Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, Csaba D. Tóth

Research output: Contribution to journalArticlepeer-review


Given a metric space ℳ = (X, δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u, v \in X, δ(u, v) ≤ δG(u, v) ≤ t \cdot δ(u, v), where δG is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s1, . . ., sn), where the points are presented one at a time (i.e., after i steps, we see Si = \{s1, . . ., si\}). The algorithm is allowed to add edges to the spanner when a new point arrives; however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner Gi for Si for all i, while minimizing the number of edges, and their total weight. We construct online (1 + ε)-spanners in the Euclidean d-space, (2k- 1)(1 + ε)-spanners for general metrics, and (2 + ε)-spanners for ultrametrics. Most notably, in the Euclidean plane, we construct a (1 + ε)-spanner with competitive ratio O(ε-3/2 log ε-1 log n), bypassing the classic lower bound \Omega(ε-2) for lightness, which compares the weight of the spanner to that of the minimum spanning tree.

Original languageEnglish
Pages (from-to)1030-1356
Number of pages327
JournalSIAM Journal on Discrete Mathematics
Issue number1
StatePublished - 2024

Bibliographical note

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  • approximation algorithms
  • metric spaces
  • online algorithms
  • online spanners
  • randomized algorithms
  • spanners


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