Near Isometric Terminal Embeddings for Doubling Metrics

Authors Michael Elkin, Ofer Neiman



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Michael Elkin
Ofer Neiman

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Michael Elkin and Ofer Neiman. Near Isometric Terminal Embeddings for Doubling Metrics. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.36

Abstract

Given a metric space (X,d), a set of terminals K subseteq X, and a parameter t >= 1, we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in K x X up to a factor of t, and have small size (e.g. number of edges for spanners, dimension for embeddings). While such terminal (aka source-wise) metric structures are known to exist in several settings, no terminal spanner or embedding with distortion close to 1, i.e., t=1+epsilon for some small 0<epsilon<1, is currently known. Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion 1+epsilon and size s(|X|) has its terminal counterpart, with distortion 1+O(epsilon) and size s(|K|)+1. In particular, for any doubling metric on n points, a set of k=o(n) terminals, and constant 0<epsilon<1, there exists - A spanner with stretch 1+epsilon for pairs in K x X, with n+o(n) edges. - A labeling scheme with stretch 1+epsilon for pairs in K x X, with label size ~~ log k. - An embedding into l_infty^d with distortion 1+epsilon for pairs in K x X, where d=O(log k). Moreover, surprisingly, the last two results apply if only K is a doubling metric, while X can be arbitrary.
Keywords
  • metric embedding
  • spanners
  • doubling metrics

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References

  1. Amir Abboud and Greg Bodwin. Reachability preservers: New extremal bounds and approximation algorithms. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1865-1883, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.122.
  2. I. Althöfer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9:81-100, 1993. Google Scholar
  3. P. Assouad. Plongements lipschitziens dans ℝⁿ. Bull. Soc. Math. France, 111(4):429-448, 1983. Google Scholar
  4. Y. Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In Proceedings of the 37th IEEE Symp. on Foundations of Computer Science, pages 184-193, 1996. Google Scholar
  5. J. Bourgain. On lipschitz embedding of finite metric spaces in hilbert space. Israel Journal of Mathematics, 52(1-2):46-52, 1985. URL: http://dx.doi.org/10.1007/BF02776078.
  6. T-H. Hubert Chan and Anupam Gupta. Small hop-diameter sparse spanners for doubling metrics. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, SODA '06, pages 70-78, Philadelphia, PA, USA, 2006. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=1109557.1109566.
  7. T.-H. Hubert Chan, Anupam Gupta, Bruce M. Maggs, and Shuheng Zhou. On hierarchical routing in doubling metrics. ACM Trans. Algorithms, 12(4):55:1-55:22, aug 2016. URL: http://dx.doi.org/10.1145/2915183.
  8. 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.
  9. Barun Chandra, Gautam Das, Giri Narasimhan, and José Soares. New sparseness results on graph spanners. In Proc. of 8th SOCG, pages 192-201, 1992. Google Scholar
  10. D. Coppersmith and M. Elkin. Sparse source-wise and pair-wise distance preservers. In SODA: ACM-SIAM Symposium on Discrete Algorithms, pages 660-669, 2005. Google Scholar
  11. Marek Cygan, Fabrizio Grandoni, and Telikepalli Kavitha. On pairwise spanners. In 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, pages 209-220, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2013.209.
  12. Gautam Das, Paul J. Heffernan, and Giri Narasimhan. Optimally sparse spanners in 3-dimensional euclidean space. In Proceedings of the Ninth Annual Symposium on Computational GeometrySan Diego, CA, USA, May 19-21, 1993, pages 53-62, 1993. URL: http://dx.doi.org/10.1145/160985.160998.
  13. Michael Elkin, Arnold Filtser, and Ofer Neiman. Prioritized metric structures and embedding. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 489-498, 2015. URL: http://dx.doi.org/10.1145/2746539.2746623.
  14. Michael Elkin, Arnold Filtser, and Ofer Neiman. Terminal embeddings. Theor. Comput. Sci., 697:1-36, 2017. URL: http://dx.doi.org/10.1016/j.tcs.2017.06.021.
  15. Michael Elkin and Shay Solomon. Optimal euclidean spanners: Really short, thin, and lanky. J. ACM, 62(5):35:1-35:45, 2015. URL: http://dx.doi.org/10.1145/2819008.
  16. Jie Gao, Leonidas J. Guibas, and An Nguyen. Deformable spanners and applications. Comput. Geom. Theory Appl., 35(1-2):2-19, 2006. URL: http://dx.doi.org/10.1016/j.comgeo.2005.10.001.
  17. Lee-Ad Gottlieb. A light metric spanner. In Proc. of 56th FOCS, pages 759-772, 2015. Google Scholar
  18. Lee-Ad Gottlieb and Liam Roditty. An optimal dynamic spanner for doubling metric spaces. In Algorithms - ESA 2008, 16th Annual European Symposium, Karlsruhe, Germany, September 15-17, 2008. Proceedings, pages 478-489, 2008. URL: http://dx.doi.org/10.1007/978-3-540-87744-8_40.
  19. Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS '03, pages 534-, Washington, DC, USA, 2003. IEEE Computer Society. URL: http://portal.acm.org/citation.cfm?id=946243.946308.
  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. William Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. In Conference in modern analysis and probability (New Haven, Conn., 1982), volume 26 of Contemporary Mathematics, pages 189-206. American Mathematical Society, 1984. Google Scholar
  22. Telikepalli Kavitha. New pairwise spanners. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, pages 513-526, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.513.
  23. N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995. Google Scholar
  24. J. Matoušek. On the distortion required for embeding finite metric spaces into normed spaces. Israel Journal of Math, 93:333-344, 1996. Google Scholar
  25. Jiri Matousek. Lectures on Discrete Geometry. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2002. Google Scholar
  26. Manor Mendel and Assaf Naor. Ramsey partitions and proximity data structures. Journal of the European Mathematical Society, 9(2):253-275, 2007. Google Scholar
  27. Giri Narasimhan and Michiel Smid. Geometric Spanner Networks. Cambridge University Press, New York, NY, USA, 2007. Google Scholar
  28. Ofer Neiman. Low dimensional embeddings of doubling metrics. Theory Comput. Syst., 58(1):133-152, 2016. URL: http://dx.doi.org/10.1007/s00224-014-9567-3.
  29. Merav Parter. Bypassing erdős' girth conjecture: Hybrid stretch and sourcewise spanners. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part II, pages 608-619, 2014. URL: http://dx.doi.org/10.1007/978-3-662-43951-7_49.
  30. Neil Robertson and P. D. Seymour. Graph minors: X. obstructions to tree-decomposition. J. Comb. Theory Ser. B, 52(2):153-190, 1991. URL: http://dx.doi.org/10.1016/0095-8956(91)90061-N.
  31. Aleksandrs Slivkins. Distance estimation and object location via rings of neighbors. Distributed Computing, 19(4):313-333, 2007. URL: http://dx.doi.org/10.1007/s00446-006-0015-8.
  32. Kunal Talwar. Bypassing the embedding: Algorithms for low dimensional metrics. In Proceedings of the Thirty-sixth Annual ACM Symposium on Theory of Computing, STOC '04, pages 281-290, New York, NY, USA, 2004. ACM. URL: http://dx.doi.org/10.1145/1007352.1007399.
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