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 O_ϵ ((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 O_ϵ (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, O_ϵ (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].