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Light Euclidean Steiner Spanners in the Plane

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Sujoy Bhore and Csaba D. Tóth. Light Euclidean Steiner Spanners in the Plane. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.15

Abstract

Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in ℝ^d. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on ε > 0 and d ∈ ℕ of the minimum lightness of a (1+ε)-spanner, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner (1+ε)-spanners of lightness O(ε^{-1}logΔ) in the plane, where Δ ≥ Ω(√n) is the spread of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness Õ(ε^{-(d+1)/2}) in dimensions d ≥ 3. Recently, Bhore and Tóth (2020) established a lower bound of Ω(ε^{-d/2}) for the lightness of Steiner (1+ε)-spanners in ℝ^d, for d ≥ 2. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions d ≥ 2. In this work, we show that for every finite set of points in the plane and every ε > 0, there exists a Euclidean Steiner (1+ε)-spanner of lightness O(ε^{-1}); this matches the lower bound for d = 2. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.

Subject Classification

ACM Subject Classification
• Mathematics of computing → Approximation algorithms
• Mathematics of computing → Paths and connectivity problems
• Theory of computation → Computational geometry
Keywords
• Geometric spanner
• lightness
• minimum weight

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References

1. Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81-100, 1993.
2. Srinivasa Rao Arikati, Danny Z. Chen, L. Paul Chew, Gautam Das, Michiel H. M. Smid, and Christos D. Zaroliagis. Planar spanners and approximate shortest path queries among obstacles in the plane. In Proc. 4th European Symposium on Algorithms (ESA), volume 1136 of LNCS, pages 514-528. Springer, 1996.
3. Sunil Arya, David M Mount, and Michiel Smid. Randomized and deterministic algorithms for geometric spanners of small diameter. In Proc. 35th IEEE Symposium on Foundations of Computer Science (FOCS), pages 703-712, 1994.
4. Sunil Arya and Michiel Smid. Efficient construction of a bounded-degree spanner with low weight. Algorithmica, 17(1):33-54, 1997.
5. Baruch Awerbuch, Alan E. Baratz, and David Peleg. Cost-sensitive analysis of communication protocols. In Proc. 9th ACM Symposium on Principles of Distributed Computing (PODC), pages 177-187, 1990.
6. Sujoy Bhore and Csaba D. Tóth. Light Euclidean Steiner spanners in the plane. Preprint, 2020. URL: http://arxiv.org/abs/2012.02216.
7. Sujoy Bhore and Csaba D. Tóth. On Euclidean Steiner (1+ε)-spanners. In Proc. 38th Symposium on Theoretical Aspects of Computer Science (STACS), volume 187 of LIPIcs, pages 13:1-13:16. Schloss Dagstuhl, 2021.
8. Glencora Borradaile and David Eppstein. Near-linear-time deterministic plane Steiner spanners for well-spaced point sets. Comput. Geom., 49:8-16, 2015.
9. Glencora Borradaile, Hung Le, and Christian Wulff-Nilsen. Greedy spanners are optimal in doubling metrics. In Proc. 13th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2371-2379, 2019.
10. Prosenjit Bose and Michiel H. M. Smid. On plane geometric spanners: A survey and open problems. Comput. Geom., 46(7):818-830, 2013.
11. Paul B. Callahan. Optimal parallel all-nearest-neighbors using the well-separated pair decomposition. In Proc. 34th IEEE Symposium on Foundations of Computer Science (FOCS), pages 332-340, 1993.
12. Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones. On locality-sensitive orderings and their applications. SIAM J. Comput., 49(3):583-600, 2020.
13. Shiri Chechik and Christian Wulff-Nilsen. Near-optimal light spanners. ACM Transactions on Algorithms (TALG), 14(3):1-15, 2018.
14. L. Paul Chew. There is a planar graph almost as good as the complete graph. In Proc. 2nd Symposium on Computational Geometry, pages 169-177. ACM Press, 1986.
15. L. Paul Chew. There are planar graphs almost as good as the complete graph. J. Comput. Syst. Sci., 39(2):205-219, 1989.
16. Kenneth L. Clarkson. Approximation algorithms for shortest path motion planning. In Proc. 19th ACM Symposium on Theory of Computing (STOC), pages 56-65, 1987.
17. Gautam Das, Paul Heffernan, and Giri Narasimhan. Optimally sparse spanners in 3-dimensional Euclidean space. In Proc. 9th Symposium on Computational Geometry (SoCG), pages 53-62. ACM Press, 1993.
18. Gautam Das, Giri Narasimhan, and Jeffrey S. Salowe. A new way to weigh malnourished Euclidean graphs. In Proc. 6th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 215-222, 1995.
19. Michael J. Demmer and Maurice P. Herlihy. The arrow distributed directory protocol. In Proc. 12th Symposium on Distributed Computing (DISC), volume 1499 of LNCS, pages 119-133. Springer, 1998.
