ONLINE SPANNERS IN METRIC SPACES

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.