On the Size Overhead of Pairwise Spanners

Ofer Neiman, Idan Shabat

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Given an undirected possibly weighted n-vertex graph G = (V,E) and a set P V 2 of pairs, a subgraph S = (V,E) is called a P-pairwise α-spanner of G, if for every pair (u, v) P we have dS(u, v) α dG(u, v). The parameter α is called the stretch of the spanner, and its size overhead is define as |E| |P| . A surprising connection was recently discussed between the additive stretch of (1+, β)-spanners, to the hopbound of (1 + , β)-hopsets. A long sequence of works showed that if the spanner/hopset has size n1+1/k for some parameter k 1, then β 1 log k. In this paper we establish a new connection to the size overhead of pairwise spanners. In particular, we show that if |P| n1+1/k, then a P-pairwise (1 + )-spanner must have size at least β |P| with β 1 log k (a near matching upper bound was recently shown in [18]). That is, the size overhead of pairwise spanners has similar bounds to the hopbound of hopsets, and to the additive stretch of spanners. We also extend the connection between pairwise spanners and hopsets to the large stretch regime, by showing nearly matching upper and lower bounds for P-pairwise α-spanners. In particular, we show that if |P| n1+1/k, then the size overhead is β k α. A source-wise spanner is a special type of pairwise spanner, for which P = A × V for some A V . A prioritized spanner is given also a ranking of the vertices V = (v1, . . . , vn), and is required to provide improved stretch for pairs containing higher ranked vertices. By using a sequence of reductions: from pairwise spanners to source-wise spanners to prioritized spanners, we improve on the state-of-The-Art results for source-wise and prioritized spanners. Since our spanners can be equipped with a path-reporting mechanism, we also substantially improve the known bounds for path-reporting prioritized distance oracles. Specifically, we provide a path-reporting distance oracle, with size O(n (log log n)2), that has a constant stretch for any query that contains a vertex ranked among the first n1 vertices (for any constant 0). Such a result was known before only for non-path-reporting distance oracles.

Original languageEnglish
Title of host publication15th Innovations in Theoretical Computer Science Conference, ITCS 2024
EditorsVenkatesan Guruswami
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959773096
DOIs
StatePublished - 1 Jan 2024
Event15th Innovations in Theoretical Computer Science Conference, ITCS 2024 - Berkeley, United States
Duration: 30 Jan 20242 Feb 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume287
ISSN (Print)1868-8969

Conference

Conference15th Innovations in Theoretical Computer Science Conference, ITCS 2024
Country/TerritoryUnited States
CityBerkeley
Period30/01/242/02/24

Keywords

  • Graph Algorithms
  • Shortest Paths
  • Spanners

ASJC Scopus subject areas

  • Software

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