Abstract
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 0<ε<1, one can find (1+ε)-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.
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The conference version of the paper appeared in ACM-SIAM SODA 2006.
This research of T.-H.H. Chan was done while the author was at Carnegie Mellon University and was partly supported by the NSF grant CCR-0122581 (the ALADDIN project), the NSF CAREER award CCF-0448095, and by an Alfred P. Sloan Fellowship.
This research of A. Gupta was partly supported by the NSF grant CCR-0122581 (the ALADDIN project), the NSF CAREER award CCF-0448095, and by an Alfred P. Sloan Fellowship.
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Chan, TH.H., Gupta, A. Small Hop-diameter Sparse Spanners for Doubling Metrics. Discrete Comput Geom 41, 28–44 (2009). https://doi.org/10.1007/s00454-008-9115-5
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DOI: https://doi.org/10.1007/s00454-008-9115-5