Kadane algorithm

[SARATH-ANSIL]

Problem:

Find the contiguous subarray within a one-dimensional array of numbers that has the largest sum.

Idea / Intuition

While traversing the array, keep track of two things:

1. Current maximum subarray sum ending at this position (maxEndingHere)
2. Overall maximum sum found so far (maxSoFar)

At each element, we have two choices:

1. Start a new subarray at the current element if the previous sum is negative.
2. Extend the current subarray by including the current element.

### Formula:
maxEndingHere = max(currentElement, maxEndingHere + currentElement)
maxSoFar = max(maxSoFar, maxEndingHere)

Step-by-Step Logic

1. Initialize maxSoFar and maxEndingHere with the first element of the array.
2. Loop through the array from the second element:
3. Update maxEndingHere using the formula above.
4. Update maxSoFar if maxEndingHere is greater.
5. After traversing the array, maxSoFar contains the maximum subarray sum.

### Why It Works

1. If the sum ending at the previous element is negative, it cannot contribute to a larger sum in the future, so we start fresh.
2. If the sum is positive, including the current element will increase the subarray sum, so we continue.

Time and Space Complexity

Time Complexity: O(n) – single pass through the array.

Space Complexity: O(1) – only two variables are needed.

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