Online Spanners in Metric Spaces

. Given a metric space \ (M= (X, ) \), a weighted graph \ (G\) over \ (X\) is a metric \ (t\) -spanner of \ (M\) if for every \ (u, v X\), \ ( (u, v) G (u, v) t (u, v) \), where \ (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+) \) -spanners in the Euclidean \ (d\) -space, \ ( (2k-1) (1+) \) -spanners for general metrics, and \ ( (2+) \) -spanners for ultrametrics. Most notably, in the Euclidean plane, we construct a \ ( (1+) \) -spanner with competitive ratio \ (O (^-3/2 ^-1 n) \), bypassing the classic lower bound \ ( (^-2) \) for lightness, which compares the weight of the spanner to that of the minimum spanning tree. Keywordsspannersonline algorithmsonline spannersmetric spacesapproximation algorithmsrandomized algorithmsMSC codes68W2568W2768W4068Rxx