Two Kadane Algorithms for the Maximum Sum Subarray Problem

The maximum sum subarray problem is to find a contiguous subarray with the largest sum. The history of algorithms to address this problem is recounted, culminating in what is known as Kadane's algorithm. However, that algorithm is not the algorithm Kadane intended. Nonetheless, the algorithm known as Kadane's has found many uses, some of which are recounted here. The algorithm Kadane intended is reported here, and compared to the algorithm attributed to Kadane. They are both linear in time, employ just a few words of memory, and use a dynamic programming structure. The results proved here show that these two algorithms differ only in the case of an input consisting of only negative numbers. In that case, the algorithm Kadane intended is more informative than the algorithm attributed to him.