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Light Euclidean Spanners with Steiner Points

Authors Hung Le, Shay Solomon

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Author Details

Hung Le
  • University of Victoria, Canada
  • University of Massachusetts, Amherst, MA, USA
Shay Solomon
  • Tel-Aviv University, Israel

Funding

  • Le, Hung: Supported by a PIMS postdoctoral fellowship and an NSERC grant.
  • Solomon, Shay: Partially supported by the Israel Science Foundation grant No.1991/19 and by Len Blavatnik and the Blavatnik Family foundation.

Acknowledgements

Shay Solomon is grateful to Michael Elkin, Ofer Neiman and Michiel Smid for fruitful discussions.

Cite As Get BibTex

Hung Le and Shay Solomon. Light Euclidean Spanners with Steiner Points. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 67:1-67:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.67

Abstract

The FOCS'19 paper of Le and Solomon [Hung Le and Shay Solomon, 2019], culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1+ε)-spanner in ℝ^d is Õ(ε^{-d}) for any d = O(1) and any ε = Ω(n^{-1/(d-1)}) (where Õ hides polylogarithmic factors of 1/ε), and also shows the existence of point sets in ℝ^d for which any (1+ε)-spanner must have lightness Ω(ε^{-d}). Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points.
Our first result is a construction of Steiner spanners in ℝ² with lightness O(ε^{-1} log Δ), where Δ is the spread of the point set. In the regime of Δ ≪ 2^(1/ε), this provides an improvement over the lightness bound of [Hung Le and Shay Solomon, 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications ε often controls the precision, and it sometimes needs to be much smaller than O(1/log n). Moreover, for spread polynomially bounded in 1/ε, this upper bound provides a quadratic improvement over the non-Steiner bound of [Hung Le and Shay Solomon, 2019], We then demonstrate that such a light spanner can be constructed in O_ε(n) time for polynomially bounded spread, where O_ε hides a factor of poly(1/(ε)). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ(ε^{-(d+1)/2} + ε^{-2} log Δ) for any 3 ≤ d = O(1) and any ε = Ω(n^{-1/(d-1)}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Euclidean spanners
  • Steiner spanners
  • light spanners

