Arnold Filtser
Shay Solomon
ABSTRACT
The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive. In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is existentially optimal (or existentially near-optimal) for several important graph families. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [SODA'16]) and doubling metrics (due to Gottlieb [FOCS'15]) are complex. Plugging our observation on these results, we conclude that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area. Further, we observe that approximate-greedy algorithms are existentially near-optimal as well. Consequently, we provide an O(n log n)-time construction of (1+epsilon)-spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb's construction, whose runtime is O(n log2 n) and whose number of edges and degree are unbounded, and remarkably, it matches the state-of-the-art Euclidean result (due to Gudmundsson et al. [SICOMP'02]) in all the involved parameters (up to dependencies on epsilon and the dimension).
- I. Althfer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81--100, 1993.Google Scholar
Cross Ref
- B. Awerbuch. Communication-time trade-offs in network synchronization. In Proc. of 4th PODC, pages 272--276, 1985. Google ScholarDigital Library
- B. Awerbuch, A. Baratz, and D. Peleg. Cost-sensitive analysis of communication protocols. In Proc. of 9th PODC, pages 177--187, 1990. Google ScholarDigital Library
- B. Awerbuch, A. Baratz, and D. Peleg. Efficient broadcast and light-weight spanners. Manuscript, 1991.Google Scholar
- Y. Bartal, A. Filtser, and O. Neiman. On notions of distortion and an almost minimum spanning tree with constant average distortion. In Proc. of 27th SODA, pages 873--882, 2016. Google ScholarDigital Library
- Y. Ben-Shimol, A. Dvir, and M. Segal. SPLAST: a novel approach for multicasting in mobile wireless ad hoc networks. In Proc. of 15th PIMRC, pages 1011--1015, 2004.Google ScholarCross Ref
- P. Bose, P. Carmi, M. Farshi, A. Maheshwari, and M. H. M. Smid. Computing the greedy spanner in near-quadratic time. Algorithmica, 58(3):711--729, 2010.Google ScholarDigital Library
- R. Braynard, D. Kostic, A. Rodriguez, J. Chase, and A. Vahdat. Opus: an overlay peer utility service. In Prof. of 5th OPENARCH, 2002.Google ScholarCross Ref
- H. T.-H. Chan and A. Gupta. Small hop-diameter sparse spanners for doubling metrics. In Proc. of 17th SODA, pages 70--78, 2006. Google ScholarDigital Library
- H. T.-H. Chan, A. Gupta, B. M. Maggs, and S. Zhou. On hierarchical routing in doubling metrics. In Proc. of 16th SODA, pages 762--771, 2005. Google ScholarDigital Library
- T.-H. H. Chan, M. Li, L. Ning, and S. Solomon. New doubling spanners: Better and simpler. In Proc. of 40th ICALP, pages 315--327, 2013. Google ScholarDigital Library
- B. Chandra, G. Das, G. Narasimhan, and J. Soares. New sparseness results on graph spanners. In Proc. of 8th SOCG, pages 192--201, 1992. Google ScholarDigital Library
- S. Chechik and C. Wulff-Nilsen. Near-optimal light spanners. In Proc. of 27th SODA, pages 883--892, 2016. Google ScholarDigital Library
- L. P. Chew. There is a planar graph almost as good as the complete graph. In Proc. of 2nd SOCG, pages 169--177, 1986. Google ScholarDigital Library
- E. Cohen. Fast algorithms for constructing $t$-spanners and paths with stretch $t$. In Proc. of 34th FOCS, pages 648--658, 1993. Google ScholarDigital Library
- J. Cong, A. B. Kahng, G. Robins, M. Sarrafzadeh, and C. K. Wong. Performance-driven global routing for cell based ics. In Proc. of 9th ICCD, pages 170--173, 1991. Google ScholarDigital Library
- J. Cong, A. B. Kahng, G. Robins, M. Sarrafzadeh, and C. K. Wong. Provably good algorithms for performance-driven global routing. In Proc. of 5th ISCAS, pages 2240--2243, 1992.Google ScholarCross Ref
