Kadane's Algorithm is a dynamic programming technique used to solve the maximum subarray problem. Given an array of integers, we need to find the contiguous subarray with the largest sum. For example, in the array negative two, one, negative three, four, negative one, two, one, negative five, four, the maximum subarray is four, negative one, two, one with a sum of six.
The algorithm follows four main steps. First, we initialize two variables: global max and current max, both set to the first element of the array. Second, we iterate through the array starting from the second element. Third, for each element, we update current max to be the maximum of either the current element alone or the current element plus the previous current max. Finally, we update global max to be the maximum of the current global max and the new current max.
Let's trace through the algorithm step by step with our example array. We start with index zero, where both current max and global max are initialized to negative two. At index one, current max becomes the maximum of one or negative two plus one, which is one. Global max is updated to one. At index two, current max becomes negative two, but global max remains one. The process continues, and we can see how the algorithm efficiently tracks the maximum subarray sum, which reaches six at index six.
The key insight of Kadane's algorithm is the decision made at each position: whether to start a new subarray from the current element or extend the previous subarray. We choose whichever gives the maximum value. This elegant approach achieves linear time complexity O of n with just a single pass through the array, and constant space complexity O of one using only two variables. This makes it highly efficient for large datasets.
To summarize what we've learned: Kadane's algorithm efficiently solves the maximum subarray problem in linear time using dynamic programming. It maintains two key variables and makes optimal decisions at each step. This makes it perfect for processing large datasets and has wide applications in finance and data analysis.