On Additive Spanners in Weighted Graphs with Local Error

Reyan Ahmed, Greg Bodwin, Keaton Hamm, Stephen Kobourov, Richard Spence

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

An additive + β spanner of a graph G is a subgraph which preserves distances up to an additive + β error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al. 2019 and 2020, Ahmed et al. 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al. 2020] provided constructions of sparse spanners with global error β= cW, where W is the maximum edge weight in G and c is constant. We improve these to local error by giving spanners with additive error + cW(s, t) for each vertex pair (s, t), where W(s, t) is the maximum edge weight along the shortest s–t path in G. These include pairwise + (2 + ε) W(·, · ) and + (6 + ε) W(·, · ) spanners over vertex pairs P⊆ V× V on Oε(n| P|1 / 3) and Oε(n| P|1 / 4) edges for all ε> 0, which extend previously known unweighted results up to ε dependence, as well as an all-pairs + 4 W(·, · ) spanner on O~ (n7 / 5) edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its lightness, defined as the total edge weight of the spanner divided by the weight of an MST of G. We provide a + εW(·, · ) spanner with Oε(n) lightness, and a + (4 + ε) W(·, · ) spanner with Oε(n2 / 3) lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.

Original languageEnglish (US)
Title of host publicationGraph-Theoretic Concepts in Computer Science - 47th International Workshop, WG 2021, Revised Selected Papers
EditorsLukasz Kowalik, Michal Pilipczuk, Pawel Rzazewski
PublisherSpringer Science and Business Media Deutschland GmbH
Pages361-373
Number of pages13
ISBN (Print)9783030868376
DOIs
StatePublished - 2021
Event47th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2021 - Virtual, Online
Duration: Jun 23 2021Jun 25 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12911 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference47th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2021
CityVirtual, Online
Period6/23/216/25/21

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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