- J. Cong, A. B. Kahng, G. Robins, M. Sarrafzadeh, and C. K. Wong. Provably good performance-driven global routing. IEEE Trans. on CAD of Integrated Circuits and Sys., 11(6):739--752, 1992. Google ScholarDigital Library
- G. Das, P. J. Heffernan, and G. Narasimhan. Optimally sparse spanners in 3-dimensional Euclidean space. In Proc. of 9th SOCG, pages 53--62, 1993. Google ScholarDigital Library
- G. Das and G. Narasimhan. A fast algorithm for constructing sparse Euclidean spanners. Int. J. Comput. Geometry Appl., 7(4):297--315, 1997.Google ScholarCross Ref
- E. D. Demaine, M. Hajiaghayi, and B. Mohar. Approximation algorithms via contraction decomposition. Combinatorica, 30(5):533--552, 2010. Google ScholarDigital Library
- M. Elkin. Computing almost shortest paths. ACM Transactions on Algorithms, 1(2):283--323, 2005. Google ScholarDigital Library
- M. Elkin, O. Neiman, and S. Solomon. Light spanners. In Proc. of 41th ICALP, pages 442--452, 2014.Google ScholarCross Ref
- M. Elkin and S. Solomon. Fast constructions of light-weight spanners for general graphs. In Proc. of 24th SODA, pages 513--525, 2013. Google ScholarDigital Library
- M. Elkin and J. Zhang. Efficient algorithms for constructing (1Google Scholar
- epsilon, beta)-spanners in the distributed and streaming models. Distributed Computing, 18(5):375--385, 2006.Google ScholarDigital Library
- P. Erd\Hos. Extremal problems in graph theory. In Proc. of Sympos. Smolenice, pages 29--36, 1964.Google Scholar
- M. Farshi. A theoretical and experimental study of geometric networks. PhD thesis, 2008.Google Scholar
- M. Farshi and J. Gudmundsson. Experimental study of geometric pht-spanners. In Proc. of 13th ESA, pages 556--567, 2005. Google ScholarDigital Library
- J. Feigenbaum, S. Kannan, A. McGregor, S. Suri, and J. Zhang. Graph distances in the streaming model: the value of space. In Proc. of 16th SODA, pages 745--754, 2005. Google ScholarDigital Library
- J. Gao, L. J. Guibas, and A. Nguyen. Deformable spanners and applications. In Proc. of 20th SoCG, pages 190--199, 2004. Google ScholarDigital Library
- L. Gottlieb. A light metric spanner. In Proc. of 56th FOCS, pages 759--772, 2015. Google ScholarDigital Library
- L. Gottlieb and L. Roditty. Improved algorithms for fully dynamic geometric spanners and geometric routing. In Proc. of 19th SODA, pages 591--600, 2008. Google ScholarDigital Library
- L. Gottlieb and L. Roditty. Improved algorithms for fully dynamic geometric spanners and geometric routing. In Proc. of 19th SODA, pages 591--600, 2008. Google ScholarDigital Library
- L. Gottlieb and L. Roditty. An optimal dynamic spanner for doubling metric spaces. In Proc. of 16th ESA, pages 478--489, 2008. Google ScholarDigital Library
- M. Grigni. Approximate TSP in graphs with forbidden minors. In Proc. of 27th ICALP, pages 869--877, 2000. Google ScholarDigital Library
- M. Grigni and H. Hung. Light spanners in bounded pathwidth graphs. In Proc. of 37th MFCS, pages 467--477, 2012. Google ScholarDigital Library
- M. Grigni and P. Sissokho. Light spanners and approximate TSP in weighted graphs with forbidden minors. In Proc. of 13th SODA, pages 852--857, 2002. Google ScholarDigital Library
- J. Gudmundsson, C. Levcopoulos, and G. Narasimhan. Fast greedy algorithms for constructing sparse geometric spanners. SIAM J. Comput., 31(5):1479--1500, 2002. Google ScholarDigital Library
- A. Gupta, R. Krauthgamer, and J. R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In Proc. of 44th FOCS, pages 534--543, 2003. Google ScholarDigital Library
