Shiri Chechik
Abstract
A spanner H of a weighted undirected graph G is a “sparse” subgraph that approximately preserves distances between every pair of vertices in G. We refer to H as a δ-spanner of G for some parameter δ ≥ 1 if the distance in H between every vertex pair is at most a factor δ bigger than in G. In this case, we say that H has stretch δ. Two main measures of the sparseness of a spanner are the size (number of edges) and the total weight (the sum of weights of the edges in the spanner).
It is well-known that for any positive integer k, one can efficiently construct a (2k − 1)-spanner of G with O(n1+1/k) edges where n is the number of vertices [2]. This size-stretch tradeoff is conjectured to be optimal based on a girth conjecture of Erdős [17]. However, the current state of the art for the second measure is not yet optimal.
Recently Elkin, Neiman and Solomon [ICALP 14] presented an improved analysis of the greedy algorithm, proving that the greedy algorithm admits (2k − 1) · (1 + ϵ) stretch and total edge weight of Oϵ ((k/ log k) · ω (MST(G)) · n1/k), where ω(MST(G)) is the weight of a MST of G. The previous analysis by Chandra et al. [SOCG 92] admitted (2k − 1) · (1 + ϵ) stretch and total edge weight of Oϵ(kω(MST(G))n1/k). Hence, Elkin et al. improved the weight of the spanner by a log k factor.
In this article, we completely remove the k factor from the weight, presenting a spanner with (2k − 1) · (1 + ϵ) stretch, Oϵ(ω(MST(G))n1/k) total weight, and O(n1+1/k) edges. Up to a (1 + ϵ) factor in the stretch this matches the girth conjecture of Erdős [17].
- D. Aingworth, C. Chekuri, P. Indyk, and R. Motwani. 1999. Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput. 28, 4 (1999), 1167--1181. arXiv:http://epubs.siam.org/doi/pdf/10.1137/S0097539796303421 Google Scholar
Digital Library
- Ingo Althöfer, Gautam Das, David P. Dobkin, Deborah Joseph, and José Soares. 1993. On sparse spanners of weighted graphs. Discrete Comput. Geom. 9 (1993), 81--100.Google ScholarCross Ref
- B. Awerbuch, M. Luby, A. V. Goldberg, and S. A. Plotkin. 1989. Network decomposition and locality in distributed computation. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science (FOCS’89). 364--369. Google ScholarDigital Library
- Surender Baswana, Telikepalli Kavitha, Kurt Mehlhorn, and Seth Pettie. 2005. New constructions of (alpha, beta)-spanners and purely additive spanners. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23-25, 2005. 672--681. http://dl.acm.org/citation.cfm?id=1070432.1070526. Google ScholarDigital Library
- Surender Baswana, Telikepalli Kavitha, Kurt Mehlhorn, and Seth Pettie. 2010. Additive spanners and (alpha, beta)-spanners. ACM Trans. Algorithms 7, 1 (2010), 5. Google ScholarDigital Library
- Béla Bollobás, Don Coppersmith, and Michael Elkin. 2005. Sparse distance preservers and additive spanners. SIAM J. Discrete Math. 19, 4 (2005), 1029--1055. Google ScholarDigital Library
- Barun Chandra, Gautam Das, Giri Narasimhan, and José Soares. 1992. New sparseness results on graph spanners. In Proceedings of the 8th Annual Symposium on Computational Geometry, Berlin, Germany, June 10-12, 1992. 192--201. Google ScholarDigital Library
- Shiri Chechik. 2013. Compact routing schemes with improved stretch. In Proc. 2013 ACM Symposium on Principles of Distributed Computing (PODC’13). 33--41. Google ScholarDigital Library
- Shiri Chechik. 2013. New additive spanners. In Proc. 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’13). New Orleans, Louisiana, 498--512. Google ScholarDigital Library
- Shiri Chechik. 2014. Approximate distance oracles with constant query time. In Proc. 46th Annual Symposium on the Theory of Computing (STOC’14). New York, NY, USA, 654--663. Google ScholarDigital Library
- Shiri Chechik. 2015. Approximate distance oracles with improved bounds. In Proc. 47th Annual Symposium on the Theory of Computing (STOC’15). Portland, OR, USA, 1--10. Google ScholarDigital Library
