## 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)) · 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 (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 article, we completely 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].

Original language | English |
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Article number | 33 |

Journal | ACM Transactions on Algorithms |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2018 |

## Keywords

- Light spanner

## All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)