Online Spanners in Metric Spaces

Abstract

Given a metric space M = (X, δ), a weighted graph G over X is a metric t-spanner of M if for every, where d 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 (s 1 , . . . , s n ), where the points are presented one at a time (i.e., after i steps, we saw S i = {s 1 , . . . , s i }). 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 G i for S i for all i, while minimizing the number of edges, and their total weight.We construct online (1+ε)-spanners in Euclidean d-space, (2k −1)(1+ε)-spanners for general metrics, and (2 + ε)-spanners for ultrametrics. Most notably, in 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 Ω(ε −2 ) for lightness, which compares the weight of the spanner, to that of the MST.