- S. Har-Peled and M. Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM J. Comput., 35(5):1148--1184, 2006. Google ScholarDigital Library
- S. Har-Peled and M. Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM J. Comput., 35(5):1148--1184, 2006. Google ScholarDigital Library
- H. Hung. Light spanner and monotone tree. CoRR, abs/1207.3807, 2012.Google Scholar
- P. N. Klein. A linear-time approximation scheme for planar weighted TSP. In Proc. of 46th FOCS, pages 647--657, 2005. Google ScholarDigital Library
- D. Kostic and A. Vahdat. Latency versus cost optimizations in hierarchical overlay networks. Technical report, Duke University, (CS-2001-04), 2002.Google Scholar
- G. Narasimhan and M. Smid. Geometric Spanner Networks. Cambridge University Press, 2007. Google ScholarDigital Library
- D. Peleg. Proximity-preserving labeling schemes and their applications. In Proc. of 25th WG, pages 30--41, 1999. Google ScholarDigital Library
- D. Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, Philadelphia, PA, 2000. Google ScholarDigital Library
- D. Peleg and A. Sch$\ddot\mboxa$ffer. Graph spanners. J. Graph Theory, 13(1):99--116, 1989.Google ScholarCross Ref
- D. Peleg and J. D. Ullman. An optimal synchronizer for the hypercube. SIAM J. Comput., 18(4):740--747, 1989. Google ScholarDigital Library
- D. Peleg and E. Upfal. A trade-off between space and efficiency for routing tables. J. ACM, 36(3):510--530, 1989. Google ScholarDigital Library
- Y. Rabinovich and R. Raz. Lower bounds on the distortion of embedding finite metric spaces in graphs. Discrete & Computational Geometry, 19(1):79--94, 1998.Google ScholarCross Ref
- L. Roditty. Fully dynamic geometric spanners. In Proc. of 23rd SoCG, pages 373--380, 2007. Google ScholarDigital Library
- L. Roditty, M. Thorup, and U. Zwick. Deterministic constructions of approximate distance oracles and spanners. In Proc. of 32nd ICALP, pages 261--272, 2005. Google ScholarDigital Library
- L. Roditty and U. Zwick. On dynamic shortest paths problems. In Proc. of 32nd ESA, pages 580--591, 2004.Google ScholarCross Ref
- F. S. Salman, J. Cheriyan, R. Ravi, and S. Subramanian. Approximating the single-sink link-installation problem in network design. SIAM Journal on Optimization, 11(3):595--610, 2001. Google ScholarDigital Library
- H. Shpungin and M. Segal. Near optimal multicriteria spanner constructions in wireless ad-hoc networks. IEEE/ACM Transactions on Networking, 18(6):1963--1976, 2010. Google ScholarDigital Library
- M. H. M. Smid. The weak gap property in metric spaces of bounded doubling dimension. In Proc. of Efficient Algorithms, pages 275--289, 2009. Google ScholarDigital Library
- S. Solomon. From hierarchical partitions to hierarchical covers: optimal fault-tolerant spanners for doubling metrics. In Proc. of 46th STOC, pages 363--372, 2014. Google ScholarDigital Library
- M. Thorup and U. Zwick. Approximate distance oracles. In Proc. of 33rd STOC, pages 183--192, 2001. Google ScholarDigital Library
- M. Thorup and U. Zwick. Compact routing schemes. In Proc. of 13th SPAA, pages 1--10, 2001. Google ScholarDigital Library
- J. Vogel, J. Widmer, D. Farin, M. Mauve, and W. Effelsberg. Priority-based distribution trees for application-level multicast. In Proc. of 2nd NETGAMES, pages 148--157, 2003. Google ScholarDigital Library
- P. von Rickenbach and R. Wattenhofer. Gathering correlated data in sensor networks. In Proc. of DIALM-POMC, pages 60--66, 2004. Google ScholarDigital Library
- B. Y. Wu, K. Chao, and C. Y. Tang. Light graphs with small routing cost. Networks, 39(3):130--138, 2002.Google ScholarCross Ref
Index Terms
- The Greedy Spanner is Existentially Optimal
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