Parallel processing for efficient subdivision search

Dadoun, Norm; Kirkpatrick, David G.

doi:10.1145/41958.41980

Cited by 33 publications

(12 citation statements)

References 14 publications

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“…First, we know from [14] and [34] that, with high probability, we can find such an independent set W i in constant time using O(r ) processors (the idea is to use randomized symmetry breaking; see [21] for a clear exposition). Next, to determine V R i+1 (s ) ∩ V R i (s) for each s ∈ R i (s), we compute the pairwise intersection of the bisectors B(s , s ) for all s ∈ R i (s) in constant time; this determines any new neighbors of s from R i (s).…”

Section: Processing Vor(r) To Form the Search Structurementioning

confidence: 99%

“…Our approach does require efficient search (to determine subproblem inputs) and merge (to put together recursively solved subproblems) techniques because the larger sample sizes do not allow the use of a polynomial number of processors for these steps. We believe it is beneficial to shift the emphasis to these steps in the design of efficient algorithms, since general techniques for parallel search and merge have been well-studied in the literature [5], [14], [21].…”

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confidence: 99%

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Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane

Rajasekaran¹,

Ramaswami²

2002

Algorithmica

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We present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with high probability using O(n) processors on a CRCW PRAM. This algorithm is optimal in terms of work done since the sequential time bound for this problem is (n log n). Our algorithm improves by an O(log n) factor the previously best known deterministic parallel algorithm, given by Goodrich,Ó'Dúnlaing, and Yap, which runs in O(log 2 n) time using O(n) processors. We obtain this result by using a new "two-stage" random sampling technique. By choosing large samples in the first stage of the algorithm, we avoid the hurdle of problem-size "blow-up" that is typical in recursive parallel geometric algorithms. We combine the two-stage sampling technique with efficient search and merge procedures to obtain an optimal algorithm. This technique gives an alternative optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use the transformation to three-dimensional half-space intersection). Introduction.Voronoi diagrams are elegant and versatile geometric structures which have applications for a wide range of problems in computational geometry and other areas. For example, given the Voronoi diagram of a set of line segments, one can easily compute the minimum weight spanning tree of the segments, or the nearest neighbor of each line segment. Voronoi diagrams are, in general, very useful tools for solving various proximity problems efficiently. Certain two-dimensional motion planning problems can also be solved quickly when the Voronoi diagram is available [30]. (Surveys on Voronoi diagrams and their applications can be found in [7] and [16].) In a number of these applications, exploiting parallelism to obtain faster solutions is a desirable goal because real-time solutions are critical. In this paper we develop an optimal parallel randomized algorithm on a PRAM (Parallel Random Access Machine) for constructing the Voronoi diagram of a set of nonintersecting, except possibly at endpoints, line segments in the plane. The first sequential algorithm for this problem was given by Lee and Drysdale [24], which ran in O(n log 2 n) time. This run-time was later improved to O(n log n) in numerous papers [22], [35], [15], which is optimal since constructing the Voronoi diagram takes (n log n) time. (This follows from a simple reduction from sorting.) The

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Section: Processing Vor(r) To Form the Search Structurementioning

confidence: 99%

mentioning

confidence: 99%

Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane

Rajasekaran¹,

Ramaswami²

2002

Algorithmica

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“…Subdivision hierarchies can be used to construct efficient search data structures. Randomized pa.ralle1 algorithms to solve this problem efficiently were given independently by Dadoun and Kirkpatrick [9] and by Reif and Sen [23]. We can clearly use the same idea of fractional independent sets to build a hierarchy of Voronoi diagrams that will be very useful for our purposes.…”

Section: Processing Vor(r) To Form the Search Data Structure Vords(r)mentioning

confidence: 99%

Optimal parallel randomized algorithms for the Voronoi diagram of line segments in the plane and related problems

Rajasekaran

Ramaswami

1994

Proceedings of the Tenth Annual Symposium on Computational Geometry - SCG '94

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In this paper, we present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with very high probability and uses O(n) processors on a CRCW PRAM. This algorithm is optimal in terms of P.T bounds since the sequential time bound for this problem is Ω(n log n). Our algorithm improves by an O(log n) factor the previously best known deterministic parallel algorithm which runs in O(log 2 n) time using O(n) processors [13]. We obtain this result by using random sampling at "two stages" of our algorithm and using efficient randomized search techniques. This technique gives a direct optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use reduction from the 3-d convex hull construction). AbstractIn this paper, we present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n non-intersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(1ogn) time with very high probability and uses O(n) processors on a CRCW PRAM. This algorithm is optimal in terms of P.T bounds since the sequential time bound for this problem is R(n1og n). Our algorithm improves by an O(1og n) factor the previously best known deterministic parallel algorithm which runs in 0(log2n) time using O(n) processors [13]. We obtain this result by using random sampling at "two stages" of our algorithm and using efficient randomized search techniques. This technique gives a direct optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use reduction from the 3-d convex hull construction).

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“…Dadoun and Kirkpatrick [14] show that their algorithm runs in O(log n) expected parallel time. However, they do not extend their analysis for the high-probability bounds which is also O(log n) as shown in the previous theorem.…”

Section: A Parallel Algorithmmentioning

confidence: 99%

Optimal randomized parallel algorithms for computational geometry

Reif

Sen

1992

Algorithmica

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Abstract. We present parallel algorithms fcr some fundamental problems in computational geometry which have a running time of O(log n) using n processors, with very high probability (approaching 1 as n ~ oo). These include planar-point location, triangulation, and trapezoidal decomposition. We also present optimal algorithms for three-dimensional maxima and two-set dominance counting by an application of integer sorting. Most of these algorithms run on a CREW PRAM model and have optimal processor-time product which improve on the previously best-known algorithms of Atallah and Goodrich [5] for these problems. The crux of these algorithms is a useful data structure which emulates the plane-sweeping paradigm used for sequential algorithms. We extend some of the techniques used by Reischuk [26] and Reif and Valiant [25] for flashsort algorithm to perform divide and conquer in a plane very efficiently leading to the improved performance by our approach.

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Parallel processing for efficient subdivision search

Cited by 33 publications

References 14 publications

Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane

Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane

Optimal parallel randomized algorithms for the Voronoi diagram of line segments in the plane and related problems

Optimal randomized parallel algorithms for computational geometry

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