Document Open Access Logo

Improved Time-Space Trade-Offs for Computing Voronoi Diagrams

Authors Bahareh Banyassady, Matias Korman, Wolfgang Mulzer, André van Renssen, Marcel Roeloffzen, Paul Seiferth, Yannik Stein



PDF
Thumbnail PDF

File

LIPIcs.STACS.2017.9.pdf
  • Filesize: 0.57 MB
  • 14 pages

Document Identifiers

Author Details

Bahareh Banyassady
Matias Korman
Wolfgang Mulzer
André van Renssen
Marcel Roeloffzen
Paul Seiferth
Yannik Stein

Cite AsGet BibTex

Bahareh Banyassady, Matias Korman, Wolfgang Mulzer, André van Renssen, Marcel Roeloffzen, Paul Seiferth, and Yannik Stein. Improved Time-Space Trade-Offs for Computing Voronoi Diagrams. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 9:1-9:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.STACS.2017.9

Abstract

Let P be a planar n-point set in general position. For k between 1 and n-1, the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P. The (nearest point) Voronoi diagram (NVD) and the farthest point Voronoi diagram (FVD) are the particular cases of k=1 and k=n-1, respectively. It is known that the family of all higher-order Voronoi diagrams of order 1 to K for P can be computed in total time O(n K^2 + n log n) using O(K^2(n-K)) space. Also NVD and FVD can be computed in O(n log n) time using O(n) space. For s in {1, ..., n}, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words of Theta(log n) bits each for reading and writing intermediate data. The output can be written only once and cannot be accessed afterwards. We describe a deterministic s-workspace algorithm for computing an NVD and also an FVD for P that runs in O((n^2/s) log s) time. Moreover, we generalize our s-workspace algorithm for computing the family of all higher-order Voronoi diagrams of P up to order K in O(sqrt(s)) in total time O( (n^2 K^6 / s) log^(1+epsilon)(K) (log s / log K)^(O(1)) ) for any fixed epsilon > 0. Previously, for Voronoi diagrams, the only known s-workspace algorithm was to find an NVD for P in expected time O((n^2/s) log s + n log s log^*s). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.
Keywords
  • memory-constrained model
  • Voronoi diagram
  • time-space trade-off

