Near-Optimal Light Spanners

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].

[1]  Béla Bollobás,et al.  Extremal problems in graph theory , 1977, J. Graph Theory.

[2]  Mikkel Thorup,et al.  Deterministic Constructions of Approximate Distance Oracles and Spanners , 2005, ICALP.

[3]  Mikkel Thorup,et al.  Spanners and emulators with sublinear distance errors , 2006, SODA '06.

[4]  Kurt Mehlhorn,et al.  New constructions of (α, β)-spanners and purely additive spanners , 2005, SODA '05.

[5]  David P. Dobkin,et al.  On sparse spanners of weighted graphs , 1993, Discret. Comput. Geom..

[6]  Michiel H. M. Smid,et al.  The Weak Gap Property in Metric Spaces of Bounded Doubling Dimension , 2009, Efficient Algorithms.

[7]  Christopher M. Hartman Extremal problems in graph theory , 1997 .

[8]  Arthur M. Farley,et al.  Spanners and message distribution in networks , 2004, Discret. Appl. Math..

[9]  Seth Pettie,et al.  Low distortion spanners , 2007, TALG.

[10]  Eli Upfal,et al.  A trade-off between space and efficiency for routing tables , 1989, JACM.

[11]  Shiri Chechik,et al.  New Additive Spanners , 2013, SODA.

[12]  Giri Narasimhan,et al.  Optimally sparse spanners in 3-dimensional Euclidean space , 1993, SCG '93.

[13]  Shiri Chechik,et al.  Approximate Distance Oracles with Improved Bounds , 2015, STOC.

[14]  David Peleg,et al.  (1+epsilon, beta)-Spanner Constructions for General Graphs , 2004, SIAM J. Comput..

[15]  Béla Bollobás,et al.  Sparse distance preservers and additive spanners , 2003, SODA '03.

[16]  Christian Wulff-Nilsen,et al.  Approximate distance oracles with improved preprocessing time , 2011, SODA.

[17]  David P. Woodruff Lower Bounds for Additive Spanners, Emulators, and More , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[18]  Michael Elkin,et al.  Optimal euclidean spanners: really short, thin and lanky , 2012, STOC '13.

[19]  Uri Zwick,et al.  All-Pairs Almost Shortest Paths , 1997, SIAM J. Comput..

[20]  Mikkel Thorup,et al.  Approximate distance oracles , 2005, J. ACM.

[21]  Michael Elkin,et al.  Light Spanners , 2014, SIAM J. Discret. Math..

[22]  Nicholas Kalouptsidis,et al.  Efficient Algorithms for , 1999 .

[23]  Mikkel Thorup,et al.  Compact routing schemes , 2001, SPAA '01.

[24]  Kurt Mehlhorn,et al.  New constructions of (alpha, beta)-spanners and purely additive spanners , 2005, SODA.

[25]  Shay Solomon,et al.  The Greedy Spanner is Existentially Optimal , 2016, PODC.

[26]  Kurt Mehlhorn,et al.  Additive spanners and (α, β)-spanners , 2010, TALG.

[27]  Shiri Chechik Compact routing schemes with improved stretch , 2013, PODC '13.

[28]  David Peleg,et al.  An optimal synchronizer for the hypercube , 1987, PODC '87.

[29]  Jose Augusto Ramos Soares,et al.  Graph Spanners: a Survey , 1992 .

[30]  David P. Woodruff Additive Spanners in Nearly Quadratic Time , 2010, ICALP.

[31]  Giri Narasimhan,et al.  New sparseness results on graph spanners , 1995, Int. J. Comput. Geom. Appl..

[32]  Andrew V. Goldberg,et al.  Network decomposition and locality in distributed computation , 1989, 30th Annual Symposium on Foundations of Computer Science.

[33]  Piotr Indyk,et al.  Fast estimation of diameter and shortest paths (without matrix multiplication) , 1996, SODA '96.

[34]  Lee-Ad Gottlieb,et al.  A Light Metric Spanner , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[35]  Shiri Chechik,et al.  Approximate Distance Oracle with Constant Query Time , 2013, ArXiv.