NDLI: Near-optimal light spanners

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Near-optimal light spanners

Content Provider	ACM Digital Library
Author	Wulff-Nilsen, Christian Chechik, Shiri
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 $O(n^{1+1/k})$ 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 $O_{ε}((k/log$ k) · ω(MST(G)) · $[EQUATION]^{1/k}),$ 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 + ε) stretch and total edge weight of $O_{ε}(kω(MST(G))n^{1/k}).$ 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 + ε) stretch, $O_{ε}(ω(MST(G))n^{1/k})$ total weight, and $O(n^{1+1/k})$ edges. Up to a (1 + ε) factor in the stretch this matches the girth conjecture of Erdős [17].
Starting Page	883
Ending Page	892
Page Count	10
File Format	PDF
ISBN	9781611974331
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2016-01-10
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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