Document Open Access Logo

Light Spanners for High Dimensional Norms via Stochastic Decompositions

Authors Arnold Filtser, Ofer Neiman

PDF
Thumbnail PDF

File

LIPIcs.ESA.2018.29.pdf
  • Filesize: 0.53 MB
  • 15 pages

Document Identifiers

Author Details

Arnold Filtser
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Ofer Neiman
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Filtser, Arnold: Partially supported by the Lynn and William Frankel Center for Computer Sciences, ISF grant 1817/17, and by BSF Grant 2015813.
  • Neiman, Ofer: Partially supported by ISF grant 1817/17, and by BSF Grant 2015813.

Cite AsGet BibTex

Arnold Filtser and Ofer Neiman. Light Spanners for High Dimensional Norms via Stochastic Decompositions. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ESA.2018.29

Abstract

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with O~(n^{1+1/t^2}) edges, little is known. In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of l_p with 1<p <=2. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of l_p for 1<p <=2 has an O(t)-spanner with n^{1+O~(1/t^p)} edges and lightness n^{O~(1/t^p)}. In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)-spanner with lightness O(n^{1/t}). We exhibit the following tradeoff: metrics with decomposability parameter nu=nu(t) admit an O(t)-spanner with lightness O~(nu^{1/t}). For example, n-point Euclidean metrics have nu <=n^{1/t}, metrics with doubling constant lambda have nu <=lambda, and graphs of genus g have nu <=g. While these families do admit a (1+epsilon)-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Spanners
  • Stochastic Decompositions
  • High Dimensional Euclidean Space
  • Doubling Dimension
  • Genus Graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Advances in metric embedding theory. Advances in Mathematics, 228(6):3026-3126, 2011. URL: http://dx.doi.org/10.1016/j.aim.2011.08.003.
  2. Ittai Abraham, Shiri Chechik, Michael Elkin, Arnold Filtser, and Ofer Neiman. Ramsey spanning trees and their applications. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, Louisiana, USA, January 7-10, 2018. Google Scholar
  3. Ittai Abraham, Cyril Gavoille, Anupam Gupta, Ofer Neiman, and Kunal Talwar. Cops, robbers, and threatening skeletons: padded decomposition for minor-free graphs. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 79-88, 2014. URL: http://dx.doi.org/10.1145/2591796.2591849.
  4. Ingo Althöfer, Gautam Das, David P. Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81-100, 1993. URL: http://dx.doi.org/10.1007/BF02189308.
  5. Alexandr Andoni and Piotr Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 459-468, 2006. URL: http://dx.doi.org/10.1109/FOCS.2006.49.
  6. Baruch Awerbuch. Communication-time trade-offs in network synchronization. In Proc. of 4th PODC, pages 272-276, 1985. Google Scholar
  7. Baruch Awerbuch. Complexity of network synchronization. J. ACM, 32(4):804-823, 1985. URL: http://dx.doi.org/10.1145/4221.4227.
  8. Baruch Awerbuch, Alan E. Baratz, and David Peleg. Cost-sensitive analysis of communication protocols. In Proceedings of the Ninth Annual ACM Symposium on Principles of Distributed Computing, Quebec City, Quebec, Canada, August 22-24, 1990, pages 177-187, 1990. URL: http://dx.doi.org/10.1145/93385.93417.
  9. Baruch Awerbuch, Alan E. Baratz, and David Peleg. Efficient broadcast and light-weight spanners. Manuscript, 1991. Google Scholar
  10. Y. Bartal. Probabilistic approximations of metric spaces and its algorithmic applications. In Proc. of 37th FOCS, pages 184-193, 1996. Google Scholar
  11. Punyashloka Biswal, James R. Lee, and Satish Rao. Eigenvalue bounds, spectral partitioning, and metrical deformations via flows. J. ACM, 57(3), 2010. URL: http://dx.doi.org/10.1145/1706591.1706593.
  12. Glencora Borradaile, Hung Le, and Christian Wulff-Nilsen. Greedy spanners are optimal in doubling metrics. CoRR, abs/1712.05007, 2017. URL: http://arxiv.org/abs/1712.05007.
  13. Glencora Borradaile, Hung Le, and Christian Wulff-Nilsen. Minor-free graphs have light spanners. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 767-778, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.76.
