Time-space trade-offs for triangulations and Voronoi diagrams

Matias Korman; Wolfgang Mulzer; André Van Renssen; Marcel Roeloffzen; Paul Seiferth; Yannik Stein

doi:10.1007/978-3-319-21840-3_40

Time-space trade-offs for triangulations and Voronoi diagrams

Matias Korman, Wolfgang Mulzer, André Van Renssen, Marcel Roeloffzen, Paul Seiferth, Yannik Stein

INF - System Information Sciences

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

6 Citations (Scopus)

Abstract

Let S be a planar n-point set. A triangulation for S is a maximal plane straight-line graph with vertex set S. The Voronoi diagram for S is the subdivision of the plane into cells such that each cell has the same nearest neighbors in S. Classically, both structures can be computed in O(n log n) time and O(n) space. We study the situation when the available workspace is limited: given a parameter s ∈ {1,..., n}, an s-workspace algorithm has read-only access to an input array with the points from S in arbitrary order, and it may use only O(s) additional words of Θ(log n) bits for reading and writing intermediate data. The output should then be written to a write-only structure. We describe a deterministic s-workspace algorithm for computing a triangulation of S in time O(n²/s + n log n log s) and a randomized s-workspace algorithm for finding the Voronoi diagram of S in expected time O((n²/s) log s + n log s log^∗ s).

Original language	English
Title of host publication	Algorithms and Data Structures - 14th International Symposium, WADS 2015, Proceedings
Editors	Frank Dehne, Jorg-Rudiger Sack, Ulrike Stege
Publisher	Springer Verlag
Pages	482-494
Number of pages	13
ISBN (Print)	9783319218397
DOIs	https://doi.org/10.1007/978-3-319-21840-3_40
Publication status	Published - 2015
Event	14th International Symposium on Algorithms and Data Structures, WADS 2015 - Victoria, Canada Duration: 2015 Aug 5 → 2015 Aug 7

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	9214
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	14th International Symposium on Algorithms and Data Structures, WADS 2015
Country/Territory	Canada
City	Victoria
Period	15/8/5 → 15/8/7

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/978-3-319-21840-3_40

Cite this

Korman, M., Mulzer, W., Van Renssen, A., Roeloffzen, M., Seiferth, P., & Stein, Y. (2015). Time-space trade-offs for triangulations and Voronoi diagrams. In F. Dehne, J-R. Sack, & U. Stege (Eds.), Algorithms and Data Structures - 14th International Symposium, WADS 2015, Proceedings (pp. 482-494). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9214). Springer Verlag. https://doi.org/10.1007/978-3-319-21840-3_40

Time-space trade-offs for triangulations and Voronoi diagrams. / Korman, Matias; Mulzer, Wolfgang; Van Renssen, André et al.

Algorithms and Data Structures - 14th International Symposium, WADS 2015, Proceedings. ed. / Frank Dehne; Jorg-Rudiger Sack; Ulrike Stege. Springer Verlag, 2015. p. 482-494 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9214).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Korman, M, Mulzer, W, Van Renssen, A, Roeloffzen, M, Seiferth, P & Stein, Y 2015, Time-space trade-offs for triangulations and Voronoi diagrams. in F Dehne, J-R Sack & U Stege (eds), Algorithms and Data Structures - 14th International Symposium, WADS 2015, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9214, Springer Verlag, pp. 482-494, 14th International Symposium on Algorithms and Data Structures, WADS 2015, Victoria, Canada, 15/8/5. https://doi.org/10.1007/978-3-319-21840-3_40

Korman M, Mulzer W, Van Renssen A, Roeloffzen M, Seiferth P, Stein Y. Time-space trade-offs for triangulations and Voronoi diagrams. In Dehne F, Sack J-R, Stege U, editors, Algorithms and Data Structures - 14th International Symposium, WADS 2015, Proceedings. Springer Verlag. 2015. p. 482-494. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-21840-3_40

Korman, Matias ; Mulzer, Wolfgang ; Van Renssen, André et al. / Time-space trade-offs for triangulations and Voronoi diagrams. Algorithms and Data Structures - 14th International Symposium, WADS 2015, Proceedings. editor / Frank Dehne ; Jorg-Rudiger Sack ; Ulrike Stege. Springer Verlag, 2015. pp. 482-494 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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N2 - Let S be a planar n-point set. A triangulation for S is a maximal plane straight-line graph with vertex set S. The Voronoi diagram for S is the subdivision of the plane into cells such that each cell has the same nearest neighbors in S. Classically, both structures can be computed in O(n log n) time and O(n) space. We study the situation when the available workspace is limited: given a parameter s ∈ {1,..., n}, an s-workspace algorithm has read-only access to an input array with the points from S in arbitrary order, and it may use only O(s) additional words of Θ(log n) bits for reading and writing intermediate data. The output should then be written to a write-only structure. We describe a deterministic s-workspace algorithm for computing a triangulation of S in time O(n2/s + n log n log s) and a randomized s-workspace algorithm for finding the Voronoi diagram of S in expected time O((n2/s) log s + n log s log∗ s).

AB - Let S be a planar n-point set. A triangulation for S is a maximal plane straight-line graph with vertex set S. The Voronoi diagram for S is the subdivision of the plane into cells such that each cell has the same nearest neighbors in S. Classically, both structures can be computed in O(n log n) time and O(n) space. We study the situation when the available workspace is limited: given a parameter s ∈ {1,..., n}, an s-workspace algorithm has read-only access to an input array with the points from S in arbitrary order, and it may use only O(s) additional words of Θ(log n) bits for reading and writing intermediate data. The output should then be written to a write-only structure. We describe a deterministic s-workspace algorithm for computing a triangulation of S in time O(n2/s + n log n log s) and a randomized s-workspace algorithm for finding the Voronoi diagram of S in expected time O((n2/s) log s + n log s log∗ s).

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Time-space trade-offs for triangulations and Voronoi diagrams

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