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Alok Aggarwal, Leonidas J. Guibas, James B. Saxe, and Peter W. Shor. A linear-time algorithm for computing the voronoi diagram of a convex polygon. Discrete Comput. Geom., 4:591-604, 1989. Google Scholar
  2. Boris Aronov, Matias Korman, Simon Pratt, André van Renssen, and Marcel Roeloffzen. Time-space trade-offs for triangulating a simple polygon. In Proc. 15th Scand. Sympos. Workshops Algorithm Theory (SWAT), volume 53, pages 30:1-30:12, 2016. Google Scholar
  3. Sanjeev Arora and Boaz Barak. Computational Complexity.A modern approach. Cambridge University Press, 2009. Google Scholar
  4. Tetsuo Asano, Kevin Buchin, Maike Buchin, Matias Korman, Wolfgang Mulzer, Günter Rote, and André Schulz. Memory-constrained algorithms for simple polygons. Comput. Geom., 46(8):959-969, 2013. Google Scholar
  5. Tetsuo Asano and David Kirkpatrick. Time-space tradeoffs for all-nearest-larger-neighbors problems. In Proc. 13th Int. Symp. Algorithms and Data Structures (WADS), pages 61-72, 2013. Google Scholar
  6. Tetsuo Asano, Wolfgang Mulzer, Günter Rote, and Yajun Wang. Constant-work-space algorithms for geometric problems. J. of Comput. Geom., 2(1):46-68, 2011. Google Scholar
  7. Tetsuo Asano, Wolfgang Mulzer, and Yajun Wang. Constant-work-space algorithms for shortest paths in trees and simple polygons. J. Graph Algorithms Appl., 15(5):569-586, 2011. Google Scholar
  8. Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi diagrams and Delaunay triangulations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. Google Scholar
  9. Yeganeh Bahoo, Bahareh Banyassady, Prosenjit Bose, Stephane Durocher, and Wolfgang Mulzer. Finding the k-visibility region of a point in a simple polygon in the memory-constrained model. In Proc. 32nd European Workshop Comput. Geom. (EWCG), 2016. Google Scholar
  10. Luis Barba, Matias Korman, Stefan Langerman, Kunihiko Sadakane, and Rodrigo I. Silveira. Space-time trade-offs for stack-based algorithms. Algorithmica, 72(4):1097-1129, 2015. Google Scholar
  11. Luis Barba, Matias Korman, Stefan Langerman, and Rodrigo I. Silveira. Computing the visibility polygon using few variables. Comput. Geom., 47(9):918-926, 2013. Google Scholar
  12. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational geometry. Algorithms and applications. Springer-Verlag, third edition, 2008. Google Scholar
  13. Allan Borodin and Stephen A. Cook. A time-space tradeoff for sorting on a general sequential model of computation. SIAM J. Comput., 11:287-297, 1982. Google Scholar
  14. Hervé Brönnimann, Timothy M. Chan, and Eric Y. Chen. Towards in-place geometric algorithms and data structures. In Proc. 20th Annu. Sympos. Comput. Geom. (SoCG), pages 239-246, 2004. Google Scholar
  15. Timothy M Chan. Remarks on k-level algorithms in the plane. manuscript, Univ. of Waterloo, 1999. Google Scholar
  16. Timothy M Chan. Random sampling, halfspace range reporting, and construction of (≤ k)-levels in three dimensions. SIAM J. Comput., 30(2):561-575, 2000. Google Scholar
  17. Timothy M. Chan and Eric Y. Chen. Multi-pass geometric algorithms. Discrete Comput. Geom., 37(1):79-102, 2007. Google Scholar
  18. Omar Darwish and Amr Elmasry. Optimal time-space tradeoff for the 2D convex-hull problem. In Proc. 22nd Annu. European Sympos. Algorithms (ESA), pages 284-295, 2014. Google Scholar
  19. Amr Elmasry and Frank Kammer. Space-efficient plane-sweep algorithms. arXiv : 1507.01767, 2015. Google Scholar
  20. Sariel Har-Peled. Shortest path in a polygon using sublinear space. J. of Comput. Geom., 7(2):19-45, 2016. Google Scholar
  21. Matias Korman. Memory-constrained algorithms. In Encyclopedia of Algorithms, pages 1260-1264. Springer Berlin Heidelberg, 2016. URL: http://dx.doi.org/10.1007/978-3-642-27848-8_586-1.
  22. Matias Korman, Wolfgang Mulzer, André van Renssen, Marcel Roeloffzen, Paul Seiferth, and Yannik Stein. Time-space trade-offs for triangulations and Voronoi diagrams. In Proc. 14th Int. Symp. Algorithms and Data Structures (WADS), pages 482-494, 2015. Google Scholar
  23. Der-Tsai Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Computers, 31(6):478-487, 1982. Google Scholar
  24. J. Ian Munro and Mike Paterson. Selection and sorting with limited storage. Theoret. Comput. Sci., 12:315-323, 1980. Google Scholar
  25. J. Ian Munro and Venkatesh Raman. Selection from read-only memory and sorting with minimum data movement. Theoret. Comput. Sci., 165(2):311-323, 1996. Google Scholar
  26. J. Pagter and T. Rauhe. Optimal time-space trade-offs for sorting. In Proc. 39th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 264-268, 1998. Google Scholar
  27. Ira Pohl. A minimum storage algorithm for computing the median. Technical Report RC2701, IBM, 1969. Google Scholar
  28. John E. Savage. Models of computation - exploring the power of computing. Addison-Wesley, 1998. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail