Article

Free access

Constructing levels in arrangements and higher order Voronoi diagrams

Authors:

Pankaj K. Agarwal,

Jiří Matoušek,

Otfried SchwarzkopfAuthors Info & Claims

SCG '94: Proceedings of the tenth annual symposium on Computational geometry

Pages 67 - 75

https://doi.org/10.1145/177424.177521

Published: 10 June 1994 Publication History

Abstract

We give a simple lazy randomized incremental algorithm to compute ≤k-levels in arrangements of x-monotone Jordan curves in the plane, and in arrangements of planes in three-dimensional space. If each pair of curves intersects in at most s points, the expected running time of the algorithm is O(k²λ_s(n/k)+min(λ_s(n)log²n,k²λ_s(n/k)logn)). For the three-dimensional case the expected running time is O(nk²+min(nlog³n,nk²logn)). The algorithm also works for computing the ≤k-level in a set of discs, with an expected running time of O(nk+min(nlog²n,nklogn)). Furthermore, we give a simple algorithm for computing the order-k Voronoi diagram of a set of n points in the plane that runs in expected time O(k(n−k)logn+nlog³n).

References

[1]

F. Aurenhammer and O. Schwarzkopf. A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams. Inter'nat. J. Comput. Geom. Appl., 2:363-381, 1992.

[2]

N. Alon and J. Spencer. The Probabilistic Method. J. Wiley & Sons, 1993.

[3]

P.K. Agarwal, M. Sharir, and P. Shor. Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences. J. Combin. Theory Ser. A, 52:228-274, 1989.

Digital Library

[4]

J.-D. Boissonnat, O. Devillers, and M. Teillaud. An semidynamic construction of higher-order Voronoi diagrams and its randomized analysis. Algorithmica, 9:329-356, 1993.

[5]

B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kthorder Voronoi diagrams. IEEE Trans. Comput., C-36:1349-1354, 1987.

Digital Library

[6]

K.L. Clarkson. New applications of random sampling in computational geometry. Discrete Comput. Geom., 2:195-222, 1987.

Digital Library

[7]

K.L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Cornput., 17:830-847, 1988.

Digital Library

[8]

K.L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4:387-421, 1989.

Digital Library

[9]

R. Cole, M. Sharir, and C. K. Yap. On khulls and related problems. SIAM J. Cornput., 16:61-77, 1987.

Digital Library

[10]

M. de Berg, K. Dobrindt, and O. Schwarzkopf. On lazy randomized incremental construction. In Proc. 25th Annu. A CM Sympos. Theory Comput., 1994.

Digital Library

[11]

H. Edelsbrunner. Algorithms in Combinatorial Geometry, volume 10 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Heidelberg, West Germany, 1987.

Digital Library

[12]

H. Everett, J.-M. Robert, and M. van Kreveld. An optimal algorithm for the (<_k)-levels, with applications to separation and transversal problems. In Proc. 9th Annu. A CM Sympos. Comput. Geom., pages 38-46, 1993.

Digital Library

[13]

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Cornput. Geom., 2:127-151, 1987.

Digital Library

[14]

K. Mulmuley. A fast planar partition algorithm, ii. J. A CM, 38:74-103, 1991.

Digital Library

[15]

K. Mulmuley. On levels in arrangements and Voronoi diagrams. Discrete Comput. Geom., 6:307-338, 1991.

Digital Library

[16]

K. Mulmuley. Computational Geometry: An Introduction Through Randomized Algorithms. Prentice Hall, New 'York, 1993.

[17]

J. Pach, W. Steiger, and E. Szemer~di. An upper bound on the number of planar ksets. Discrete Comput. Geom., 7:109-123, 1992.

Digital Library

[18]

M. Sharir. On k-sets in arrangements of curves and surfaces. Discrete Comput. Geom., 6:593-613, 1991.

Digital Library

Cited By

Claverol Mde las Heras Parrilla AHuemer CMartínez-Moraian A(2024)The edge labeling of higher order Voronoi diagramsJournal of Global Optimization10.1007/s10898-024-01386-090:2(515-549)Online publication date: 21-May-2024
https://doi.org/10.1007/s10898-024-01386-0
Dewri RThurimella R(2014)Exploiting Service Similarity for Privacy in Location-Based Search QueriesIEEE Transactions on Parallel and Distributed Systems10.1109/TPDS.2013.3425:2(374-383)Online publication date: 1-Feb-2014
https://dl.acm.org/doi/10.1109/TPDS.2013.34
Weller RWeller R(2013)Sphere Packings for Arbitrary ObjectsNew Geometric Data Structures for Collision Detection and Haptics10.1007/978-3-319-01020-5_4(91-112)Online publication date: 2013
https://doi.org/10.1007/978-3-319-01020-5_4
Show More Cited By

