Near-optimal light spanners

Shiri Chechik, Christian Wulff-Nilsen

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


A spanner if of a weighted undirected graph G is a "sparse" subgraph that approximately preserves distances between every pair of vertices in G. We refer to if 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-l)-spanner of G with 0(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 Erdös [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 + ∈) stretch and total edge weight of Oc((k/logk) · ω(MST(G)) · n1/k), where ω(MST(G)) is the weight of a minimum spanning tree of G. The previous analysis by Chandra et al. [SOCG 92] admitted (2k-1) · (1 + ∈) stretch and total edge weight of Ot(kω(MST(G))nl/k). Hence, Elkin et al. improved the weight of the spanner by a log k factor. In this work, we complectly remove the k factor from the weight, presenting a spanner with (2k-1) · (1 + ∈) stretch, O∈(ω(MST(G))n1/k) total weight, and O(n1+1/k) edges. Up to a (1 + ∈) factor in the stretch this matches the girth conjecture of Erdös [17].

Original languageEnglish
Title of host publication27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016
EditorsRobert Krauthgamer
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Electronic)9781510819672
StatePublished - 2016
Event27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States
Duration: 10 Jan 201612 Jan 2016

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016
Country/TerritoryUnited States


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