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Orthogonal Point Location and Rectangle Stabbing Queries in 3-d

Authors Timothy M. Chan, Yakov Nekrich, Saladi Rahul, Konstantinos Tsakalidis

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Author Details

Timothy M. Chan
  • Dept. of Computer Science, University of Illinois at Urbana-Champaign, USA
Yakov Nekrich
  • Cheriton School of Computer Science, University of Waterloo, Canada
Saladi Rahul
  • Dept. of Computer Science, University of Illinois at Urbana-Champaign, USA
Konstantinos Tsakalidis
  • Dept. of Computer and Information Science, Tandon School of Engineering, New York University, USA

Funding

  • Tsakalidis, Konstantinos: Partially supported by NSF grants CCF-1319648 and CCF-1533564

Cite AsGet BibTex

Timothy M. Chan, Yakov Nekrich, Saladi Rahul, and Konstantinos Tsakalidis. Orthogonal Point Location and Rectangle Stabbing Queries in 3-d. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.31

Abstract

In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing. - Orthogonal point location. We give the first linear-space data structure that supports 3-d point location queries on n disjoint axis-aligned boxes with optimal O(log n) query time in the (arithmetic) pointer machine model. This improves the previous O(log^{3/2} n) bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time. - Rectangle stabbing. We give the first linear-space data structure that supports 3-d 4-sided and 5-sided rectangle stabbing queries in optimal O(log_wn+k) time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-k rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing. For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe's grid-based recursive technique (FOCS 2000), combined with a number of new ideas.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • geometric data structures
  • orthogonal point location
  • rectangle stabbing
  • pointer machines
  • I/O model
  • word RAM model

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References

  1. Peyman Afshani. On dominance reporting in 3D. In Proceedings of European Symposium on Algorithms (ESA), pages 41-51, 2008. Google Scholar
  2. Peyman Afshani, Lars Arge, and Kasper Dalgaard Larsen. Orthogonal range reporting: query lower bounds, optimal structures in 3-d, and higher-dimensional improvements. In Proceedings of Symposium on Computational Geometry (SoCG), pages 240-246, 2010. Google Scholar
  3. Peyman Afshani, Lars Arge, and Kasper Green Larsen. Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. In Proceedings of Symposium on Computational Geometry (SoCG), pages 323-332, 2012. Google Scholar
  4. Peyman Afshani, Gerth Stølting Brodal, and Norbert Zeh. Ordered and unordered top-k range reporting in large data sets. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 390-400, 2011. Google Scholar
  5. Stephen Alstrup, Gerth Stølting Brodal, and Theis Rauhe. New data structures for orthogonal range searching. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 198-207, 2000. Google Scholar
  6. Lars Arge, Andrew Danner, and Sha-Mayn Teh. I/O-efficient point location using persistent B-trees. ACM Journal of Experimental Algorithmics, 8, 2003. Google Scholar
  7. J.L. Bentley. Solutions to Klee’s rectangle problems. Technical Report, Carnegie-Mellon University, Pittsburgh, PA, 1977. Google Scholar
  8. Gerth Stølting Brodal. External memory three-sided range reporting and top-k queries with sublogarithmic updates. In Proceedings of Symposium on Theoretical Aspects of Computer Science (STACS), volume 47, pages 23:1-23:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  9. Gerth Stølting Brodal, Rolf Fagerberg, Mark Greve, and Alejandro Lopez-Ortiz. Online sorted range reporting. In International Symposium on Algorithms and Computation (ISAAC), pages 173-182, 2009. Google Scholar
  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 Pǎtraşcu. Orthogonal range searching on the RAM, revisited. In Proceedings of Symposium on Computational Geometry (SoCG), pages 1-10, 2011. Google Scholar
  12. Bernard Chazelle. Filtering search: A new approach to query-answering. SIAM Journal of Computing, 15(3):703-724, 1986. Google Scholar
  13. Mark de Berg, Marc J. van Kreveld, and Jack Snoeyink. Two- and three-dimensional point location in rectangular subdivisions. Journal of Algorithms, 18(2):256-277, 1995. Google Scholar
  14. Herbert Edelsbrunner, Leonidas J. Guibas, and Jorge Stolfi. Optimal point location in a monotone subdivision. SIAM Journal of Computing, 15(2):317-340, 1986. Google Scholar
  15. Herbert Edelsbrunner, G. Haring, and D. Hilbert. Rectangular point location in d dimensions with applications. Comput. J., 29(1):76-82, 1986. Google Scholar
  16. Michael T. Goodrich, Jyh-Jong Tsay, Darren Erik Vengroff, and Jeffrey Scott Vitter. External-memory computational geometry. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 714-723, 1993. Google Scholar
  17. John Iacono and Stefan Langerman. Dynamic point location in fat hyperrectangles with integer coordinates. In Proceedings of the Canadian Conference on Computational Geometry (CCCG), 2000. Google Scholar
  18. David G. Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal of Computing, 12(1):28-35, 1983. Google Scholar
  19. Richard J. Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. SIAM Journal of Computing, 9(3):615-627, 1980. Google Scholar
  20. Yakov Nekrich. I/O-efficient point location in a set of rectangles. In Latin American Symposium on Theoretical Informatics (LATIN), pages 687-698, 2008. Google Scholar
  21. Mihai Pǎtraşcu. Unifying the landscape of cell-probe lower bounds. SIAM Journal of Computing, 40(3):827-847, 2011. Google Scholar
  22. Mihai Pǎtraşcu and Mikkel Thorup. Time-space trade-offs for predecessor search. In Proceedings of ACM Symposium on Theory of Computing (STOC), pages 232-240, 2006. Google Scholar
  23. Saladi Rahul. Improved bounds for orthogonal point enclosure query and point location in orthogonal subdivisions in ℝ³. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 200-211, 2015. Google Scholar
  24. Saladi Rahul and Ravi Janardan. A general technique for top-k geometric intersection query problems. IEEE Transactions on Knowledge and Data Engineering (TKDE), 26(12):2859-2871, 2014. Google Scholar
  25. Saladi Rahul and Yufei Tao. On top-k range reporting in 2d space. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 265-275, 2015. Google Scholar
  26. Saladi Rahul and Yufei Tao. Efficient top-k indexing via general reductions. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), 2016. Google Scholar
  27. Neil Sarnak and Robert Endre Tarjan. Planar point location using persistent search trees. Communications of the ACM (CACM), 29(7):669-679, 1986. Google Scholar
  28. Cheng Sheng and Yufei Tao. Dynamic top-k range reporting in external memory. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), 2012. Google Scholar
  29. Qingmin Shi and Joseph JáJá. Novel transformation techniques using q-heaps with applications to computational geometry. SIAM Journal of Computing, 34(6):1474-1492, 2005. Google Scholar
  30. Jack Snoeyink. Point location. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, pages 767-787. CRC Press, 2nd edition, 2004. Google Scholar
  31. Yufei Tao. A dynamic I/O-efficient structure for one-dimensional top-k range reporting. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 256-265, 2014. Google Scholar
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