Index Terms

Constructing levels in arrangements and higher order Voronoi diagrams
1. Mathematics of computing
  1. Probability and statistics
    1. Probabilistic algorithms
    2. Probabilistic reasoning algorithms
      1. Markov-chain Monte Carlo methods
      2. Sequential Monte Carlo methods
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Recommendations

Constructing Levels in Arrangements and Higher Order Voronoi Diagrams
Constructing Levels in Arrangements and Higher Order Voronoi Diagrams

We give simple randomized incremental algorithms for computing the Amk-level in an arrangement of n lines in the plane or in an arrangement of n planes in $\Reals^3$. The expected running time of our algorithms is $O(nk+n\alpha(n)\log n)$ for the planar ...
Constructing arrangements optimally in parallel

We give two optimal parallel algorithms for constructing the arrangement ofn lines in the plane. The first nethod is quite simple and runs inO(log2n) time usingO(n2) work, and the second method, which is more sophisticated, runs inO(logn) time usingO(n2)...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SCG '94: Proceedings of the tenth annual symposium on Computational geometry

June 1994

400 pages

ISBN:0897916484

DOI:10.1145/177424

Chairman:
Kurt Mehlhorn
U.S. Army Research Office

Copyright © 1994 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 10 June 1994

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

SCG94

Sponsor:

SCG94: Tenth Symposium on Computational Geometry

June 6 - 8, 1994

New York, Stony Brook, USA

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

12
Total Citations
View Citations
517
Total Downloads

Downloads (Last 12 months)68
Downloads (Last 6 weeks)5

Reflects downloads up to 24 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Claverol Mde las Heras Parrilla AHuemer CMartínez-Moraian A(2024)The edge labeling of higher order Voronoi diagramsJournal of Global Optimization10.1007/s10898-024-01386-090:2(515-549)Online publication date: 21-May-2024
https://doi.org/10.1007/s10898-024-01386-0
Dewri RThurimella R(2014)Exploiting Service Similarity for Privacy in Location-Based Search QueriesIEEE Transactions on Parallel and Distributed Systems10.1109/TPDS.2013.3425:2(374-383)Online publication date: 1-Feb-2014
https://dl.acm.org/doi/10.1109/TPDS.2013.34
Weller RWeller R(2013)Sphere Packings for Arbitrary ObjectsNew Geometric Data Structures for Collision Detection and Haptics10.1007/978-3-319-01020-5_4(91-112)Online publication date: 2013
https://doi.org/10.1007/978-3-319-01020-5_4
Palmer J(2009)Higher Order Voronoi Diagrams and Distance Functions in Art and VisualizationGeneralized Voronoi Diagram: A Geometry-Based Approach to Computational Intelligence10.1007/978-3-540-85126-4_12(267-283)Online publication date: 2009
https://doi.org/10.1007/978-3-540-85126-4_12
Palmer J(2007)High Order Voronoi SculptureProceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering10.1109/ISVD.2007.28(110-116)Online publication date: 9-Jul-2007
https://dl.acm.org/doi/10.1109/ISVD.2007.28
Palmer J(2006)Using Line and Texture to Visualize Higher-Order Voronoi DiagramsProceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering10.1109/ISVD.2006.36(166-172)Online publication date: 2-Jul-2006
https://dl.acm.org/doi/10.1109/ISVD.2006.36
Basch JGuibas LRamkumar G(2005)Reporting red-blue intersections between two sets of connected line segmentsAlgorithms — ESA '9610.1007/3-540-61680-2_64(302-319)Online publication date: 6-Jun-2005
https://doi.org/10.1007/3-540-61680-2_64
Nagai TTokura N(2001)Tight Error Bound of Goemetric Problems on Convex Objects with Imprecise CoordinatesDiscrete and Computational Geometry10.1007/3-540-47738-1_24(252-263)Online publication date: 20-Sep-2001
https://doi.org/10.1007/3-540-47738-1_24
Agarwal PArge LErickson JFranciosa PVitter JMendelson AParedaens J(1998)Efficient searching with linear constraintsProceedings of the seventeenth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems10.1145/275487.275506(169-178)Online publication date: 1-May-1998
https://dl.acm.org/doi/10.1145/275487.275506
Agarwal PEfrat ASharir MSnoeyink J(1995)Vertical decomposition of shallow levels in 3-dimensional arrangements and its applicationsProceedings of the eleventh annual symposium on Computational geometry10.1145/220279.220284(39-50)Online publication date: 1-Sep-1995
https://dl.acm.org/doi/10.1145/220279.220284
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Table of Contents