20. Adrian Dumitrescu and Csaba D. Tóth. Light orthogonal networks with constant geometric dilation. J. Discrete Algorithms, 7(1):112-129, 2009.
21. Herbert Edelsbrunner, Joseph O'Rourke, and Emmerich Welzl. Stationing guards in rectilinear art galleries. Computer Vision, Graphics, and Image Processing, 27(2):167-176, 1984.
22. Michael Elkin, Ofer Neiman, and Shay Solomon. Light spanners. SIAM Journal on Discrete Mathematics, 29(3):1312-1321, 2015.
23. Michael Elkin and Shay Solomon. Steiner shallow-light trees are exponentially lighter than spanning ones. SIAM Journal on Computing, 44(4):996-1025, 2015.
24. Jie Gao, Leonidas J. Guibas, and An Nguyen. Deformable spanners and applications. Comput. Geom., 35(1-2):2-19, 2006.
25. Lee-Ad Gottlieb. A light metric spanner. In 2015 IEEE 56th Symposium on Foundations of Computer Science, pages 759-772, 2015.
26. Lee-Ad Gottlieb, Aryeh Kontorovich, and Robert Krauthgamer. Efficient regression in metric spaces via approximate Lipschitz extension. IEEE Transactions on Information Theory, 63(8):4838-4849, 2017.
27. Joachim Gudmundsson, Christos Levcopoulos, and Giri Narasimhan. Fast greedy algorithms for constructing sparse geometric spanners. SIAM J. Comput., 31(5):1479-1500, 2002.
28. Joachim Gudmundsson, Christos Levcopoulos, Giri Narasimhan, and Michiel Smid. Approximate distance oracles for geometric spanners. ACM Transactions on Algorithms (TALG), 4(1):1-34, 2008.
29. Maurice Herlihy, Srikanta Tirthapura, and Rogert Wattenhofer. Competitive concurrent distributed queuing. In Proc. 20th ACM Symposium on Principles of Distributed Computing (PODC), pages 127-133, 2001.
30. J. Mark Keil. Approximating the complete Euclidean graph. In Proc. 1st Scandinavian Workshop on Algorithm Theory (SWAT), volume 318 of LNCS, pages 208-213. Springer, 1988.
31. J. Mark Keil and Carl A. Gutwin. Classes of graphs which approximate the complete Euclidean graph. Discrete & Computational Geometry, 7:13-28, 1992.
32. Samir Khuller, Balaji Raghavachari, and Neal E. Young. Balancing minimum spanning and shortest path trees. In Proc. 4th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 243-250, 1993.
33. Hung Le and Shay Solomon. Truly optimal Euclidean spanners. In Proc. 60th IEEE Symposium on Foundations of Computer Science (FOCS), pages 1078-1100, 2019.
34. Hung Le and Shay Solomon. Light Euclidean spanners with Steiner points. In Proc. 28th European Symposium on Algorithms (ESA), volume 173 of LIPIcs, pages 67:1-67:22. Schloss Dagstuhl, 2020.
35. Hung Le and Shay Solomon. A unified and fine-grained approach for light spanners. CoRR, abs/2008.10582, 2020. URL: http://arxiv.org/abs/2008.10582.
36. Christos Levcopoulos. Heuristics for Minimum Decompositions of Polygons. PhD thesis, Linköping, 1987. No. 74 of Linköping Studies in Science and Technology.
37. Anil Maheshwari, Jörg-Rüdiger Sack, and Hristo N. Djidjev. Link distance problems. In Jörg-Rüdiger Sacks and Jorge Urutia, editors, Handbook of Computational Geometry, chapter 12, pages 519-558. North-Holland, 2000.
38. Anil Maheshwari, Michiel H. M. Smid, and Norbert Zeh. I/O-efficient algorithms for computing planar geometric spanners. Comput. Geom., 40(3):252-271, 2008.
39. Giri Narasimhan and Michiel Smid. Geometric Spanner Networks. Cambridge University Press, 2007.
40. David Peleg and Alejandro A. Schäffer. Graph spanners. Journal of Graph Theory, 13(1):99-116, 1989.
41. David Peleg and Jeffrey D. Ullman. An optimal synchronizer for the hypercube. SIAM J. Comput., 18(4):740-747, 1989.
42. David Peleg and Eli Upfal. A trade-off between space and efficiency for routing tables. Journal of the ACM (JACM), 36(3):510-530, 1989.
43. Satish B. Rao and Warren D. Smith. Approximating geometrical graphs via “spanners” and “banyans”. In Proc. 13th ACM Symposium on Theory of Computing (STOC), pages 540-550, 1998.
44. Jim Ruppert and Raimund Seidel. Approximating the d-dimensional complete Euclidean graph. In Proc. 3rd Canadian Conference on Computational Geometry (CCCG), pages 207-210, 1991.
45. Christian Schindelhauer, Klaus Volbert, and Martin Ziegler. Geometric spanners with applications in wireless networks. Comput. Geom., 36(3):197-214, 2007.
46. Michiel Smid. The weak gap property in metric spaces of bounded doubling dimension. In Efficient Algorithms, pages 275-289. Springer, 2009.
47. Shay Solomon. Euclidean Steiner shallow-light trees. J. Comput. Geom., 6(2):113-139, 2015.
48. Subhash Suri. On some link distance problems in a simple polygon. IEEE Trans. Robotics Autom., 6(1):108-113, 1990.
49. Andrew Chi-Chih Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comput., 11(4):721-736, 1982.
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