, 4, 0, -1, -2 };
        int N = 6;
        System.out.println(maximumSubarraySum(arr, N));
    }
}

// This code is contributed by dwivediyash

# Python3 program for the above approach
import sys

# Function to find the maximum sum
# of sub-array
def maximumSubarraySum(arr) :

    # Initialize the variables
    N = len(arr);
    memo = {};
    res = -(sys.maxsize - 1);
    currsum = 0;
    currval = 0;

    # Traverse over the range
    for i in range(N) :

        # Add the current value to the
        # variable currsum for prefix sum
        currval = arr[i];
        currsum = currsum + currval;

        # Calculate the result
        if currval in memo :
            
            if (currval > 0) :
                res = max(res,currsum - memo[currval] + currval);
            else :
                res = max(res,currsum - memo[currval] + 2 * currval);
        
        else :
            memo[currval] = currsum;
            
        if (currval < 0) :
            currsum = currsum - currval;

    # Return the answer
    return res;


# Driver Code
if __name__ == "__main__" :

    arr = [ -1, -3, 4, 0, -1, -2 ];
    print(maximumSubarraySum(arr));

    # This code is contributed by AnkThon

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG{

    // Function to find the maximum sum
    // of sub-array
    static int maximumSubarraySum(int[] arr, int N)
    {

        // Initialize the variables
        Dictionary<int, int> memo = new
                Dictionary<int, int>();
        int res = Int32.MinValue;
        int currsum = 0, currval = 0;

        // Traverse over the range
        for (int i = 0; i < N; ++i) {

            // Add the current value to the
            // variable currsum for prefix sum
            currval = arr[i];
            currsum = currsum + currval;

            // Calculate the result
            if (memo.ContainsKey(currval)) {
                if (currval > 0)
                    res = Math.Max(
                        res, currsum - memo[(currval)]
                                 + currval);
                else
                    res = Math.Max(
                        res, currsum - memo[(currval)]
                                 + 2 * currval);
            }
            else
                memo.Add(currval, currsum);
            if (currval < 0)
                currsum = currsum - currval;
        }

        // Return the answer
        return res;
    }

// Driver Code
public static void Main()
{
    int[] arr = { -1, -3, 4, 0, -1, -2 };
    int N = 6;
    Console.WriteLine(maximumSubarraySum(arr, N));
}
}

// This code is contributed by sanjoy_62.

<script>
        // JavaScript Program to implement
        // the above approach

        // Function to find the maximum sum
        // of sub-array
        function maximumSubarraySum(arr) {

            // Initialize the variables
            let N = arr.length;
            let memo = new Map();
            let res = -99999999;
            let currsum = 0, currval = 0;

            // Traverse over the range
            for (let i = 0; i < N; ++i) {

                // Add the current value to the
                // variable currsum for prefix sum
                currval = arr[i];
                currsum = currsum + currval;

                // Calculate the result
                if (memo.has(currval)) {
                    if (currval > 0)
                        res = Math.max(
                            res,
                            currsum - memo.get(currval)
                            + currval);
                    else
                        res = Math.max(
                            res,
                            currsum - memo.get(currval)
                            + 2 * currval);
                }
                else
                    memo.set(currval, currsum);
                if (currval < 0)
                    currsum = currsum - currval;
            }

            // Return the answer
            return res;
        }

        // Driver Code
        let arr = [-1, -3, 4, 0, -1, -2];
        document.write(maximumSubarraySum(arr));

// This code is contributed by Potta Lokesh
    </script>

2

Time Complexity: O(N)
Auxiliary Space: O(N)