Kadane's Algorithm

• Dynamic Programming Approach: Kadane's Algorithm is a classic dynamic programming algorithm used to find the maximum sum of a contiguous subarray within a one-dimensional array of numbers.
• Efficiency: It offers an efficient linear time complexity solution (O(n)), significantly improving over brute force methods that examine every possible subarray.

How Does It Work?

The core idea of Kadane's Algorithm is this:

1. Track Two Sums:

   • max_ending_here: The maximum sum of a subarray ending at the current position.
   • max_so_far: The overall maximum sum of a subarray found so far.

2. Iteration and Update:

   • Iterate through the array.
   • At each position, update max_ending_here as the maximum between:
     • The current element itself (starting a new subarray)
     • The current element + max_ending_here from the previous position (extending the existing subarray)
   • After updating max_ending_here, check if it's greater than max_so_far and update max_so_far if necessary.

3. Result: After the entire array is processed, max_so_far holds the maximum contiguous subarray sum.

Applications

• Finance: Finding periods of maximum profit in stock price data.
• Image Processing: Detecting regions with maximum brightness.
• Bioinformatics: Identifying gene sequences with the highest scores.
• General Problem Solving: Numerous scenarios where finding the maximum sum of a contiguous segment is beneficial.

Key Points

• Kadane's Algorithm works effectively even if the array contains all negative numbers.
• It focuses on finding contiguous subarrays, not any combination of elements.