Two Approaches to Building Time-Windowed Geometric Data Structures

Authors Timothy M. Chan, Simon Pratt



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Timothy M. Chan
Simon Pratt

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Timothy M. Chan and Simon Pratt. Two Approaches to Building Time-Windowed Geometric Data Structures. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.SoCG.2016.28

Abstract

Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.'s O(n log^2 n) and O(n log n loglog n) solutions respectively. Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.
Keywords
  • time window
  • geometric data structures
  • range searching
  • dynamic convex hull

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References

  1. Pankaj K. Agarwal and Jirı Matoušek. Dynamic half-space range reporting and its applications. Algorithmica, 13(4):325-345, 1995. Google Scholar
  2. Pankaj K. Agarwal and Micha Sharir. Off-line dynamic maintenance of the width of a planar point set. Computational Geometry: Theory and Applications, 1:65-78, 1991. URL: http://dx.doi.org/10.1016/0925-7721(91)90001-U.
  3. Nancy M. Amato, Michael T. Goodrich, and Edgar A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 683-694, 1994. URL: http://dx.doi.org/10.1109/SFCS.1994.365724.
  4. Michael J. Bannister, William E. Devanny, Michael T. Goodrich, Joseph A. Simons, and Lowell Trott. Windows into geometric events: Data structures for time-windowed querying of temporal point sets. In Proceedings of the 26th Canadian Conference on Computational Geometry (CCCG), 2014. Google Scholar
  5. Drago Bokal, Sergio Cabello, and David Eppstein. Finding all maximal subsequences with hereditary properties. In Proceedings of the 31st International Symposium on Computational Geometry (SoCG), pages 240-254, 2015. Google Scholar
  6. Gerth Stølting Brodal and Riko Jacob. Dynamic planar convex hull. In Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 617-626, 2002. Google Scholar
  7. Timothy M. Chan. Dynamic planar convex hull operations in near-logarithmaic amortized time. Journal of the ACM, 48(1):1-12, 2001. URL: http://dx.doi.org/10.1145/363647.363652.
  8. Timothy M. Chan. A fully dynamic algorithm for planar width. In Proceedings of the 17th Annual Symposium on Computational Geometry (SoCG), pages 172-176, 2001. Google Scholar
  9. Timothy M. Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. J. ACM, 57(3):16:1-16:15, 2010. URL: http://dx.doi.org/10.1145/1706591.1706596.
  10. Timothy M. Chan. Persistent predecessor search and orthogonal point location on the word ram. ACM Transactions on Algorithms, 9(3):22, 2013. Google Scholar
  11. Timothy M. Chan, Kasper Green Larsen, and Mihai Pătraşcu. Orthogonal range searching on the RAM, revisited. In Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG), pages 1-10, 2011. Google Scholar
  12. Timothy M. Chan and Simon Pratt. Time-windowed closest pair. In Proceedings of the 27th Canadian Conference on Computational Geometry (CCCG), 2015. Google Scholar
  13. Timothy M. Chan and Konstantinos Tsakalidis. Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. In Proceedings of the 31st International Symposium on Computational Geometry (SoCG), pages 719-732, 2015. Google Scholar
  14. Bernard Chazelle. On the convex layers of a planar set. IEEE Transactions on Information Theory, 31(4):509-517, 1985. Google Scholar
  15. Bernard Chazelle and Leonidas J. Guibas. Fractional cascading: I. A data structuring technique. Algorithmica, 1(1-4):133-162, 1986. URL: http://dx.doi.org/10.1007/BF01840440.
  16. Bernard Chazelle and Leonidas J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1(1-4):163-191, 1986. URL: http://dx.doi.org/10.1007/BF01840441.
  17. Kenneth L. Clarkson and Peter W. Shor. Application of random sampling in computational geometry, II. Discrete & Computational Geometry, 4:387-421, 1989. URL: http://dx.doi.org/10.1007/BF02187740.
  18. David Eppstein. Incremental and decremental maintenance of planar width. Journal of Algorithms, 37(2):570-577, 2000. Google Scholar
  19. Sergiu Hart and Micha Sharir. Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes. Combinatorica, 6(2):151-178, 1986. URL: http://dx.doi.org/10.1007/BF02579170.
  20. John Hershberger and Subhash Suri. Offline maintenance of planar configurations. In Proceedings of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 32-41, 1991. Google Scholar
  21. John Hershberger and Subhash Suri. Applications of a semi-dynamic convex hull algorithm. BIT, 32(2):249-267, 1992. Google Scholar
  22. Ketan Mulmuley. Randomized multidimensional search trees: Lazy balancing and dynamic shuffling. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 180-196, 1991. Google Scholar
  23. J. Ian Munro. Tables. In Proceedings of the 16th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 37-42. Springer, 1996. Google Scholar
  24. Mark H. Overmars and Jan van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23(2):166-204, 1981. Google Scholar
  25. Franco P. Preparata. An optimal real-time algorithm for planar convex hulls. Communications of the ACM, 22(7):402-405, 1979. URL: http://dx.doi.org/10.1145/359131.359132.
  26. Franco P. Preparata and Michael I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, New York, NY, USA, 1985. Google Scholar
  27. Otfried Schwarzkopf. Dynamic maintenance of geometric structures made easy. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 197-206, 1991. Google Scholar
  28. Micha Sharir and Pankaj K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, 1995. Google Scholar
  29. Arie Tamir. Improved complexity bounds for center location problems on networks by using dynamic data structures. SIAM Journal on Discrete Mathematics, 1(3):377-396, 1988. URL: http://dx.doi.org/10.1137/0401038.
  30. Gelin Zhou. Two-dimensional range successor in optimal time and almost linear space. Information Processing Letters, 2015. Google Scholar
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