On Locality-Sensitive Orderings and Their Applications

Authors Timothy M. Chan, Sariel Har-Peled, Mitchell Jones



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Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, USA
Sariel Har-Peled
  • Department of Computer Science, University of Illinois at Urbana-Champaign, USA
Mitchell Jones
  • Department of Computer Science, University of Illinois at Urbana-Champaign, USA

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Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones. On Locality-Sensitive Orderings and Their Applications. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ITCS.2019.21

Abstract

For any constant d and parameter epsilon > 0, we show the existence of (roughly) 1/epsilon^d orderings on the unit cube [0,1)^d, such that any two points p, q in [0,1)^d that are close together under the Euclidean metric are "close together" in one of these linear orderings in the following sense: the only points that could lie between p and q in the ordering are points with Euclidean distance at most epsilon | p - q | from p or q. These orderings are extensions of the Z-order, and they can be efficiently computed.
Functionally, the orderings can be thought of as a replacement to quadtrees and related structures (like well-separated pair decompositions). We use such orderings to obtain surprisingly simple algorithms for a number of basic problems in low-dimensional computational geometry, including (i) dynamic approximate bichromatic closest pair, (ii) dynamic spanners, (iii) dynamic approximate minimum spanning trees, (iv) static and dynamic fault-tolerant spanners, and (v) approximate nearest neighbor search.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Approximation algorithms
  • Data structures
  • Computational geometry

