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 such short path also uses at most D edges. We consider the problem of constructing sparse (1 + ε)-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 < ε < 1, one can find a (1 + ε)-spanner for the metric with nearly linear number of edges (i.e., only O(n log * n + nε −O(k) ) edges) and a constant hop diameter, and also a (1 + ε)-spanner with linear number of edges (i.e., only nε edges) which achieves a hop diameter that grows like the functional inverse of the Ackermann's function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal.
Tight lower bounds on the hop-diameter of Euclidean spanners with a given number of edges have been recently established by Chan and Gupta REF .
Small hop-diameter sparse spanners for doubling metrics
SODA '06
Computer Science Mathematics