Kadane's Algorithm in Java

The Maximum Subarray Problem constitutes one of the efficient algorithms in algorithmic problems that can be solved through Kadane's Algorithm. Here the problem is to find the largest sum of contiguous sub array which can be solved in O(n) time complexity in one dimensional array.

This algorithm is highly useful in fields such as business and finance as it can be used to deduce most profitable trading days, or in deep analytical fields to identify higher activity patterns. Its applicability is not limited to the usual algorithm problems and can be applied to any other dynamic programming problem that can be solved using subarrays or sequences.

The Kadane's Algorithm simply forges the maximum sum of a subarray up to the last element at a time implying either to continue adding to the previous subarray or start with the element in case the resultant sum is better than the previous subarray's sum.

This approach is just as efficient and effective in terms of space and time and the use of simple logical reasoning to arrive at the solution of what looks like a complicated problem; this why Kadane's algorithm is vital tool in efficient data analytic and dynamic programming.

Problem Statement

The motivation of the Maximum Subarray Problem is to determine which contiguous subarray of a given array yields the greatest sum. For example, given the array:

We will keep track of the maximum sum along with the limits of its subarray becoming [4, −1, 2, 1] with maximum sum 6. And this problem can be solved by using Kadane's Algorithm in time complexity O(n) which makes the process efficient.

Algorithm Intuition

I find that the most basic idea around which Kadane's Algorithm is constructed is the concept of the "sliding window." The algorithm examines each element in the array while maintaining two variables:

• currentSum helps to keep track of sum of elements in array as calculated until it is reached the subarray from left of the given array.
• Maximum Sum encountered till now (maxSum) - This variable stores the maximum sum verified until now.

It moves from each element of the array, by adding to currentSum either extending it with the current array element or starting with the current element in a new subarray. If currentSum goes beyond maxSum then the value of maxSum is overwritten with current Sum.

Steps of Kadane's Algorithm

Kadane's Algorithm is essentially about deciding at each element whether to:

• Add the element to current Sum and continue the current subarray.
• Begin a new subarray from the current figure.

By doing the two steps above and keeping record of the global maximum then it guarantees that the algorithm will yield to the maximum sum capable in any given contiguous subarray situation. Now, let us dissect the algorithm to get a clearer understanding of the mathematical process here involved.

File Name: KadaneAlgorithm.java

Output:

 Maximum subarray sum is: 6 

Complexity Analysis

Time Complexity: O(n), because each and every one of the implemented functions takes a number of operations proportional to the size of the input array.

Space Complexity: O(1), because we need a constant amount of extra space for maxsum and currentsum.

Conclusion

Kadane's Algorithm is another optimal solution towards the Maximum Subarray Problem. It uses the concept of running sum, that is, the sum of each element while choosing dynamically the best choice whether to add current element to the sum in the current subarray.

If it is wiser to initialize a new subarray since current element is smaller than previous elements, depending on which choice yields the overall biggest sum value.

Therefore, the proposed approach has the time complexity of O(n) and the space complexity of O(1), which must be the best solution for this problem when a large volume of data is processed. Learning of Kadane's Algorithm is a good start up in dynamic programming and solving sub-problems of contiguous subarray type.