Fast sequential and parallel algorithms for finding the largest rectangle separating two sets
01 Jan 1990-International Journal of Computer Mathematics (Gordon and Breach Science Publishers)-Vol. 37, pp 49-61
TL;DR: This work considers two limiting cases of this problem when the cardinalities of set A is much greater than that of set B, and presents efficient sequential and parallel algorithms for these two problems.
Abstract: Given a bounding isothetic rectangle BR and two sets of points A and B with cardinalities n and m inside it, we have to find an isothetic rectangle containing maximum number of points from set A and no point from set B. We consider two limiting cases of this problem when the cardinalities of set A (resp. set B) is much greater than that of set B (resp. set ,A). We present efficient sequential and parallel algorithms for these two problems. Our sequential algorithms run in O((n + m)log m + m 2) and O((m+ n) log n + n 2) time respectively. The parallel algorithms in CREW PRAM run in o(log n) ando(log m 2) time using O(max(n,m 2/logm)) and O(max(m,n 2/logn)) processors respectively. Our sequential algorithms are faster than a previous algorithm under these constraints on cardinality. No previous parallel algorithm was known for this problem. We also present an optimal systolic algorithm for the original problem.
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17 Dec 2002
TL;DR: An O(log log N) bus cycles parallel algorithm for the medial axis transform of an N/spl times/N binary image on a linear array with a reconfigurable pipelined bus system using N/sup 2/ processors is provided.
Abstract: In this paper based on the advantages of both optical transmission and electronic computation, we first provide an O(log log N) bus cycles parallel algorithm for the medial axis transform of an N/spl times/N binary image on a linear array with a reconfigurable pipelined bus system using N/sup 2/ processors. By increasing the number of processors, the proposed algorithm can be modified to run in O(log log/sub q/ N) and O(1) bus cycles using qN/sup 2/ and N/sup 2+1//spl isin// processors respectively, where 1/spl les/q/spl les//spl radic/N, /spl isin/ is a constant and /spl isin//spl ges/1. These results improve on previously known algorithms developed on various parallel computation models. Key Words: Medial axis transform, image processing, image compression, computer vision, parallel algorithms, linear array with a reconfigurable pipelined bus system.
11 citations
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TL;DR: This paper shows that the essential condition for distribution-free learnability is finiteness of the Vapnik-Chervonenkis dimension, a simple combinatorial parameter of the class of concepts to be learned.
Abstract: Valiant's learnability model is extended to learning classes of concepts defined by regions in Euclidean space En. The methods in this paper lead to a unified treatment of some of Valiant's results, along with previous results on distribution-free convergence of certain pattern recognition algorithms. It is shown that the essential condition for distribution-free learnability is finiteness of the Vapnik-Chervonenkis dimension, a simple combinatorial parameter of the class of concepts to be learned. Using this parameter, the complexity and closure properties of learnable classes are analyzed, and the necessary and sufficient conditions are provided for feasible learnability.
1,967 citations
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TL;DR: A parallel implementation of merge sort on a CREW PRAM that uses n processors and O(logn) time; the constant in the running time is small.
Abstract: We give a parallel implementation of merge sort on a CREW PRAM that uses n processors and $O(\log n)$ time; the constant in the running time is small. We also give a more complex version of the algorithm for the EREW PRAM; it also uses n processors and $O(\log n)$ time. The constant in the running time is still moderate, though not as small.
847 citations
01 Dec 1985
242 citations
TL;DR: This study assumes the weakest PRAM model, where shared memory locations can only be exclusively read or written (the EREW model) to solve the prefix computation problem, when the order of the elements is specified by a linked list.
Abstract: The prefix computation problem is to compute all n initial products a1* . . . *a1,i=1, . . ., n of a set of n elements, where * is an associative operation. An O(((logn) log(2n/p))XI(n/p)) time deterministic parallel algorithm using p≤n processors is presented to solve the prefix computation problem, when the order of the elements is specified by a linked list. For p≤O(n1-e)(e〉0 any constant), this algorithm achieves linear speedup. Such optimal speedup was previously achieved only by probabilistic algorithms. This study assumes the weakest PRAM model, where shared memory locations can only be exclusively read or written (the EREW model).
205 citations