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 ∈ X, δ;(u, v) ≤; δ;G(u, v) ≤; t 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 have seen 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. Under the L2-norm in Rd for arbitrary constant d ∈ N, we present an online (1 + ϵ)-spanner algorithm with competitive ratio Od(ϵ-d log n), improving the previous bound of Od(ϵ-(d+1) log n). Moreover, the spanner maintained by the algorithm has Od(ϵ1-d log ϵ-1) n edges, almost matching the (offline) optimal bound of Od(ϵ1-d) n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ϵ-3/2 log ϵ-1 log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ωd(ϵ-d) lower bound for the competitive ratio for online (1+ϵ)-spanner algorithms in Rd under the L1-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k - 1)(1 + ϵ) for k ≥ 2 and ϵ ∈ (0, 1), we show that it maintains a spanner with O(ϵ-1 log ϵ-1) n1+1k edges and O(ϵ-1n 1k log2 n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω ( 1 k n1/k) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2 + ϵ)-spanner for ultrametrics with O(ϵ-1 log ϵ-1) n edges and O(ϵ-2) lightness.