Optimal Parallel Randomized Algorithms for Three-Dimensional Convex Hulls and Related Problems

Further applications of random sampling techniques which have been used for deriving efficient parallel algorithms are presented by J. H. Reif and S. Sen [Proc. 16th International Conference on Parallel Processing, 1987]. This paper presents an optimal parallel randomized algorithm for computing int...

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Vydané v:SIAM journal on computing Ročník 21; číslo 3; s. 466 - 485
Hlavní autori: Reif, John H., Sen, Sandeep
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Philadelphia, PA Society for Industrial and Applied Mathematics 01.06.1992
Predmet:
Algorithms
Applied sciences
Artificial intelligence
Computer science
Computer science; control theory; systems
Exact sciences and technology
Geometry
Linear programming
Pattern recognition. Digital image processing. Computational geometry
Intersection
Multiprocessor
Computational geometry
Parallel algorithm
Convex hull
Randomization
Optimal algorithm
Three dimensional space
Computer system
ISSN:0097-5397, 1095-7111
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Abstract Further applications of random sampling techniques which have been used for deriving efficient parallel algorithms are presented by J. H. Reif and S. Sen [Proc. 16th International Conference on Parallel Processing, 1987]. This paper presents an optimal parallel randomized algorithm for computing intersection of half spaces in three dimensions. Because of well-known reductions, these methods also yield equally efficient algorithms for fundamental problems like the convex hull in three dimensions, Voronoi diagram of point sites on a plane, and Euclidean minimal spanning tree. The algorithms run in time $T = O(\log n)$ for worst-case inputs and use $P = O(n)$ processors in a CREW PRAM model where $n$ is the input size. They are randomized in the sense that they use a total of only polylogarithmic number of random bits and terminate in the claimed time bound with probability $1 - n^{ - \alpha } $ for any fixed $\alpha > 0$. They are also optimal in $P\cdot T$ product since the sequential time bound for all these problems is $\Omega (n\log n)$. The best known deterministic parallel algorithms for two-dimensional Voronoi-diagram and three-dimensional convex hull run in $O(\log ^2 n)$ and $O(\log ^2 n\log ^ * n)$ time, respectively, while using $O(n/\log n)$ and $O(n)$ processors, respectively.
AbstractList Further applications of random sampling techniques which have been used for deriving efficient parallel algorithms are presented by J. H. Reif and S. Sen [Proc. 16th International Conference on Parallel Processing, 1987]. This paper presents an optimal parallel randomized algorithm for computing intersection of half spaces in three dimensions. Because of well-known reductions, these methods also yield equally efficient algorithms for fundamental problems like the convex hull in three dimensions, Voronoi diagram of point sites on a plane, and Euclidean minimal spanning tree. The algorithms run in time $T = O(\log n)$ for worst-case inputs and use $P = O(n)$ processors in a CREW PRAM model where $n$ is the input size. They are randomized in the sense that they use a total of only polylogarithmic number of random bits and terminate in the claimed time bound with probability $1 - n^{ - \alpha } $ for any fixed $\alpha > 0$. They are also optimal in $P\cdot T$ product since the sequential time bound for all these problems is $\Omega (n\log n)$. The best known deterministic parallel algorithms for two-dimensional Voronoi-diagram and three-dimensional convex hull run in $O(\log ^2 n)$ and $O(\log ^2 n\log ^ * n)$ time, respectively, while using $O(n/\log n)$ and $O(n)$ processors, respectively.
Author Reif, John H.
Sen, Sandeep
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10.1016/0196-6774(85)90007-0
10.1016/0020-0190(79)90074-7
10.1007/BF02187879
10.1007/BF01840357
10.1137/0205015
10.1007/BF01762120
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10.1007/978-3-642-61568-9
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Multiprocessor
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SubjectTerms Algorithms
Applied sciences
Artificial intelligence
Computer science
Computer science; control theory; systems
Exact sciences and technology
Geometry
Linear programming
Pattern recognition. Digital image processing. Computational geometry
Title Optimal Parallel Randomized Algorithms for Three-Dimensional Convex Hulls and Related Problems
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