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References

  1. M. A. Abam and S. Har-Peled. New constructions of sspds and their applications. In Proceedings of the 26th Annual Symposium on Computational Geometry, SoCG `10, page 192–200, 2010. URL: https://doi.org/10.1145/1810959.1810993.
  2. P. K. Agarwal, K. Fox, D. Panigrahi, K. R. Varadarajan, and A. Xiao. Faster algorithms for the geometric transportation problem. In 33rd International Symposium on Computational Geometry, volume 77 of SoCG `17, pages 7:1-7:16, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.7.
  3. Stephen Alstrup, Søren Dahlgaard, Arnold Filtser, Morten Stöckel, and Christian Wulff-Nilsen. Constructing light spanners deterministically in near-linear time. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 4:1-4:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  4. I. Althöfer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Computational Geometry, 9(1):81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  5. B Aronov, M. Berg, O. Cheong, J. Gudmundsson, H. Haverkort, and A. Vigneron. Sparse geometric graphs with small dilation. In International Symposium on Algorithms and Computation, ISAAC `05, pages 50-59, 2005. URL: https://doi.org/10.1007/11602613_7.
  6. S. Arya and M. H. M. Smid. Efficient construction of a bounded degree spanner with low weight. Algorithmica, 17(1):33-54, 1997. Google Scholar
  7. Sunil Arya, David M. Mount, and Michiel H. M. Smid. Randomized and deterministic algorithms for geometric spanners of small diameter. In 35th Annual Symposium on Foundations of Computer Science, Santa Fe, New Mexico, USA, 20-22 November 1994, pages 703-712. IEEE Computer Society, 1994. Google Scholar
  8. Gali Bar-On and Paz Carmi. backslashdelta -greedy t-spanner. In Faith Ellen, Antonina Kolokolova, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures - 15th International Symposium, WADS 2017, St. John’s, NL, Canada, July 31 - August 2, 2017, Proceedings, volume 10389 of Lecture Notes in Computer Science, pages 85-96. Springer, 2017. Google Scholar
  9. J. Beardwood, J. H. Halton, and J. M. Hammersley. The shortest path through many points. Mathematical Proceedings of the Cambridge Philosophical Society, 55(4):299-327, 1959. URL: https://doi.org/10.1017/s0305004100034095.
  10. Marc Benkert, Alexander Wolff, Florian Widmann, and Takeshi Shirabe. The minimum manhattan network problem: Approximations and exact solutions. Comput. Geom., 35(3):188-208, 2006. Google Scholar
  11. G. Bodwin and V. V. Williams. Better distance preservers and additive spanners. In Proceedings of the Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 855-872, 2016. Google Scholar
  12. G. Borradaile, H. Le, and C. Wulff-Nilsen. Minor-free graphs have light spanners. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science, FOCS '17, pages 767-778, 2017. URL: https://doi.org/10.1109/FOCS.2017.76.
  13. G. Borradaile, H. Le, and C. Wulff-Nilsen. Greedy spanners are optimal in doubling metrics. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA `19, pages 2371-2379, 2019. URL: https://doi.org/10.1137/1.9781611975482.145.
  14. Prosenjit Bose, Paz Carmi, Mathieu Couture, Michiel H. M. Smid, and Daming Xu. On a family of strong geometric spanners that admit local routing strategies. In Frank K. H. A. Dehne, Jörg-Rüdiger Sack, and Norbert Zeh, editors, Algorithms and Data Structures, 10th International Workshop, WADS 2007, Halifax, Canada, August 15-17, 2007, Proceedings, volume 4619 of Lecture Notes in Computer Science, pages 300-311. Springer, 2007. Google Scholar
  15. Prosenjit Bose, Luc Devroye, Maarten Löffler, Jack Snoeyink, and Vishal Verma. Almost all delaunay triangulations have stretch factor greater than pi/2. Comput. Geom., 44(2):121-127, 2011. Google Scholar
  16. K. Buchin. Delaunay triangulations in linear time? (part I). arXiv preprint, 2008. URL: http://arxiv.org/abs/arXiv:0812.0387.
  17. K. Buchin and W. Mulzer. Linear-time delaunay triangulations simplified. In 25th European Workshop on Computational Geometry, EuroCG '09, pages 235-238, 2009. Google Scholar
  18. M. Bundefineddoiu, J. Chuzhoy, P. Indyk, and A. Sidiropoulos. Low-distortion embeddings of general metrics into the line. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, STOC `05, page 225–233, 2005. URL: https://doi.org/10.1145/1060590.1060624.
  19. T. Carpenter, F. V. Fomin, D. Lokshtanov, S. Saurabh, and A. Sidiropoulos. Algorithms for low-distortion embeddings into arbitrary 1-dimensional spaces. In 34th International Symposium on Computational Geometry, SoCG `2018, pages 21:1-21:14, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.21.
  20. T. Chan. Well-separated pair decomposition in linear time? Information Processing Letters, 107(5):138-141, 2008. URL: https://doi.org/10.1016/j.ipl.2008.02.008.
  21. T.-H. H. Chan, M. Li, L. Ning, and S. Solomon. New doubling spanners: Better and simpler. SIAM Journal on Computing, 44(1):37-53, 2015. URL: https://doi.org/10.1137/130930984.
