# Maximum equilibrium sum in an array

## Introduction

In computer science and programming, arrays are used to store collections of elements as basic data structures. Finding the maximum equilibrium sum-a location within the array where the sum of the elements on the left and right sides is equal-is one of the intriguing ideas related to arrays. This idea highlights the symmetry and balance that can exist in arrays, providing new opportunities for investigation in algorithm design and problem-solving.

### Understanding Equilibrium in Arrays

Equilibrium in the context of arrays refers to the condition where the elements on both sides maintain balance with respect to their cumulative sum. Formally, an equilibrium point i is defined by the equation given an array arr of length n:

arr[0] + arr[1] + ... + arr[i-1] = arr[i+1] + arr[i+2] + ... + arr[n-1]

According to this equation, the total of the elements on the left and right sides of the equilibrium point are equal. Finding equilibrium points can help identify array divisions that produce equal cumulative sums on both sides.

### Acquiring Knowledge of Equilibrium in an Array

It's crucial to understand the idea of equilibrium in an array before delving into the intricacies of the maximum equilibrium sum problem. Equilibrium is a location in the array where the total of the elements on the left and right sides equals one another. This idea lays the groundwork for solving the trickier issue of determining the maximum equilibrium sum.

### Problem of Maximum Equilibrium Sum

Finding the highest equilibrium sum from an array of integers is the difficult task posed by the maximum equilibrium sum problem. To get the best results, this calls for a calculated strategy that strikes a balance between the factors on both sides.

### Prefix and Suffix Sums-Based Optimal Method

We can use the strength of prefix and suffix sums to address the efficiency issues with the naive approach. We can greatly reduce the time complexity of our solution by computing the cumulative sums from the beginning and end of the array beforehand. We can quickly calculate the sum on either side of a potential equilibrium point using this optimisation, allowing us to make defensible decisions.

### Challenges in Finding Maximum Equilibrium Sum

A thorough examination of potential equilibrium points is required to determine the array's maximum equilibrium sum. Simple brute-force solutions are possible, but they may not be the most effective for larger arrays. A method that avoids repeating calculations and iterates through the array just once is necessary for an optimal approach.

### Effective Approach: Prefix Sum Method

The prefix sum concept can be used to efficiently find the maximum equilibrium sum. The total of all elements from index 0 to i is the prefix sum of an element at index i. We quickly ascertain the existence of an equilibrium point by computing the prefix sums of the array.

Here is a step-by-step explanation of the strategy:

• For the specified array arr, determine the prefix sum array prefix_sum.
• Go through each index i in the array one by one.
• Compare the left sum (the sum of the elements up to index i-1) and the right sum (the sum of the elements from index i+1 to the end) for each index i.
• If the left sum equals the right sum, i is a potential equilibrium point.
• Note the highest i value at which equilibrium is reached.

Code:

Output:

```Maximum Equilibrium Sum: 9
```

Here's a brief explanation of the code:

1. The function max_equilibrium_sum(arr) takes a single argument: arr, which is the input array.
2. Two arrays, prefix_sum and suffix_sum, are initialized to store the cumulative sums of the array elements from the left and right, respectively.
3. A loop iterates through the array to calculate the prefix_sum. It sums up elements from the beginning of the array and stores each sum in the prefix_sum
4. Another loop iterates in reverse to calculate the suffix_sum. It sums up elements from the end of the array and stores each sum in the suffix_sum
5. A variable named max_sum is initialized to store the maximum equilibrium sum found.
6. A loop iterates through the array from the second element to the second-to-last element (indices 1 to n-2).
7. For each index i, the code checks if the sum of elements to the left (up to index i-1) is equal to the sum of elements to the right (starting from index i+1). If they are equal, the sum is considered a potential equilibrium sum, and the maximum is updated if it's larger.
8. After iterating through the array, the function returns the max_sum, which represents the maximum equilibrium sum found.
9. An example usage is provided using the array arr = [1, 5, 3, 8, 2, 7]. The max_equilibrium_sum function is called with this array, and the resulting maximum equilibrium sum is printed.

### Further Enhancements and Variations

The maximum equilibrium sum problem offers many opportunities for improvements and modifications. For larger datasets, investigating various data structures, dynamic programming strategies, and parallel computing can result in even more effective solutions.

### Future Prospects of Equilibrium-based Algorithms

The importance of equilibrium-based algorithms keeps expanding as technology develops. Finding the ideal balances between different components remains a crucial aspect of problem-solving, from financial modelling to artificial intelligence.