Merging free trees in parallel for efficient voronoi diagram construction

Richard Cole, Michael T. Goodrich, Colm Dúnlaing

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper describes a new approach for constructing the Voronoi diagram of n points in the plane in parallel. Our approach is based on a divide-and-conquer procedure where we implement the “marry” step by merging forests of free trees (to build the “contour” between the subproblem solutions) in O(log log n) time. This merging procedure is based an a √n -divide-and-merge technique reminiscent of the list-merging approach of Valiant. Our method also involves an optimal parallel method for computing the proximity envelope of a point set with respect to a given line. This structure facilitates the use of our fast mering procedure, for it allows the divide-and-conquer procedure to continue without needing to explicitly remove edges of recursively constructed diagrams that are not part of the final diagram. We use this approach to derive two results regarding the deterministic parallel construction of a Voronoi diagram. Specifically, we show that one can solve the Voronoi diagram problem in O(log n log log n) time and O(n log2n) work (which improves the previous time bound while maintaining the same work bound) or, alternatively, in O(log2n) time and O(n log n) work (which improves the previous work bound while maintaining the same time bound). Our model of computation is the CREW PRAM.

Original languageEnglish (US)
Title of host publicationAutomata, Languages and Programming - l7th International Colloquium, Proceedings
PublisherSpringer Verlag
Pages432-445
Number of pages14
Volume443 LNCS
ISBN (Print)9783540528265
StatePublished - 1990
Event17th International Colloquium on Automata, Languages and Programming, 1990 - Warwick, United Kingdom
Duration: Jul 16 1990Jul 20 1990

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume443 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other17th International Colloquium on Automata, Languages and Programming, 1990
CountryUnited Kingdom
CityWarwick
Period7/16/907/20/90

Fingerprint

Voronoi Diagram
Merging
Divide and conquer
Diagram
Parallel Methods
Models of Computation
Point Sets
Envelope
Proximity
Divides
Continue
Computing
Line

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Cole, R., Goodrich, M. T., & Dúnlaing, C. (1990). Merging free trees in parallel for efficient voronoi diagram construction. In Automata, Languages and Programming - l7th International Colloquium, Proceedings (Vol. 443 LNCS, pp. 432-445). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 443 LNCS). Springer Verlag.

Merging free trees in parallel for efficient voronoi diagram construction. / Cole, Richard; Goodrich, Michael T.; Dúnlaing, Colm.

Automata, Languages and Programming - l7th International Colloquium, Proceedings. Vol. 443 LNCS Springer Verlag, 1990. p. 432-445 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 443 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Cole, R, Goodrich, MT & Dúnlaing, C 1990, Merging free trees in parallel for efficient voronoi diagram construction. in Automata, Languages and Programming - l7th International Colloquium, Proceedings. vol. 443 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 443 LNCS, Springer Verlag, pp. 432-445, 17th International Colloquium on Automata, Languages and Programming, 1990, Warwick, United Kingdom, 7/16/90.
Cole R, Goodrich MT, Dúnlaing C. Merging free trees in parallel for efficient voronoi diagram construction. In Automata, Languages and Programming - l7th International Colloquium, Proceedings. Vol. 443 LNCS. Springer Verlag. 1990. p. 432-445. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
Cole, Richard ; Goodrich, Michael T. ; Dúnlaing, Colm. / Merging free trees in parallel for efficient voronoi diagram construction. Automata, Languages and Programming - l7th International Colloquium, Proceedings. Vol. 443 LNCS Springer Verlag, 1990. pp. 432-445 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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