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Proceedings ArticleDOI

Polling: a new randomized sampling technique for computational geometry

01 Feb 1989-pp 394-404
TL;DR: A new randomized sampling technique, called Polling, is introduced which has applications to deriving efficient parallel algorithms for fundamental problems like the convex hull in three dimensions, Voronoi diagram of point sites on a plane and Euclidean minimal spanning tree.
Abstract: We introduce a new randomized sampling technique, called Polling which has applications to deriving efficient parallel algorithms. As an example of its use in computational geometry, we present an optimal parallel randomized algorithm for intersection of half-spaces in three dimensions. Because of well-known reductions, our 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. Our algorithms run in time T = O(logn) for worst-case inputs and uses 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 O(log2n) random bits and terminate in the claimed time bound with probability 1 - n-a for any a > 0. They are also optimal in P . T product since the sequential time bound for all these problems is O(nlogn). The best known determistic parallel algorithms for 2-D Voronoi-diagram and 3-D Convex hull run in O(log2n) and O(log2nlog * n) time respectively while using O(n) processors.

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Citations
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Proceedings ArticleDOI
Kenneth L. Clarkson1
06 Jan 1988
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.

1,163 citations


Cites methods from "Polling: a new randomized sampling ..."

  • ...The results in this paper have been used in an algorithm for triangulating simple polygons [17], for small-dimensional linear and integer programming [13], for an optimal parallel algorithm for Voronoi diagrams [ 39 ], and in various combinatorial results on arrangements [15]....

    [...]

Book ChapterDOI
02 Jan 1991
TL;DR: In this paper, the authors discuss parallel algorithms for shared-memory machines and discuss the theoretical foundations of parallel algorithms and parallel architectures, and present a theoretical analysis of the appropriate logical organization of a massively parallel computer.
Abstract: Publisher Summary This chapter discusses parallel algorithms for shared-memory machines. Parallel computation is rapidly becoming a dominant theme in all areas of computer science and its applications. It is estimated that, within a decade, virtually all developments in computer architecture, systems programming, computer applications and the design of algorithms will be taking place within the context of parallel computation. In preparation for this revolution, theoretical computer scientists have begun to develop a body of theory centered on parallel algorithms and parallel architectures. As there is no consensus yet on the appropriate logical organization of a massively parallel computer, and as the speed of parallel algorithms is constrained as much by limits on interprocessor communication as it is by purely computational issues, it is not surprising that a variety of abstract models of parallel computation have been pursued. Closest to the hardware level are the VLSI models, which focus on the technological limits of today's chips, in which gates and wires are packed into a small number of planar layers.

812 citations

01 Jan 1993
TL;DR: In this paper, the authors describe an efficient approach to partitioning unstructured meshes that occur naturally in the finite element and finite difference methods, making use of the underlying geometric structure of a given mesh and finding a provably good partition in random O(n) time.
Abstract: This paper describes an efficient approach to partitioning unstructured meshes that occur naturally in the finite element and finite difference methods. The approach makes use of the underlying geometric structure of a given mesh and finds a provably good partition in random O(n) time. It applies to meshes in both two and three dimensions. The new method has applications in efficient sequential and parallel algorithms for large-scale problems in scientific computing. This is an overview paper written with emphasis on the algorithmic aspects of the approach. Many detailed proofs can be found in companion papers.

152 citations

Journal ArticleDOI
30 Oct 1989
TL;DR: The set cover algorithm is applied to learning theory, providing an NC algorithm for learning the concept class obtained by taking the closure under finite union or finite intersection of any concept class of finite VC dimension which has an NC hypothesis finder.
Abstract: NC approximation algorithms are given for the unweighted and weighted set cover problems. The algorithms use a linear number of processors and give a cover that has at most log n times the optimal size/weight, thus matching the performance of the best sequential algorithms. The set cover algorithm is applied to learning theory, providing an NC algorithm for learning the concept class obtained by taking the closure under finite union or finite intersection of any concept class of finite VC dimension which has an NC hypothesis finder. In addition, a linear-processor NC algorithm is given for a variant of the set cover problem and used to obtain NC algorithms for several problems in computational geometry. >

139 citations


Cites methods from "Polling: a new randomized sampling ..."

  • ...Randomization is a tool that has been used extensively to construct algorithms in computational geometry [9-12, 14, 24, 25]....

    [...]

Journal ArticleDOI
TL;DR: The design and implementation of a practical parallel algorithm for Delaunay triangulation that works well on general distributions, achieves significantly better speedups over good sequential code, does not assume a uniform distribution of points, and is widely portable due to its use of MPI as a communication mechanism.
Abstract: This paper describes the design and implementation of a practical parallel algorithm for Delaunay triangulation that works well on general distributions. Although there have been many theoretical parallel algorithms for the problem, and some implementations based on bucketing that work well for uniform distributions, there has been little work on implementations for general distributions. We use the well known reduction of 2D Delaunay triangulation to find the 3D convex hull of points on a paraboloid. Based on this reduction we developed a variant of the Edelsbrunner and Shi 3D convex hull algorithm, specialized for the case when the point set lies on a paraboloid. This simplification reduces the work required by the algorithm (number of operations) from O(n log 2 n) to O(n log n) . The depth (parallel time) is O( log 3 n) on a CREW PRAM. The algorithm is simpler than previous O(n log n) work parallel algorithms leading to smaller constants. Initial experiments using a variety of distributions showed that our parallel algorithm was within a factor of 2 in work from the best sequential algorithm. Based on these promising results, the algorithm was implemented using C and an MPI-based toolkit. Compared with previous work, the resulting implementation achieves significantly better speedups over good sequential code, does not assume a uniform distribution of points, and is widely portable due to its use of MPI as a communication mechanism. Results are presented for the IBM SP2, Cray T3D, SGI Power Challenge, and DEC AlphaCluster.

103 citations

References
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Book
01 Jan 1987
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

2,284 citations

Proceedings ArticleDOI
Kenneth L. Clarkson1
06 Jan 1988
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.

1,163 citations

Proceedings ArticleDOI
13 Oct 1975
TL;DR: The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.
Abstract: A number of seemingly unrelated problems involving the proximity of N points in the plane are studied, such as finding a Euclidean minimum spanning tree, the smallest circle enclosing the set, k nearest and farthest neighbors, the two closest points, and a proper straight-line triangulation. For most of the problems considered a lower bound of O(N log N) is shown. For all of them the best currently-known upper bound is O(N2) or worse. The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space. The Voronoi diagram is used to obtain O(N log N) algorithms for all of the problems.

1,140 citations

Journal ArticleDOI
Richard Cole1
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

Journal ArticleDOI
TL;DR: The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.
Abstract: We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ . These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.

799 citations