Small Hop-diameter Sparse Spanners for Doubling Metrics
Given a metric M = (V, d), a graph G = (V, E) is a t-spanner for M if every pair of nodes in V has a "short" path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every pair of nodes has such a short path that also uses at most D edges. We consider the problem of constructing sparse (1 + epsilon)-spanners with small hop diameter for metrics of low doubling dimension. In this paper, we show that given any metric with constant doubling dimension k and any 0 < epsilon < 1, one can find (1 + epsilon)-spanner for the metric with nearly linear number of edges (i.e., only O(n log* n + n epsilon-(O(k))) edges) and constant hop diameter; we can also obtain a (1 + epsilon)-spanner with linear number of edges (i.e., only n epsilon-(O(k)) edges) that achieves a hop diameter that grows like the functional inverse of Ackermann's function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal.
sparse spanners doubling metrics hop diameter algorithms