  22. 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, 2016. Preliminary version appeared in SODA 2005. Google Scholar
  23. B. Chandra, G. Das, G. Narasimhan, and J. Soares. New sparseness results on graph spanners. In Proceedings of the Eighth Annual Symposium on Computational Geometry, 1992. Google Scholar
  24. Barun Chandra. Constructing sparse spanners for most graphs in higher dimensions. Inf. Process. Lett., 51(6):289-294, 1994. Google Scholar
  25. S. Chechik and C. Wulff-Nilsen. Near-optimal light spanners. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'16, pages 883-892, 2016. Google Scholar
  26. Shiri Chechik and Christian Wulff-Nilsen. Near-optimal light spanners. ACM Trans. Algorithms, 14(3):33:1-33:15, 2018. preliminary version published in SODA 2016. URL: https://doi.org/10.1145/3199607.
  27. L. P. Chew. There is a planar graph almost as good as the complete graph. In Proceedings of the Second Annual Symposium on Computational Geometry, SCG `86, pages 169-177, 1986. Google Scholar
  28. L. P. Chew. There are planar graphs almost as good as the complete graph. Journal of Computer and System Sciences, 39(2):205-219, 1989. Google Scholar
  29. K. Clarkson. Approximation algorithms for shortest path motion planning. In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, STOC `87, pages 56-65, 1987. Google Scholar
  30. V. Cohen-Addad and C. Mathieu. Effectiveness of local search for geometric optimization. In 31st International Symposium on Computational Geometry, volume 35 of SoCG `2015, pages 329-343, 2015. URL: https://doi.org/10.4230/LIPIcs.SOCG.2015.329.
  31. G. Das, P. Heffernan, and G. Narasimhan. Optimally sparse spanners in 3-dimensional euclidean space. In Proceedings of the 9th Annual Symposium on Computational Geometry, SCG '93, pages 53-62, 1993. Google Scholar
  32. G. Das, G. Narasimhan, and J. Salowe. A new way to weigh malnourished euclidean graphs. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '95, pages 215-222, 1995. Google Scholar
  33. Michael Elkin, Ofer Neiman, and Shay Solomon. Light spanners. SIAM J. Discret. Math., 29(3):1312-1321, 2015. preliminary version published in ICALP 2014. Google Scholar
  34. Michael Elkin and Shay Solomon. Narrow-shallow-low-light trees with and without steiner points. SIAM J. Discret. Math., 25(1):181-210, 2011. preliminary version published in ESA 2009. Google Scholar
  35. Michael Elkin and Shay Solomon. Steiner shallow-light trees are exponentially lighter than spanning ones. SIAM J. Comput., 44(4):996-1025, 2015. preliminary version published in FOCS 2011. URL: https://doi.org/10.1137/13094791X.
  36. J. Erickson. Dense point sets have sparse delaunay triangulations or "... but not too nasty". Discrete & Computational Geometry, 33(1):83-115, 2004. URL: https://doi.org/10.1007/s00454-004-1089-3.
  37. Mohammad Farshi and Joachim Gudmundsson. Experimental study of geometric t-spanners. In Gerth Stølting Brodal and Stefano Leonardi, editors, Algorithms - ESA 2005, 13th Annual European Symposium, Palma de Mallorca, Spain, October 3-6, 2005, Proceedings, volume 3669 of Lecture Notes in Computer Science, pages 556-567. Springer, 2005. Google Scholar
  38. Mohammad Farshi and Joachim Gudmundsson. Experimental study of geometric t-spanners: A running time comparison. In Camil Demetrescu, editor, Experimental Algorithms, 6th International Workshop, WEA 2007, Rome, Italy, June 6-8, 2007, Proceedings, volume 4525 of Lecture Notes in Computer Science, pages 270-284. Springer, 2007. Google Scholar
  39. Arnold Filtser and Ofer Neiman. Light spanners for high dimensional norms via stochastic decompositions. In 26th Annual European Symposium on Algorithms, ESA 2018, August 20-22, 2018, Helsinki, Finland, pages 29:1-29:15, 2018. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.29.
  40. J. Gao, L. J. Guibas, and A. Nguyen. Deformable spanners and applications. Computational Geometry, 35(1):2-19, 2006. URL: https://doi.org/10.1016/j.comgeo.2005.10.001.
  41. Lee-Ad Gottlieb. A light metric spanner. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 759-772, 2015. URL: https://doi.org/10.1109/FOCS.2015.52.
  42. J. Gudmundsson, C. Levcopoulos, G. Narasimhan, and M. H. M. Smid. Approximate distance oracles for geometric graphs. In Proc. of 13th SODA, pages 828-837, 2002. Google Scholar
  43. J. Gudmundsson, C. Levcopoulos, G. Narasimhan, and M. H. M. Smid. Approximate distance oracles for geometric spanners. ACM Transactions on Algorithms, 4(1), 2008. Google Scholar
  44. J. Gudmundsson, G. Narasimhan, and M. H. M. Smid. Fast pruning of geometric spanners. In Proc. of 22nd STACS, pages 508-520, 2005. Google Scholar
  45. Joachim Gudmundsson, Christos Levcopoulos, Giri Narasimhan, and Michiel H. M. Smid. Approximate distance oracles revisited. In Proc. of 13th ISAAC, pages 357-368, 2002. Google Scholar
