## Abstract

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) ≤ δ_{G}(u, v) ≤ t \cdot δ(u, v), where δ_{G} 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 language | English |
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Pages (from-to) | 1030-1356 |

Number of pages | 327 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 38 |

Issue number | 1 |

DOIs | |

State | Published - 2024 |

### Bibliographical note

Publisher Copyright:© 2024 Society for Industrial and Applied Mathematics Publications. All rights reserved.

## Keywords

- approximation algorithms
- metric spaces
- online algorithms
- online spanners
- randomized algorithms
- spanners