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.

The maximum subarray problem (MSP) involves selection of a segment of consecutive array elements that has the largest possible sum over all other segments in a given array. The efficient algorithms for the MSP and related problems are expected to contribute to various applications in genomic sequence analysis, data mining or in computer vision etc. The MSP is a conceptually simple problem, and several linear time optimal algorithms for 1D version of the problem are already known. For 2D version, the currently known upper bounds are cubic or near-cubic time. For the wider applications, it would be interesting if multiple maximum subarrays are computed instead of just one, which motivates the work in the first half of the thesis. The generalized problem of K-maximum subarray involves finding K segments of the largest sum in sorted order. Two subcategories of the problem can be defined, which are K-overlapping maximum subarray problem (K-OMSP), and K-disjoint maximum subarray problem (K-DMSP). Studies on the K-OMSP have not been undertaken previously, hence the thesis explores various techniques to speed up the computation, and several new algorithms. The first algorithm for the 1D problem is of O(Kn) time, and increasingly efficient algorithms of O(K² + n logK) time, O((n+K) logK) time and O(n+K logmin(K, n)) time are presented. Considerations on extending these results to higher dimensions are made, which contributes to establishing O(n³) time for 2D version of the problem where K is bounded by a certain range. Ruzzo and Tompa studied the problem of all maximal scoring subsequences, whose definition is almost identical to that of the K-DMSP with a few subtle differences. Despite slight differences, their linear time algorithm is readily capable of computing the 1D K-DMSP, but it is not easily extended to higher dimensions. This observation motivates a new algorithm based on the tournament data structure, which is of O(n+K logmin(K, n)) worst-case time. The extended version of the new algorithm is capable of processing a 2D problem in O(n³ + min(K, n) · n² logmin(K, n)) time, that is O(n³) for K ≤ n/log n For the 2D MSP, the cubic time sequential computation is still expensive for practical purposes considering potential applications in computer vision and data mining. The second half of the thesis investigates a speed-up option through parallel computation. Previous parallel algorithms for the 2D MSP have huge demand for hardware resources, or their target parallel computation models are in the realm of pure theoretics. A nice compromise between speed and cost can be realized through utilizing a mesh topology. Two mesh algorithms for the 2D MSP with O(n) running time that require a network of size O(n²) are designed and analyzed, and various techniques are considered to maximize the practicality to their full potential.

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