  46. Joachim Gudmundsson, Christos Levcopoulos, Giri Narasimhan, and Michiel H. M. Smid. Approximate distance oracles for geometric spanners. ACM Trans. Algorithms, 4(1):10:1-10:34, 2008. Google Scholar
  47. Joachim Gudmundsson, Giri Narasimhan, and Michiel H. M. Smid. Fast pruning of geometric spanners. In Volker Diekert and Bruno Durand, editors, STACS 2005, 22nd Annual Symposium on Theoretical Aspects of Computer Science, Stuttgart, Germany, February 24-26, 2005, Proceedings, volume 3404 of Lecture Notes in Computer Science, pages 508-520. Springer, 2005. Google Scholar
  48. S. Har-Peled. Clustering motion. Discrete and Computational Geometry, 31(4):545-565, 2004. URL: https://doi.org/10.1007/s00454-004-2822-7.
  49. S. Har-peled. Geometric Approximation Algorithms. American Mathematical Society, 2011. URL: https://doi.org/10.5555/2031416.
  50. S. Har-Peled and B. Raichel. Net and prune: A linear time algorithm for euclidean distance problems. Journal of the ACM, 62(6):1-35, 2015. URL: https://doi.org/10.1145/2831230.
  51. S. Har-Peled and B. Sadri. How fast is the k-means method? In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA `05, page 877–885, 2006. Google Scholar
  52. Y. Hassin and D. Peleg. Sparse communication networks and efficient routing in the plane. In Proc. of 19th PODC, pages 41-50, 2000. Google Scholar
  53. R. M. Karp. Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Mathematics of Operations Research, 2(3):209-224, 1977. URL: https://doi.org/10.1287/moor.2.3.209.
  54. J. M. Keil. Approximating the complete euclidean graph. In Proceedings of the first Scandinavian Workshop on Algorithm Theory, SWAT `88, pages 208-213, 1988. Google Scholar
  55. J. M. Keil and C. A. Gutwin. Classes of graphs which approximate the complete Euclidean graph. Discrete and Computational Geometry, 7(1):13-28, 1992. Google Scholar
  56. Michael Kerber and Arnur Nigmetov. Metric spaces with expensive distances. CoRR, abs/1901.08805, 2019. URL: http://arxiv.org/abs/1901.08805.
  57. P. N. Klein. Subset spanner for planar graphs, with application to subset TSP. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, STOC '06, pages 749-756, 2006. URL: https://doi.org/10.1145/1132516.1132620.
  58. Hung Le. A PTAS for subset TSP in minor-free graphs. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2279-2298, 2020. Full version: https://arxiv.org/abs/1804.01588. URL: https://doi.org/10.1137/1.9781611975994.140.
  59. Hung Le and Shay Solomon. Truly optimal euclidean spanners. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 1078-1100, 2019. Full version at https://arxiv.org/abs/1904.12042. URL: https://doi.org/10.1109/FOCS.2019.00069.
  60. Y. Mansour and D. Peleg. An approximation algorithm for min-cost network design. DIMACS Series in Discr. Math and TCS, 53:97-106, 2000. Google Scholar
  61. G. Narasimhan and M. Smid. Geometric Spanner Networks. Cambridge University Press, 2007. Google Scholar
  62. A. Nayyeri and B. Raichel. Reality distortion: Exact and approximate algorithms for embedding into the line. In Proceedings of the 56th Annual Symposium on Foundations of Computer Science, FOCS `15, page 729–747, 2015. URL: https://doi.org/10.1109/FOCS.2015.50.
  63. A. Nayyeri and B. Raichel. A treehouse with custom windows: Minimum distortion embeddings into bounded treewidth graphs. In Proceedings of the 28 th Annual ACM-SIAM Symposium on Discrete Algorithm, SODA `17, page 724–736, 2017. URL: https://doi.org/10.1137/1.9781611974782.46.
  64. A. Nayyeri and B. Raichel. Viewing the rings of a tree: Minimum distortion embeddings into trees. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA `19, pages 2380-2399, 2019. URL: https://doi.org/10.1137/1.9781611975482.146.
  65. S. B. Rao and W. D. Smith. Approximating geometrical graphs via "spanners" and "banyans". In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, STOC '98, pages 540-550, 1998. Full version at http://graphics.stanford.edu/courses/cs468-06-winter/Papers/rs-tsp.pdf. URL: https://doi.org/10.1145/276698.276868.
  66. J. Ruppert and R. Seidel. Approximating the d-dimensional complete Euclidean graph. In Proceedings of the 3rd Canadian Conference on Computational Geometry, CCCG `91, page 207–210, 1991. Google Scholar
  67. Jeffrey S. Salowe. On euclidean spanner graphs with small degree. In Proceedings of the Eighth Annual Symposium on Computational Geometry, Berlin, Germany, June 10-12, 1992, pages 186-191, 1992. Google Scholar
  68. S. Solomon. Euclidean Steiner shallow-light trees. Journal of Computational Geometry, 6(2), 2015. preliminary version published in SoCG 2014. URL: https://doi.org/10.20382/JOCG.V6I2A7.
  69. S. Solomon and M. Elkin. Balancing degree, diameter and weight in euclidean spanners. In Proceedings of the 18th Annual European Symposium on Algorithms, ESA `10, pages 48-59, 2010. URL: https://doi.org/10.1007/978-3-642-15775-2_5.
  70. A. C. Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM Journal on Computing, 11(4):721-736, 1982. Google Scholar
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