Java Program for Equilibrium index of an array

 The Equilibrium Index of an array is an index in the array where the sum of the elements to the left of it is equal to the sum of the elements to the right of it. In other words, for an array arr[], the equilibrium index i satisfies the condition:

$$\text{sum(arr[0...i-1])} = \text{sum(arr[i+1...n-1])}$$

Where:

• arr[0...i-1] represents the elements to the left of the index i.
• arr[i+1...n-1] represents the elements to the right of the index i.

Explanation:

• Left sum: The sum of elements before the equilibrium index.
• Right sum: The sum of elements after the equilibrium index.

If an equilibrium index exists, the array is said to have a point of equilibrium. If no such index exists, then the array doesn't have an equilibrium point.

Steps to Solve the Problem:

1. Find the Total Sum of the Array: First, calculate the total sum of the array.
2. Iterate Through the Array: Traverse the array, maintaining the sum of the elements on the left side (left sum).
3. Compare Left and Right Sums: For each element, subtract the left sum from the total sum to get the right sum (right sum = total sum - left sum - current element).
4. Check for Equilibrium: If at any point the left sum equals the right sum, that index is the equilibrium index.

Intuition Behind the Approach:

Instead of recalculating the left and right sums repeatedly, you can optimize the process by maintaining a running sum of the left side. This reduces the complexity of the solution, allowing for a more efficient approach.

Time Complexity:

The time complexity of this approach is O(n), where n is the length of the array. This is because you are just traversing the array once while calculating the left sum and comparing it with the right sum.

Example:

Consider the array: [1, 3, 5, 2, 2].

• Total sum = 1 + 3 + 5 + 2 + 2 = 13.
• Initially, the left sum is 0. The right sum would be 13 - 0 - arr[0] = 12.

Then you would move to the next element and update the left and right sums, checking for an equilibrium index at each step.

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