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1	Sub-cubic Time Algorithm for the k-disjoint Maximum subarray Problem Lee, Sang Myung (Chris) January 2011 (has links) The maximum subarray problem is to find the array portion that maximizes the sum of array elements in it. This problem was first introduced by Grenander and brought to computer science by Bentley in 1984. This problem has been branched out into other problems based on their characteristics. k-overlapping maximum subarray problem where the overlapping solutions are allowed, and k-disjoint maximum subarray problem where all the solutions are disjoint from each other are those. For k-overlapping maximum subarray problems, significant improvement have been made since the problem was first introduced. For k-disjoint maximum subarrsy, Ruzzo and Tompa gave an O(n) time solution for one-dimension. This solution is, however, difficult to extend to two-dimensions. While a trivial solution of O(kn^3) time is easily obtainable for two-dimensions, little study has been undertaken to better this. This paper introduces a faster algorithm for the k-disjoint maximum sub-array problem under the conventional RAM model, based on distance matrix multiplication. Also, DMM reuse technique is introduced for the maximum subarray problem based on recursion for space optimization.
2	Average case analysis of algorithms for the maximum subarray problem Bashar, Mohammad Ehsanul January 2007 (has links) Maximum Subarray Problem (MSP) is to find the consecutive array portion that maximizes the sum of array elements in it. The goal is to locate the most useful and informative array segment that associates two parameters involved in data in a 2D array. It's an efficient data mining method which gives us an accurate pattern or trend of data with respect to some associated parameters. Distance Matrix Multiplication (DMM) is at the core of MSP. Also DMM and MSP have the worst-case complexity of the same order. So if we improve the algorithm for DMM that would also trigger the improvement of MSP. The complexity of Conventional DMM is O(n³). In the average case, All Pairs Shortest Path (APSP) Problem can be modified as a fast engine for DMM and can be solved in O(n² log n) expected time. Using this result, MSP can be solved in O(n² log² n) expected time. MSP can be extended to K-MSP. To incorporate DMM into K-MSP, DMM needs to be extended to K-DMM as well. In this research we show how DMM can be extended to K-DMM using K-Tuple Approach to solve K-MSP in O(Kn² log² n log K) time complexity when K ≤ n/log n. We also present Tournament Approach which solves K-MSP in O(n² log² n + Kn²) time complexity and outperforms the K-Tuple Maximum Subarray Problem Distance Matrix Multiplication All Pairs Shortest Path Problem Shortest Path Expected Time Average Time Average Case Analysis Binary Heap Prefix Sum Directed Acyclic Graph X + Y Problem Binary Tournament Tree Solution Set MSP K-MSP DMM K-DMM APSP K-Tuple

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Sub-cubic Time Algorithm for the k-disjoint Maximum subarray Problem

Average case analysis of algorithms for the maximum subarray problem