External Memory Planar Point Location with Fast Updates

Authors John Iacono, Ben Karsin, Grigorios Koumoutsos

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Author Details

John Iacono
  • Université Libre de Bruxelles, Belgium
  • New York University, USA
Ben Karsin
  • Université Libre de Bruxelles, Belgium
Grigorios Koumoutsos
  • Université Libre de Bruxelles, Belgium

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John Iacono, Ben Karsin, and Grigorios Koumoutsos. External Memory Planar Point Location with Fast Updates. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We study dynamic planar point location in the External Memory Model or Disk Access Model (DAM). Previous work in this model achieves polylog query and polylog amortized update time. We present a data structure with O(log_B^2 N) query time and O(1/B^(1-epsilon) log_B N) amortized update time, where N is the number of segments, B the block size and epsilon is a small positive constant, under the assumption that all faces have constant size. This is a B^(1-epsilon) factor faster for updates than the fastest previous structure, and brings the cost of insertion and deletion down to subconstant amortized time for reasonable choices of N and B. Our structure solves the problem of vertical ray-shooting queries among a dynamic set of interior-disjoint line segments; this is well-known to solve dynamic planar point location for a connected subdivision of the plane with faces of constant size.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Computational geometry
  • Theory of computation → Models of computation
  • point location
  • data structures
  • dynamic algorithms
  • computational geometry


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  1. Pankaj K. Agarwal, Lars Arge, Gerth Stølting Brodal, and Jeffrey Scott Vitter. I/O-Efficient Dynamic Point Location in Monotone Planar Subdivisions. In Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 11-20, 1999. URL: http://dl.acm.org/citation.cfm?id=314500.314525.
  2. Alok Aggarwal and Jeffrey Scott Vitter. The Input/Output Complexity of Sorting and Related Problems. Commun. ACM, 31(9):1116-1127, 1988. URL: https://doi.org/10.1145/48529.48535.
  3. Lars Arge. The Buffer Tree: A Technique for Designing Batched External Data Structures. Algorithmica, 37(1):1-24, 2003. URL: https://doi.org/10.1007/s00453-003-1021-x.
  4. Lars Arge, Gerth Stølting Brodal, and Loukas Georgiadis. Improved Dynamic Planar Point Location. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 305-314, 2006. URL: https://doi.org/10.1109/FOCS.2006.40.
  5. Lars Arge, Gerth Stølting Brodal, and S. Srinivasa Rao. External Memory Planar Point Location with Logarithmic Updates. Algorithmica, 63(1-2):457-475, 2012. URL: https://doi.org/10.1007/s00453-011-9541-2.
  6. Lars Arge, Vasilis Samoladas, and Jeffrey Scott Vitter. On Two-Dimensional Indexability and Optimal Range Search Indexing. In PODS, pages 346-357. ACM Press, 1999. Google Scholar
  7. Lars Arge and Jan Vahrenhold. I/O-efficient dynamic planar point location. Comput. Geom., 29(2):147-162, 2004. URL: https://doi.org/10.1016/j.comgeo.2003.04.001.
  8. Lars Arge, Darren Erik Vengroff, and Jeffrey Scott Vitter. External-Memory Algorithms for Processing Line Segments in Geographic Information Systems. Algorithmica, 47(1):1-25, 2007. URL: https://doi.org/10.1007/s00453-006-1208-z.
  9. Lars Arge and Jeffrey Scott Vitter. Optimal External Memory Interval Management. SIAM J. Comput., 32(6):1488-1508, 2003. URL: https://doi.org/10.1137/S009753970240481X.
  10. Hanna Baumgarten, Hermann Jung, and Kurt Mehlhorn. Dynamic Point Location in General Subdivisions. J. Algorithms, 17(3):342-380, 1994. URL: https://doi.org/10.1006/jagm.1994.1040.
  11. Rudolf Bayer and Edward M. McCreight. Organization and Maintenance of Large Ordered Indices. Acta Inf., 1:173-189, 1972. URL: https://doi.org/10.1007/BF00288683.
  12. Jon Louis Bentley. Decomposable Searching Problems. Inf. Process. Lett., 8(5):244-251, 1979. URL: https://doi.org/10.1016/0020-0190(79)90117-0.
  13. Gerth Stølting Brodal. External Memory Three-Sided Range Reporting and Top-k Queries with Sublogarithmic Updates. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS), pages 23:1-23:14, 2016. URL: https://doi.org/10.4230/LIPIcs.STACS.2016.23.
  14. Gerth Stølting Brodal and Rolf Fagerberg. Lower bounds for external memory dictionaries. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 546-554, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644201.
  15. Timothy M. Chan and Yakov Nekrich. Towards an Optimal Method for Dynamic Planar Point Location. In IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 390-409, 2015. URL: https://doi.org/10.1109/FOCS.2015.31.
  16. Bernard Chazelle. Computational Geometry for the Gourmet: Old Fare and New Dishes. In Automata, Languages and Programming, 18th International Colloquium (ICALP), pages 686-696, 1991. URL: https://doi.org/10.1007/3-540-54233-7_174.
  17. Bernard Chazelle. Computational geometry: a retrospective. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing (STOC), pages 75-94, 1994. URL: https://doi.org/10.1145/195058.195110.
  18. Siu-Wing Cheng and Ravi Janardan. New Results on Dynamic Planar Point Location. SIAM J. Comput., 21(5):972-999, 1992. URL: https://doi.org/10.1137/0221057.
  19. Yi-Jen Chiang and Roberto Tamassia. Dynamization of the trapezoid method for planar point location in monotone subdivisions. Int. J. Comput. Geometry Appl., 2(3):311-333, 1992. Google Scholar
  20. Herbert Edelsbrunner and Hermann A. Maurer. On the Intersection of Orthogonal Objects. Inf. Process. Lett., 13(4/5):177-181, 1981. URL: https://doi.org/10.1016/0020-0190(81)90053-3.
  21. Michael T. Goodrich and Roberto Tamassia. Dynamic Trees and Dynamic Point Location. SIAM J. Comput., 28(2):612-636, 1998. Google Scholar
  22. J. Ian Munro and Yakov Nekrich. Dynamic Planar Point Location in External Memory. In SoCG, volume 129 of LIPIcs, pages 52:1-52:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019. Google Scholar
  23. Eunjin Oh and Hee-Kap Ahn. Point Location in Dynamic Planar Subdivisions. In 34th International Symposium on Computational Geometry, SoCG 2018, pages 63:1-63:14, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.63.
  24. Jack Snoeyink. Point Location. In Handbook of Discrete and Computational Geometry, Second Edition, pages 767-785. Chapman & Hall/CRC, 2004. URL: https://doi.org/10.1201/9781420035315.pt4.
  25. Jeffrey Scott Vitter. External memory algorithms and data structures. ACM Comput. Surv., 33(2):209-271, 2001. URL: https://doi.org/10.1145/384192.384193.
  26. Jeffrey Scott Vitter. Algorithms and Data Structures for External Memory. Foundations and Trends in Theoretical Computer Science, 2(4):305-474, 2006. URL: https://doi.org/10.1561/0400000014.
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