Optimal randomized parallel algorithms for computational geometry

Reif, H J; Sen, Sandeep

doi:10.1007/bf01758753

Cited by 37 publications

(11 citation statements)

References 29 publications

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“…Subdivision hierarchies are used to construct efficient search data structures. Randomized algorithms to construct such hierarchies were given independently by Dadoun and Kirkpatrick [14] and by Reif and Sen [34]. We can clearly use the same idea of fractional independent sets to build a hierarchy of Voronoi diagrams that is very useful for our purposes.…”

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

confidence: 99%

“…The following is an important theorem. THEOREM 2.1 [34]. Given a process tree that has the property that a procedure at depth i from the root takes time T i such that, for fixed positive constants k and α, and some 0 < ε < 1, The number of levels in the process tree is O(log log n).…”

Section: Preliminaries Definitions Andmentioning

confidence: 99%

“…Cole and Goodrich give further applications of this technique in [12]. Reif and Sen [34] also obtained optimal randomized parallel algorithms for a number of these problems; these algorithms use n processors and run in O(log n) time with high probability.…”

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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%

Section: Preliminaries Definitions Andmentioning

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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“…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%

“…Cole and Goodrich give further applications of this technique in [8]. Reif and Sen [23] also obtained optimal randomized parallel algorithms for a number of these problems; these algorithms use n processors and run in O(log n) time with very high probability.…”

Section: Introductionmentioning

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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Derandomization in Computational Geometry

Matoušek

1996

Journal of Algorithms

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Optimal randomized parallel algorithms for computational geometry

Cited by 37 publications

References 29 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

Derandomization in Computational Geometry

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