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References

  1. Brian Alspach. The wonderful Walecki construction. Bull. Inst. Combin. Appl., 52:7-20, 2008. Google Scholar
  2. Sanjeev Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. J. Assoc. Comput. Mach., 45(5):753-782, September 1998. URL: http://www.cs.princeton.edu/~arora/pubs/tsp.ps.
  3. Marshall W. Bern. Approximate Closest-Point Queries in High Dimensions. Inform. Process. Lett., 45(2):95-99, 1993. URL: http://dx.doi.org/10.1016/0020-0190(93)90222-U.
  4. Paul B. Callahan and S. Rao Kosaraju. Faster Algorithms for Some Geometric Graph Problems in Higher Dimensions. In Vijaya Ramachandran, editor, Proc. 4th ACM-SIAM Sympos. Discrete Alg. (SODA), pages 291-300. ACM/SIAM, 1993. URL: http://dl.acm.org/citation.cfm?id=313559.313777.
  5. T.-H. Hubert Chan, Mingfei Li, Li Ning, and Shay Solomon. New Doubling Spanners: Better and Simpler. SIAM J. Comput., 44(1):37-53, 2015. URL: http://dx.doi.org/10.1137/130930984.
  6. Timothy M. Chan. Approximate nearest neighbor queries revisited. Discrete Comput. Geom., 20(3):359-373, 1998. URL: http://dx.doi.org/10.1007/PL00009390.
  7. Timothy M. Chan. Closest-point problems simplified on the RAM. In Proc. 13th ACM-SIAM Sympos. Discrete Alg. (SODA), pages 472-473. SIAM, 2002. URL: http://dl.acm.org/citation.cfm?id=545381.545444.
  8. Timothy M. Chan. A minimalist’s implementation of an approximate nearest neighbor algorithm in fixed dimensions, 2006. URL: http://tmc.web.engr.illinois.edu/sss.ps.
  9. Timothy M. Chan. Well-separated pair decomposition in linear time? Inform. Process. Lett., 107(5):138-141, 2008. URL: http://dx.doi.org/10.1016/j.ipl.2008.02.008.
  10. Timothy M. Chan and Dimitrios Skrepetos. Dynamic data structures for approximate Hausdorff distance in the word RAM. Comput. Geom. Theory Appl., 60:37-44, 2017. URL: http://dx.doi.org/10.1016/j.comgeo.2016.08.002.
  11. Artur Czumaj and Hairong Zhao. Fault-Tolerant Geometric Spanners. Discrete Comput. Geom., 32(2):207-230, 2004. URL: http://www.springerlink.com/index/10.1007/s00454-004-1121-7.
  12. David Eppstein. Dynamic Euclidean minimum spanning trees and extrema of binary functions. Discrete Comput. Geom., 13:111-122, 1995. URL: http://www.ics.uci.edu/~eppstein/pubs/Epp-DCG-95.pdf.
  13. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Sys. Sci., 69(3):485-497, 2004. URL: http://dx.doi.org/10.1016/j.jcss.2004.04.011.
  14. Uriel Feige and Robert Krauthgamer. Stereoscopic families of permutations, and their applications. In Proc. 5th Israel Symp. Theo. Comput. and Systems (ISTCS), pages 85-95. IEEE Computer Society, 1997. URL: http://dx.doi.org/10.1109/ISTCS.1997.595160.
  15. Michael L. Fredman and Dan E. Willard. Surpassing the Information Theoretic Bound with Fusion Trees. J. Comput. Sys. Sci., 47(3):424-436, 1993. URL: http://dx.doi.org/10.1016/0022-0000(93)90040-4.
  16. Irene Gargantini. An Effective Way to Represent Quadtrees. Commun. ACM, 25(12):905-910, 1982. URL: http://dx.doi.org/10.1145/358728.358741.
  17. Lee-Ad Gottlieb and Liam Roditty. An Optimal Dynamic Spanner for Doubling Metric Spaces. In Proc. 16th Annu. Euro. Sympos. Alg. (ESA), pages 478-489, 2008. URL: http://dx.doi.org/10.1007/978-3-540-87744-8_40.
  18. Lee-Ad Gottlieb and Liam Roditty. Improved algorithms for fully dynamic geometric spanners and geometric routing. In Proc. 19th ACM-SIAM Sympos. Discrete Alg. (SODA), pages 591-600, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347148.
  19. Sariel Har-Peled. Geometric Approximation Algorithms, volume 173 of Math. Surveys &Monographs. Amer. Math. Soc., Boston, MA, USA, 2011. URL: http://dx.doi.org/10.1090/surv/173.
  20. Sariel Har-Peled and Manor Mendel. Fast Construction of Nets in Low Dimensional Metrics, and Their Applications. SIAM J. Comput., 35(5):1148-1184, 2006. URL: http://dx.doi.org/10.1137/S0097539704446281.
  21. Dorit S. Hochbaum and Wolfgang Maass. Approximation Schemes for Covering and Packing Problems in Image Processing and VLSI. J. Assoc. Comput. Mach., 32(1):130-136, 1985. URL: http://dx.doi.org/10.1145/2455.214106.
  22. Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilsen. Faster Fully-Dynamic Minimum Spanning Forest. In Nikhil Bansal and Irene Finocchi, editors, Proc. 23rd Annu. Euro. Sympos. Alg. (ESA), volume 9294 of Lect. Notes in Comp. Sci., pages 742-753. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_62.
  23. Piotr Indyk and Rajeev Motwani. Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality. In Proc. 30th ACM Sympos. Theory Comput. (STOC), pages 604-613, 1998. URL: http://dx.doi.org/10.1145/276698.276876.
  24. Ibrahim Kamel and Christos Faloutsos. On Packing R-trees. In Bharat K. Bhargava, Timothy W. Finin, and Yelena Yesha, editors, Proc. 2nd Intl. Conf. Info. Knowl. Mang., pages 490-499. ACM, 1993. URL: http://dx.doi.org/10.1145/170088.170403.
  25. Sanjiv Kapoor and Xiang-Yang Li. Efficient Construction of Spanners in d-Dimensions. CoRR, abs/1303.7217, 2013. URL: http://arxiv.org/abs/1303.7217.
  26. Christos Levcopoulos, Giri Narasimhan, and Michiel H. M. Smid. Efficient Algorithms for Constructing Fault-Tolerant Geometric Spanners. In Jeffrey Scott Vitter, editor, Proc. 30th ACM Sympos. Theory Comput. (STOC), pages 186-195. ACM, 1998. URL: http://dx.doi.org/10.1145/276698.276734.
  27. Swanwa Liao, Mario A. López, and Scott T. Leutenegger. High Dimensional Similarity Search With Space Filling Curves. In Proc. 17th Int. Conf. on Data Eng. (ICDE), pages 615-622, 2001. URL: http://dx.doi.org/10.1109/ICDE.2001.914876.
  28. Tamás Lukovszki. New Results of Fault Tolerant Geometric Spanners. In Frank K. H. A. Dehne, Arvind Gupta, Jörg-Rüdiger Sack, and Roberto Tamassia, editors, Proc. 6th Workshop Alg. Data Struct. (WADS), volume 1663 of Lect. Notes in Comp. Sci., pages 193-204. Springer, 1999. URL: http://dx.doi.org/10.1007/3-540-48447-7_20.
  29. Tamás Lukovszki. New results on geometric spanners and their applications. PhD thesis, University of Paderborn, Germany, 1999. URL: http://d-nb.info/958256713.
  30. Liam Roditty. Fully Dynamic Geometric Spanners. Algorithmica, 62(3-4):1073-1087, 2012. URL: http://dx.doi.org/10.1007/s00453-011-9504-7.
  31. Shay Solomon. From hierarchical partitions to hierarchical covers: Optimal fault-tolerant spanners for doubling metrics. In David B. Shmoys, editor, Proc. 46th ACM Sympos. Theory Comput. (STOC), pages 363-372. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591864.
  32. Peter van Emde Boas. Preserving Order in a Forest in Less Than Logarithmic Time and Linear Space. Inf. Process. Lett., 6(3):80-82, 1977. URL: http://dx.doi.org/10.1016/0020-0190(77)90031-X.
  33. Jean-Louis Verger-Gaugry. Covering a Ball with Smaller Equal Balls in ℝⁿ. Discrete Comput. Geom., 33(1):143-155, 2005. URL: http://dx.doi.org/10.1007/s00454-004-2916-2.
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