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International Symposium on Algorithms and Computation

ISAAC 2015: Algorithms and Computation pp 14–22Cite as

Adaptive Point Location in Planar Convex Subdivisions

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9472))

Abstract

We present a planar point location structure for a convex subdivision S. Given a query sequence of length m, the total running time is \(O(\mathrm {OPT} + m\log \log n + n)\), where n is the number of vertices in S and \(\mathrm {OPT}\) is the minimum running time to process the same query sequence by any linear decision tree for answering planar point location queries in S. The running time includes the preprocessing time. Therefore, for \(m \ge n\), our running time is only worse than the best possible bound by \(O(\log \log n)\) per query, which is much smaller than the \(O(\log n)\) query time offered by an worst-case optimal planar point location structure.

Supported by FSGRF14EG26, HKUST.

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Notes

  1. 1.

    Methods that employ indexing (e.g. [15]) and bit tricks (e.g. [12]) do not fall under the linear decision tree model.

References

  1. Afshani, P., Barbay, J., Chan, T.: Instance optimal geometric algorithms. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 129–138 (2009)

    Google Scholar 

  2. Adamy, U., Seidel, R.: On the exact worst case query complexity of planar point location. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 609–618 (1998)

    Google Scholar 

  3. Aronov, B., de Berg, M., Roeloffzen, M., Speckmann, B.: Distance-sensitive planar point location. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 49–60. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  4. Arya, S., Cheng, S.W., Mount, D.M., Ramesh, H.: Efficient expected-case algorithms for planar point location. In: Proceedings of the 7th Scandinavian Workshop on Algorithm Theory, pp. 353–366 (2000)

    Google Scholar 

  5. Arya, S., Malamatos, T., Mount, D.M.: Nearly optimal expected-case planar point location. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 208–218 (2000)

    Google Scholar 

  6. Arya, S., Malamatos, T., Mount, D.M.: A simple entropy-based algorithm for planar point location. ACM Trans. Algorithms 3(2), article 17 (2007)

    Google Scholar 

  7. Arya, S., Malamatos, T., Mount, D., Wong, K.: Optimal expected-case planar point location. SIAM J. Comput. 37(2), 584–610 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boissonnat, J.D., Yvinec, M.: Algorithmic Geometry. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  9. Bose, P., Devroye, L., Douïeb, K., Dujmovic, V., King, J., Morin, P.: Point location in disconnected planar subdivisions. arXiv:1001.2763v1 [cs.CG], 15 January 2010

  10. Bose, P., Devroye, L., Douïeb, K., Dujmovic, V., King, J., Morin, P.: Odds-On Trees, arXiv:1002.1092v1 [cs.CG], 5 February 2010

  11. de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications. Springer, New York (2008)

    Book  MATH  Google Scholar 

  12. Chan, T.M., Pătraşcu, M.: Transdichotomous results in computational geometry, I: point location in sublogarithmic time. SIAM J. Comput. 39(2), 703–729 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Collette, S., Dujmović, V., Iacono, J., Langerman, S., Morin, P.: Entropy, triangulation, and point location in planar subdivisions. ACM Trans. Algorithms 8(3), article 29 (2012)

    Google Scholar 

  14. Dobkin, D.P., Kirkpatrick, D.G.: Determining the separation of preprocessed polyhedra–a unified approach. In: Proceedings of the 17th International Colloquium on Automata, Languages and Programming, pp. 400–413 (1990)

    Google Scholar 

  15. Edahiro, M., Kokubo, I., Asano, T.: A new point-location algorithm and its practical efficiency–comparison with existing algorithms. ACM Trans. Graph. 3(2), 86–109 (1984)

    Article  MATH  Google Scholar 

  16. Edelsbrunner, H., Guibas, L.J., Stolfi, J.: Optimal point location in a monotone subdivision. SIAM J. Comput. 15(2), 317–340 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Iacono, J.: Expected asymptotically optimal planar point location. Comput. Geom. Theory Appl. 29(1), 19–22 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Iacono, J., Langerman, S.: Proximate planar point location. In: Proceedings of the 19th Annual Symposium on Computational Geometry, pp. 220–226 (2003)

    Google Scholar 

  19. Iacono, J., Mulzer, W.: A static optimality transformation with applications to planar point location. Int. J. Comput. Geom. Appl. 22(4), 327–340 (2012)

    Article  MathSciNet  Google Scholar 

  20. Kirkpatrick, D.G.: Optimal search in planar subdivisions. SIAM J. Comput. 12(1), 28–35 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lee, D.T., Preparata, F.P.: Location of a point in a planar subdivision and its applications. SIAM J. Comput. 6(3), 594–606 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mulmuley, K.: A fast planar partition algorithm, I. J. Symbolic Comput. 10(3–4), 253–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Preparata, F.P.: A new approach to planar point location. SIAM J. Comput. 10(3), 473–483 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sarnak, N., Tarjan, R.E.: Planar point location using persistent search trees. Commun. ACM 29(7), 669–679 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Seidel, R.: A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Comput. Geom. Theory Appl. 1(1), 51–64 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shannon, C.E.: A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev. 5(1), 3–55 (2001)

    Article  MathSciNet  Google Scholar 

  27. Sleator, D.D., Tarjan, R.E.: Self-adjusting binary search trees. J. ACM 32(3), 652–686 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgment

We thank the anonymous referees for their helpful comments.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, HKUST, Hong Kong, China

    Siu-Wing Cheng & Man-Kit Lau

Authors
  1. Siu-Wing Cheng
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  2. Man-Kit Lau
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Correspondence to Siu-Wing Cheng .

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Editors and Affiliations

  1. Masdar Institute, Abu Dhabi, United Arab Emirates

    Khaled Elbassioni

  2. Kyoto University, Kyoto, Japan

    Kazuhisa Makino

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Cheng, SW., Lau, MK. (2015). Adaptive Point Location in Planar Convex Subdivisions. In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_2

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  • DOI: https://doi.org/10.1007/978-3-662-48971-0_2

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  • Online ISBN: 978-3-662-48971-0

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