- Gautam Das, Paul Heffernan, and Giri Narasimhan. 1993. Optimally sparse spanners in 3-dimensional euclidean space. In Proceedings of the 9th Annual Symposium on Computational Geometry (SCG’93). ACM, 53--62. Google ScholarDigital Library
- D. Dor, S. Halperin, and U. Zwick. 2000. All-pairs almost shortest paths. SIAM J. Comput. 29, 5 (2000), 1740--1759. Google ScholarDigital Library
- Michael Elkin, Ofer Neiman, and Shay Solomon. 2014. Light spanners. In Automata, Languages, and Programming—41st International Colloquium (ICALP’14), Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I. 442--452.Google Scholar
- Michael Elkin and David Peleg. 2004. (1+epsilon, beta)-spanner constructions for general graphs. SIAM J. Comput. 33, 3 (2004), 608--631. Google ScholarDigital Library
- Michael Elkin and Shay Solomon. 2013. Optimal Euclidean spanners: Really short, thin and lanky. In Proceedings of the Symposium on Theory of Computing Conference (STOC’13). 645--654. Google ScholarDigital Library
- P. Erdős. 1964. Extremal problems in graph theory. In Theory of Graphs and Its Applications (Proc. Sympos. Smolenice, 1963). Publ. House Czechoslovak Acad. Sci., Prague, 29--36.Google Scholar
- Arthur M. Farley, Andrzej Proskurowski, Daniel Zappala, and Kurt J. Windisch. 2004. Spanners and message distribution in networks. Discrete Appl. Math. 137, 2 (2004), 159--171. Google ScholarDigital Library
- Arnold Filtser and Shay Solomon. 2016. The greedy spanner is existentially optimal. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing (PODC’16), Chicago, IL, USA, July 25-28, 2016. 9--17. Google ScholarDigital Library
- Lee-Ad Gottlieb. 2015. A light metric spanner. In Proceedings of the Symposium on Foundations of Computer Science (FOCS’15). CoRR abs/1505.03681 http://arxiv.org/abs/1505.03681 Google ScholarDigital Library
- David Peleg and Alejandro A. Schäffer. 1989. Graph spanners. J. Graph Theor. 13, 1 (1989), 99--116.Google ScholarCross Ref
- David Peleg and Jeffrey D. Ullman. 1989. An optimal synchronizer for the hypercube. SIAM J. Comput. 18, 4 (1989), 740--747. Google ScholarDigital Library
- David Peleg and Eli Upfal. 1989. A trade-off between space and efficiency for routing tables. J. ACM 36, 3 (1989), 510--530. Google ScholarDigital Library
- Seth Pettie. 2009. Low distortion spanners. ACM Trans. Algorithms 6, 1 (2009). Google ScholarDigital Library
- Liam Roditty, Mikkel Thorup, and Uri Zwick. 2005. Deterministic constructions of approximate distance oracles and spanners. In Proc. 32nd International Colloquium on Automata, Languages and Programming (ICALP’05). Lisboa, Portugal, 261--272. Google ScholarDigital Library
- Michiel Smid. 2009. Efficient algorithms. 275--289.Google Scholar
- Mikkel Thorup and Uri Zwick. 2001. Compact routing schemes. In Proc. 13th ACM Symposium on Parallel Algorithms and Architectures (SPAA’01). Crete Island, Greece, 1--10. Google ScholarDigital Library
- Mikkel Thorup and Uri Zwick. 2005. Approximate distance oracles. J. ACM 52, 1 (2005), 1--24. Google ScholarDigital Library
- Mikkel Thorup and Uri Zwick. 2006. Spanners and emulators with sublinear distance errors. In Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA’06). Miami, FL, USA, 802--809. Google ScholarDigital Library
- David P. Woodruff. 2006. Lower bounds for additive spanners, emulators, and more. In Proc. 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06). Berkeley, CA, USA, 389--398. Google ScholarDigital Library
- David P. Woodruff. 2010. Additive spanners in nearly quadratic time. In Proc. 37th International Colloquium on Automata, Languages and Programming (ICALP’10) (1). Bordeaux, France, 463--474. Google ScholarDigital Library
- Christian Wulff-Nilsen. 2012. Approximate distance oracles with improved preprocessing time. In Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA’12). Kyoto, Japan, 202--208. Google ScholarDigital Library
Index Terms
- Near-Optimal Light Spanners
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