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11
Randomized Algorithms for Binary Search and Load Balancing on Fixed Connection Networks With Geometric Applications
, 1990
"... There are now a number of fundamental problems in computational geometry that have optimal algorithms on PRAM models. We present randomized parallel algorithms which execute on an nprocessor buttery interconnection network in O(log n) time for the following problems of input size n: trapezoidal ..."
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Cited by 20 (3 self)
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There are now a number of fundamental problems in computational geometry that have optimal algorithms on PRAM models. We present randomized parallel algorithms which execute on an nprocessor buttery interconnection network in O(log n) time for the following problems of input size n: trapezoidal decomposition, visibility, triangulation and 2D convex hull. These algorithms involve tackling some of the very basic problems like binary search and loadbalancing that we take for granted in PRAM models. Apart from a 2D convex hull algorithm, these are the rst nontrivial geometric algorithms which attain this performance on xed connection networks. Our techniques use a number of ideas from Flashsort which have to be modied to handle more dicult situations; it seems likely that they will have wider applications.
Implementing data structures on a hypercube multiprocessor, and applications in parallel computational geometry
 Journal of Parallel and Distributed Computing
, 1990
"... Abstract, In this paper, we study the problem of implementing standard data structures on a hypercube multiprocessor. We present a technique for fficiently executing multiple independant search processes on a class of graphs called ordered hlevel graphs. We show how this technique can be utilized t ..."
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Cited by 18 (6 self)
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Abstract, In this paper, we study the problem of implementing standard data structures on a hypercube multiprocessor. We present a technique for fficiently executing multiple independant search processes on a class of graphs called ordered hlevel graphs. We show how this technique can be utilized to implement a segment tree on a hypercube, thereby obtainig O(logZn) time algorithms for sotving the next element search problem, the trapezoidal decomposition probtem, the triangulation problem, and the (multiple) planar point location problem. 1
Lower bounds for intersection searching and fractional cascading in higher dimension
, 2003
"... Given an nedge convex subdivision of the plane, is it possible to report its k intersections with a query line segment in Oðk þ polylogðnÞÞ time, using subquadratic storage? If the query is a plane and the input is a polytope with n vertices, can one achieve Oðk þ polylogðnÞÞ time with subcubic sto ..."
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Cited by 11 (0 self)
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Given an nedge convex subdivision of the plane, is it possible to report its k intersections with a query line segment in Oðk þ polylogðnÞÞ time, using subquadratic storage? If the query is a plane and the input is a polytope with n vertices, can one achieve Oðk þ polylogðnÞÞ time with subcubic storage? Does any convex polytope have a boundary dominant Dobkin–Kirkpatrick hierarchy? Can fractional cascading be generalized to planar maps instead of linear lists? We prove that the answer to all of these questions is no, and we derive nearoptimal solutions to these classical problems.
Parallel Computational Geometry: An approach using randomization
 IN HANDBOOK OF COMPUTATIONAL GCOMETRY, EDITED BY J.R. SACK AND
, 1998
"... 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. ..."
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Cited by 5 (0 self)
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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 that result in very general strategies for solving a wide class of fundamental problems from computational geometry like convex hulls, voronoi diagrams, triangulation, pointlocation and arrangements. Our description emphasizes the algorithmic techniques rather than a detailed treatment of the individual problems.
PARALLEL ALGORITHMS IN GEOMETRY
"... The goal of parallel algorithm design is to develop parallel computational methods that run very fast with as few processors as possible, and there is an extensive literature of such algorithms for computational geometry problems. There are several different parallel computing models, and in order t ..."
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Cited by 3 (0 self)
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The goal of parallel algorithm design is to develop parallel computational methods that run very fast with as few processors as possible, and there is an extensive literature of such algorithms for computational geometry problems. There are several different parallel computing models, and in order to maintain a focus in this
Highly Parallelizable Problems (Extended Abstract)
"... We establish that several problems are highly parallelizable. For each of these problems, we design an optimal O (loglogn ) time parallel algorithm on the Common CRCW PRAM model which is the weakest among the CRCW PRAM models. These problems include: # all nearest smaller values, # preprocessing ..."
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Cited by 3 (0 self)
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We establish that several problems are highly parallelizable. For each of these problems, we design an optimal O (loglogn ) time parallel algorithm on the Common CRCW PRAM model which is the weakest among the CRCW PRAM models. These problems include: # all nearest smaller values, # preprocessing for answering range maxima queries, # several problems in Computational Geometry, # string matching. Until recently, such algorithms were known only for finding the maximum and merging. A new lower bound technique is presented showing that some of the new O (loglogn ) upper bounds cannot be improved even when non optimal algorithms are used. The technique extends Ramseylike lower bound argumentation due to auf der Heide and Wigderson [MW85]. Its most interesting applications are for Computational Geometry problems for which no previous lower bounds are known.
(Preliminary Version)
"... Randomized algorithms for binary search and load balancing on fixed connection networks ..."
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Randomized algorithms for binary search and load balancing on fixed connection networks
Summary of Results
, 1990
"... In this paper we show that it is impossible to solve a number of "natural" 2dimensional geometric problems in paIyIog time with a polynomial number of processors (unless P = NC). Thus, we disprove a popular belief that there are no natural Pcomplete geometric problems in the plane. The p ..."
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In this paper we show that it is impossible to solve a number of "natural" 2dimensional geometric problems in paIyIog time with a polynomial number of processors (unless P = NC). Thus, we disprove a popular belief that there are no natural Pcomplete geometric problems in the plane. The problems we address include instances of polygon triangulation, planar partitioning, and geometric layering. Our results are based on nontrivial reductions from the monotone circuit value and planar circuit value problems. 1
Parallel Algorithms for Evaluating Sequences of SetManipulation Operations
"... Given an offline sequence S of n setmanipulation operations, we investigate the parallel complexity of evaluating S (Le., finding the response to every operation in S and returning the resulting set). We show that the problem of evaluating S is in N C for various combinations of common setmanipu ..."
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Given an offline sequence S of n setmanipulation operations, we investigate the parallel complexity of evaluating S (Le., finding the response to every operation in S and returning the resulting set). We show that the problem of evaluating S is in N C for various combinations of common setmanipulation operations. Once we establish membership in NO (or, if membership in NO is obvious), we develop techniques for improving the time and/or processor