Optimal, output-sensitive algorithms for constructing planar hulls in parallel
01 Aug 1997-Computational Geometry: Theory and Applications (Elsevier Science Publishers B. V.)-Vol. 8, Iss: 3, pp 151-166
TL;DR: This paper describes a very simple O(lognlogH) time optimal deterministic algorithm for the planar hulls which is an improvement over the previously known Ω( log 2 n) time algorithm for small outputs and presents a fast randomized algorithm that runs in expected time O(logH·loglogn) and does optimal O(n logH) work.
Abstract: In this paper we focus on the problem of designing very fast parallel algorithms for the planar convex hull problem that achieve the optimal O(nlogH) work-bound for input size n and output size H. Our algorithms are designed for the arbitrary CRCW PRAM model. We first describe a very simple O(lognlogH) time optimal deterministic algorithm for the planar hulls which is an improvement over the previously known Ω( log 2 n) time algorithm for small outputs. For larger values of H, we can achieve a running time of O(lognloglogn) steps with optimal work. We also present a fast randomized algorithm that runs in expected time O(logH·loglogn) and does optimal O(nlogH) work. For log H = Ω( loglog n) , we can achieve the optimal running time of O(logH) while simultaneously keeping the work optimal. When logH is o(logn), our results improve upon the previously best known Θ(logn) expected time randomized algorithm of Ghouse and Goodrich. The randomized algorithms do not assume any input distribution and the running times hold with high probability.
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01 Jan 2000
TL;DR: Very general methods for designing efficient parallel algorithms for problems in computational geometry, including the PRAM, are described, providing strong evidence that these techniques yield equally efficient algorithms in more concrete computing models like Butterfly networks.
Abstract: We describe very general methods for designing efficient parallel algorithms for problems in computational geometry. Although our main focus is the PRAM, we provide strong evidence that these techniques yield equally efficient algorithms in more concrete computing models like Butterfly networks. The algorithms exploit random sampling and randomized techniques for solving a wide class of fundamental problems from computational geometry like convex hulls, Voronoi diagrams, triangulation, point-location and arrangements. In addition, the algorithms on the Butterfly network rely critically on an efficient randomized multisearching algorithm. Our description emphasizes the algorithmic techniques rather than a detailed treatment of the individual problems.
15 citations
Journal Article•
TL;DR: In this article, the authors investigate a new paradigm of algorithm design for geometric problems that can be termed distribution-sensitive and derive fast and efficient parallel algorithms for sorting multisets along with the geometric problems.
Abstract: We investigate a new paradigm of algorithm design for geometric problems that can be termed distribution-sensitive. Our notion of distribution is more combinatorial in nature than spatial. We illustrate this on problems like planar-hulls and 2D-maxima where some of the previously known output-sensitive algorithms are recast in this setting. In a number of cases, the distribution-sensitive analysis yields superior results for the above problems. Moreover these bounds are shown to be tight in the linear decision tree model.Our approach owes its spirit to the results known for sorting multisets and we exploit this relationship further to derive fast and efficient parallel algorithms for sorting multisets along with the geometric problems.
12 citations
08 Jul 1998
TL;DR: A new paradigm of algorithm design for geometric problems that can be termed distribution-sensitive is investigated, which yields superior results for problems like planar-hulls and 2D-maxima.
Abstract: We investigate a new paradigm of algorithm design for geometric problems that can be termed distribution-sensitive. Our notion of distribution is more combinatorial in nature than spatial. We illustrate this on problems like planar-hulls and 2D-maxima where some of the previously known output-sensitive algorithms are recast in this setting. In a number of cases, the distribution-sensitive analysis yields superior results for the above problems. Moreover these bounds are shown to be tight for a certain class of algorithms.
11 citations
TL;DR: An optimal speed-up (with respect to the input size only) sublogarithmic time algorithm that uses superlinear number of processors for vector maxima in three dimensions that is faster than previously known algorithms.
11 citations
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"Optimal, output-sensitive algorithm..." refers background in this paper
...Several sequential algorithms have been proposed for planar hull with the worst case time bound O(n log n) [22] [30] [31]....
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