Fault-Tolerant Spanners for Doubling Metrics: Better and Simpler

In STOC’95 Arya et al. [2] conjectured that for any constant dimensional n-point Euclidean space, a (1+ǫ)-spanner with constant degree, diameter O(log n) and weight O(log n) ·ω(MST ) can be built in O(n · log n) time. Recently Elkin and Solomon [10] (technical report, April 2012) proved this conjecture of Arya et al. in the affirmative. In fact, the proof of [10] is more general in two ways. First, it applies to arbitrary doubling metrics. Second, it provides a complete tradeoff between the three involved parameters that is tight (up to constant factors) in the entire range. Subsequently, Chan et al. [5] (technical report, July 2012]) provided another proof for Arya et al.’s conjecture, which is simpler than the proof of Elkin and Solomon [10]. Moreover, Chan et al. [5] also showed that one can build a fault-tolerant (FT) spanner with similar properties. Specifically, they showed that there exists a k-FT (1 + ǫ)-spanner with degree O(k), diameter O(log n) and weight O(k · logn) · ω(MST ). The running time of the construction of [5] was not analyzed. In this work we improve the results of Chan et al. [5], using a simpler proof. Specifically, we present a simple proof which shows that a k-FT (1 + ǫ)-spanner with degree O(k), diameter O(log n) and weight O(k · log n) · ω(MST ) can be built in O(n · (logn+ k)) time. Similarly to the constructions of [10] and [5], our construction applies to arbitrary doubling metrics. However, in contrast to the construction of Elkin and Solomon [10], our construction fails to provide a complete (and tight) tradeoff between the three involved parameters. The construction of Chan et al. [5] has this drawback too. For random point sets in R, we “shave” a factor of logn from the weight bound. Specifically, in this case our construction provides within the same time O(n · (log n + k)), a k-FT (1 + ǫ)-spanner with degree O(k), diameter O(log n) and weight that is with high probability O(k) · ω(MST ). ∗Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected]. Part of this work was done while this author was a graduate student in the Department of Computer Science, Ben-Gurion University of the Negev, under the support of the Clore Fellowship grant No. 81265410, the BSF grant No. 2008430, and the ISF grant No. 87209011.