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Efficient Algorithms for the Maximum Sum Problems

Algorithm Research Institute, Christchurch 8053, New Zealand
Computer Science and Software Engineering, University of Canterbury, Christchurch 8140, New Zealand
Author to whom correspondence should be addressed.
Academic Editors: Bruno Carpentieri and Spyros Kontogiannis
Algorithms 2017, 10(1), 5;
Received: 9 August 2016 / Revised: 2 December 2016 / Accepted: 26 December 2016 / Published: 4 January 2017
PDF [1175 KB, uploaded 4 January 2017]


We present efficient sequential and parallel algorithms for the maximum sum (MS) problem, which is to maximize the sum of some shape in the data array. We deal with two MS problems; the maximum subarray (MSA) problem and the maximum convex sum (MCS) problem. In the MSA problem, we find a rectangular part within the given data array that maximizes the sum in it. The MCS problem is to find a convex shape rather than a rectangular shape that maximizes the sum. Thus, MCS is a generalization of MSA. For the MSA problem, O ( n ) time parallel algorithms are already known on an ( n , n ) 2D array of processors. We improve the communication steps from 2 n 1 to n, which is optimal. For the MCS problem, we achieve the asymptotic time bound of O ( n ) on an ( n , n ) 2D array of processors. We provide rigorous proofs for the correctness of our parallel algorithm based on Hoare logic and also provide some experimental results of our algorithm that are gathered from the Blue Gene/P super computer. Furthermore, we briefly describe how to compute the actual shape of the maximum convex sum. View Full-Text
Keywords: maximum sub-array; maximum convex sum; parallel algorithm maximum sub-array; maximum convex sum; parallel algorithm

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Bae, S.E.; Shinn, T.-W.; Takaoka, T. Efficient Algorithms for the Maximum Sum Problems. Algorithms 2017, 10, 5.

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