Sparse fault-tolerant spanners for doubling metrics with bounded hop-diameter or degree
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
New doubling spanners: better and simpler
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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+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 0εε)-spanner for the metric with nearly linear number of edges (i.e., only O(nlog * n+n ε −O(k)) edges) and constant hop diameter; we can also obtain a (1+ε)-spanner with linear number of edges (i.e., only n ε −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.