  14. R. Braynard, D. Kostic, A. Rodriguez, J. Chase, and A. Vahdat. Opus: an overlay peer utility service. In Prof. of 5th OPENARCH, 2002. Google Scholar
  15. Gruia Calinescu, Howard Karloff, and Yuval Rabani. Approximation algorithms for the 0-extension problem. SIAM J. Comput., 34(2):358-372, 2005. URL: http://dx.doi.org/10.1137/S0097539701395978.
  16. P. B. Callahan and S. R. Kosaraju. A decomposition of multi-dimensional point-sets with applications to k-nearest-neighbors and n-body potential fields. In Proc. of 24th STOC, pages 546-556, 1992. Google Scholar
  17. B. Chandra, G. Das, G. Narasimhan, and J. Soares. New sparseness results on graph spanners. Int. J. Comput. Geometry Appl., 5:125-144, 1995. Google Scholar
  18. Shiri Chechik and Christian Wulff-Nilsen. Near-optimal light spanners. In Proc. of 27th SODA, pages 883-892, 2016. Google Scholar
  19. Edith Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput., 28(1):210-236, 1998. URL: http://dx.doi.org/10.1137/S0097539794261295.
  20. Don Coppersmith and Michael Elkin. Sparse sourcewise and pairwise distance preservers. SIAM J. Discrete Math., 20(2):463-501, 2006. URL: http://dx.doi.org/10.1137/050630696.
  21. 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.
  22. Amin Vahdat Dejan Kostic. Latency versus cost optimizations in hierarchical overlay networks. Technical Report CS-2001-04, Duke University, 2002. Google Scholar
  23. Michael Elkin. Computing almost shortest paths. ACM Trans. Algorithms, 1(2):283-323, 2005. URL: http://dx.doi.org/10.1145/1103963.1103968.
  24. Michael Elkin, Ofer Neiman, and Shay Solomon. Light spanners. In Proc. of 41th ICALP, pages 442-452, 2014. Google Scholar
  25. Michael Elkin and Jian Zhang. Efficient algorithms for constructing (1+epsilon, beta)-spanners in the distributed and streaming models. Distributed Computing, 18(5):375-385, 2006. URL: http://dx.doi.org/10.1007/s00446-005-0147-2.
  26. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, STOC '03, pages 448-455, New York, NY, USA, 2003. ACM. URL: http://dx.doi.org/10.1145/780542.780608.
  27. Uriel Feige and Gideon Schechtman. On the optimality of the random hyperplane rounding technique for max cut. Random Struct. Algorithms, 20(3):403-440, 2002. URL: http://dx.doi.org/10.1002/rsa.10036.
  28. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. Graph distances in the streaming model: the value of space. In Proc. of 16th SODA, pages 745-754, 2005. Google Scholar
  29. Arnold Filtser and Shay Solomon. The greedy spanner is existentially optimal. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25-28, 2016, pages 9-17, 2016. URL: http://dx.doi.org/10.1145/2933057.2933114.
  30. Lee-Ad Gottlieb. A light metric spanner. In Proc. of 56th FOCS, pages 759-772, 2015. Google Scholar
  31. Michelangelo Grigni. Approximate TSP in graphs with forbidden minors. In Proc. of 27th ICALP, pages 869-877, 2000. Google Scholar
  32. Jonathan L. Gross and Thomas W. Tucker. Topological Graph Theory. Wiley-Interscience, New York, NY, USA, 1987. Google Scholar
  33. Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In Proc. of 44th FOCS, pages 534-543, 2003. Google Scholar
  34. Sariel Har-Peled, Piotr Indyk, and Anastasios Sidiropoulos. Euclidean spanners in high dimensions. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 804-809, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.57.
  35. 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.
  36. Jonathan A. Kelner, James R. Lee, Gregory N. Price, and Shang-Hua Teng. Higher eigenvalues of graphs. In FOCS, pages 735-744, 2009. URL: http://dx.doi.org/10.1109/FOCS.2009.69.
  37. Philip N. Klein, Serge A. Plotkin, and Satish Rao. Excluded minors, network decomposition, and multicommodity flow. In STOC, pages 682-690, 1993. URL: http://dx.doi.org/10.1145/167088.167261.
  38. Robert Krauthgamer, James R. Lee, Manor Mendel, and Assaf Naor. Measured descent: A new embedding method for finite metrics. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pages 434-443, Washington, DC, USA, 2004. IEEE Computer Society. URL: http://dx.doi.org/10.1109/FOCS.2004.41.
