Online Spanners in Metric Spaces
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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 ∈ X, δ(u,v)≤ d_G(u,v)≤ t·δ(u,v), 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^-3/2logε^-1log n), bypassing the classic lower bound Ω(ε^-2) for lightness, which compares the weight of the spanner, to that of the MST.
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