skip to main content
10.1145/1998196.1998254acmconferencesArticle/Chapter ViewAbstractPublication PagessocgConference Proceedingsconference-collections
research-article

Kinetic mesh refinement in 2D

Published: 13 June 2011 Publication History

Abstract

We provide a kinetic data structure (KDS) to the planar kinetic mesh refinement problem, which concerns computation of meshes of continuously moving points. Our KDS computes the Delaunay triangulation of a size-optimal well-spaced superset of a set of moving points with algebraic trajectories of constant degree. Our KDS is compact, requiring linear space in the size of the output. It is local, using a point in O(log Delta) certificates. It is responsive, repairing itself in O(log Delta) time per event. It is efficient, processing O(n2 log3 Delta) events in the worst case; this is optimal up to a polylogarithmic factor. Also, our KDS is dynamic, responding to point insertions and deletions in O(log Delta) time. In our bounds Delta stands for the geometric spread, the ratio of the diameter to the closest pair distance. To the best of our knowledge, this is the first KDS for mesh refinement.

References

[1]
U. A. Acar, G. E. Blelloch, M. Blume, R. Harper, and K. Tangwongsan. An experimental analysis of self-adjusting computation. ACM Trans. Prog. Lang. Sys.}, 32(1):3:1--3:53, 2009.
[2]
U. A. Acar, G. E. Blelloch, R. Ley-Wild, K. Tangwongsan, and D. Türkoglu. Traceable data types for self-adjusting computation. In Programming Language Design and Implementation, 2010.
[3]
U. A. Acar, G. E. Blelloch, K. Tangwongsan, and D. Türkoglu. Robust kinetic convex hulls in 3D. In European Symposium on Algorithms, September 2008.
[4]
U. A. Acar, A. Cotter, B. Hudson, and D. Türkoglu. Dynamic well-spaced point sets. In Symposium on Computational Geometry, 2010.
[5]
U. A. Acar, B. Hudson, G. L. Miller, and T. Phillips. SVR: Practical engineering of a fast 3d meshing algorithm. Proceedings of the Sixteenth International Meshing Roundtable (IMR).
[6]
P. K. Agarwal, J. Gao, L. J. Guibas, H. Kaplan, V. Koltun, N. Rubin, and M. Sharir. Kinetic stable delaunay graphs. In Symposium on Computational Geometry, 2010.
[7]
P. K. Agarwal, Y. Wang, and H. Yu. A two-dimensional kinetic triangulation with near-quadratic topological changes. Discrete & Computational Geometry, 36(4):573--592, 2006.
[8]
J. Basch, L. J. Guibas, and J. Hershberger. Data structures for mobile data. Journal of Algorithms, 31(1):1--28, 1999.
[9]
M. Bern, D. Eppstein, and J. R. Gilbert. Provably Good Mesh Generation. Journal of Computer and System Sciences, 48(3):384--409, 1994.
[10]
S. W. Cheng, T. K. Dey, H. Edelsbrunner, M. A. Facello, and S.-H. Teng. Sliver Exudation. Journal of the ACM, 47(5):883--904, 2000.
[11]
T. K. Dey. Curve and Surface Reconstruction. Cambridge Univ. Press, 2006.
[12]
H. Edelsbrunner. Triangulations and meshes in computational geometry. Acta Numerica, 9:133--213, 2000.
[13]
J. J. Fu and R. C. T. Lee. Voronoi diagrams of moving points in the plane. In FST and TC 10, pages 238--254, New York, NY, USA, 1990.
[14]
J. Gao, L. J. Guibas, and A. Nguyen. Deformable spanners and applications. Computational Geometry: Theory and Applications, 35(1):2--19, 2006.
[15]
L. J. Guibas. Kinetic data structures: a state of the art report. In Workshop on algo. found. of robotics, pages 191--209, Natick, MA, USA, 1998.
[16]
L. J. Guibas, J. S. B. Mitchell, and T. Roos. Voronoi diagrams of moving points in the plane. In 17th Intl. Workshop Graph-Theoretic Concepts Computer Science, pages 113--209. Springer-Verlag, Inc., 1992.
[17]
M. A. Hammer, U. A. Acar, and Y. Chen. CEAL: a C-based language for self-adjusting computation. In Programming Language Design and Imp., June 2009.
[18]
B. Hudson, G. L. Miller, and T. Phillips. Sparse Voronoi Refinement. In 15th International Meshing Roundtable, pages 339--356, 2006.
[19]
B. Hudson, G. L. Miller, T. Phillips, and D. R. Sheehy. Size complexity of volume meshes vs. surface meshes. In Symposium on Discrete Algorithms, 2009.
[20]
H. Kaplan, N. Rubin, and M. Sharir. A kinetic triangulation scheme for moving points in the plane. In Symposium on Computational Geometry, 2010.
[21]
R. Ley-Wild, U. A. Acar, and M. Fluet. A cost semantics for self-adjusting computation. In Principles of Programming Languages, 2009.
[22]
J. Ruppert. A Delaunay refinement algorithm for quality 2-dimensional mesh generation. Journal of Algorithms, 18(3):548--585, 1995.
[23]
D. Talmor. Well-Spaced Points for Numerical Methods. PhD thesis, Carnegie Mellon University, August 1997.

Cited By

View all
  • (2018)An Efficient Kinetic Range Query for One Dimensional Axis Parallel SegmentsInternational Journal of Intelligent Information Technologies10.4018/IJIIT.201801010414:1(48-62)Online publication date: 1-Jan-2018
  • (2017)Brief AnnouncementProceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3087556.3087595(275-277)Online publication date: 24-Jul-2017
  • (2011)Parallelism in dynamic well-spaced point setsProceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures10.1145/1989493.1989498(33-42)Online publication date: 4-Jun-2011

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences
SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry
June 2011
532 pages
ISBN:9781450306829
DOI:10.1145/1998196
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: 13 June 2011

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. deformable spanners
  2. kinetic data structures
  3. mesh refinement
  4. self-adjusting computation
  5. well-spaced point sets

Qualifiers

  • Research-article

Conference

SoCG '11
SoCG '11: Symposium on Computational Geometry
June 13 - 15, 2011
Paris, France

Acceptance Rates

Overall Acceptance Rate 625 of 1,685 submissions, 37%

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)1
  • Downloads (Last 6 weeks)0
Reflects downloads up to 22 Feb 2025

Other Metrics

Citations

Cited By

View all
  • (2018)An Efficient Kinetic Range Query for One Dimensional Axis Parallel SegmentsInternational Journal of Intelligent Information Technologies10.4018/IJIIT.201801010414:1(48-62)Online publication date: 1-Jan-2018
  • (2017)Brief AnnouncementProceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3087556.3087595(275-277)Online publication date: 24-Jul-2017
  • (2011)Parallelism in dynamic well-spaced point setsProceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures10.1145/1989493.1989498(33-42)Online publication date: 4-Jun-2011

View Options

Login options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Figures

Tables

Media

Share

Share

Share this Publication link

Share on social media