  39. J. R. Lee and A. Naor. Extending lipschitz functions via random metric partitions. Inventiones Mathematicae, 160(1):59-95, 2005. Google Scholar
  40. James R. Lee and Anastasios Sidiropoulos. Genus and the geometry of the cut graph. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 193-201, 2010. URL: http://dx.doi.org/10.1137/1.9781611973075.18.
  41. Tom Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM, 46:787-832, November 1999. URL: http://dx.doi.org/10.1145/331524.331526.
  42. 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
  43. Nathan Linial and Michael Saks. Low diameter graph decompositions. Combinatorica, 13(4):441-454, 1993. (Preliminary version in 2nd SODA, 1991). Google Scholar
  44. Manor Mendel and Assaf Naor. Ramsey partitions and proximity data structures. Journal of the European Mathematical Society, 9(2):253-275, 2007. Google Scholar
  45. Giri Narasimhan and Michiel H. M. Smid. Geometric spanner networks. Cambridge University Press, 2007. Google Scholar
  46. Huy L. Nguyen. Approximate nearest neighbor search in 𝓁_p. CoRR, abs/1306.3601, 2013. URL: http://arxiv.org/abs/1306.3601.
  47. David Peleg. Proximity-preserving labeling schemes and their applications. In Graph-Theoretic Concepts in Computer Science, 25th International Workshop, WG '99, Ascona, Switzerland, June 17-19, 1999, Proceedings, pages 30-41, 1999. URL: http://dx.doi.org/10.1007/3-540-46784-X_5.
  48. David Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, Philadelphia, PA, 2000. Google Scholar
  49. David Peleg and Jeffrey D. Ullman. An optimal synchronizer for the hypercube. SIAM J. Comput., 18(4):740-747, 1989. URL: http://dx.doi.org/10.1137/0218050.
  50. David Peleg and Eli Upfal. A trade-off between space and efficiency for routing tables. J. ACM, 36(3):510-530, 1989. Google Scholar
  51. Satish B. Rao. Small distortion and volume preserving embeddings for planar and Euclidean metrics. In SOCG, pages 300-306, 1999. Google Scholar
  52. L. Roditty and U. Zwick. On dynamic shortest paths problems. In Proc. of 32nd ESA, pages 580-591, 2004. Google Scholar
  53. Liam Roditty, Mikkel Thorup, and Uri Zwick. Deterministic constructions of approximate distance oracles and spanners. In Automata, Languages and Programming, 32nd International Colloquium, ICALP 2005, Lisbon, Portugal, July 11-15, 2005, Proceedings, pages 261-272, 2005. URL: http://dx.doi.org/10.1007/11523468_22.
  54. J. S. Salowe. Construction of multidimensional spanner graphs, with applications to minimum spanning trees. In Proc. of 7th SoCG, pages 256-261, 1991. Google Scholar
  55. Michiel H. M. Smid. The weak gap property in metric spaces of bounded doubling dimension. In Efficient Algorithms, Essays Dedicated to Kurt Mehlhorn on the Occasion of His 60th Birthday, pages 275-289, 2009. URL: http://dx.doi.org/10.1007/978-3-642-03456-5_19.
  56. Kunal Talwar. Bypassing the embedding: algorithms for low dimensional metrics. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 281-290, 2004. URL: http://dx.doi.org/10.1145/1007352.1007399.
  57. Mikkel Thorup and Uri Zwick. Compact routing schemes. In Proc. of 13th SPAA, pages 1-10, 2001. Google Scholar
  58. Mikkel Thorup and Uri Zwick. Approximate distance oracles. J. ACM, 52(1):1-24, 2005. URL: http://dx.doi.org/10.1145/1044731.1044732.
  59. P. M. Vaidya. A sparse graph almost as good as the complete graph on points in k dimensions. Discrete &Computational Geometry, 6:369-381, 1991. Google Scholar
  60. Jürgen Vogel, Jörg Widmer, Dirk Farin, Martin Mauve, and Wolfgang Effelsberg. Priority-based distribution trees for application-level multicast. In Proceedings of the 2nd Workshop on Network and System Support for Games, NETGAMES 2003, Redwood City, California, USA, May 22-23, 2003, pages 148-157, 2003. URL: http://dx.doi.org/10.1145/963900.963914.
  61. Bang Ye Wu, Kun-Mao Chao, and Chuan Yi Tang. Light graphs with small routing cost. Networks, 39(3):130-138, 2002. URL: http://dx.doi.org/10.1002/net.10019.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact