Near-Optimal Light Spanners

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 (2 k − 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 (2 k − 1) · (1 + ϵ) stretch and total edge weight of O ϵ (( k / log k ) · ω ( MST ( G )) · n 1/ k ), where ω( MST ( G )) is the weight of a MST of G . The previous analysis by Chandra et al. [SOCG 92] admitted (2 k − 1) · (1 + ϵ) stretch and total edge weight of O ϵ ( k ω( MST ( G )) n 1/ 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 (2 k − 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].