The norms of graph spanners

Eden Chlamtáč, Michael Dinitz, Thomas Robinson

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

2 Scopus citations


A t-spanner of a graph G is a subgraph H in which all distances are preserved up to a multiplicative t factor. A classical result of Althöfer et al. is that for every integer k and every graph G, there is a (2k − 1)-spanner of G with at most O(n1+1/k) edges. But for some settings the more interesting notion is not the number of edges, but the degrees of the nodes. This spurred interest in and study of spanners with small maximum degree. However, this is not necessarily a robust enough objective: we would like spanners that not only have small maximum degree, but also have “few” nodes of “large” degree. To interpolate between these two extremes, in this paper we initiate the study of graph spanners with respect to the `p-norm of their degree vector, thus simultaneously modeling the number of edges (the `1-norm) and the maximum degree (the `∞-norm). We give precise upper bounds for all ranges of p and stretch t: we prove that the greedy (2k− 1)-spanner has `p norm of at k+p most max(O(n), O(n kp )), and that this bound is tight (assuming the Erdős girth conjecture). We also study universal lower bounds, allowing us to give “generic” guarantees on the approximation ratio of the greedy algorithm which generalize and interpolate between the known approximations for the `1 and `∞ norm. Finally, we show that at least in some situations, the `p norm behaves fundamentally differently from `1 or `∞: there are regimes (p = 2 and stretch 3 in particular) where the greedy spanner has a provably superior approximation to the generic guarantee.

Original languageEnglish
Title of host publication46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
EditorsChristel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771092
StatePublished - 1 Jul 2019
Event46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 - Patras, Greece
Duration: 9 Jul 201912 Jul 2019

Publication series

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


Conference46th International Colloquium on Automata, Languages, and Programming, ICALP 2019


  • Approximations
  • Spanners

ASJC Scopus subject areas

  • Software

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