Optimal Vertex Fault Tolerant Spanners (for fixed stretch) [chapter]

Greg Bodwin, Michael Dinitz, Merav Parter, Virginia Vassilevska Williams
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
greedy algorithm used to build spanners in the non-faulty setting. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case k = 2 (and hence we close the EFT problem for 3-approximations), but it falls to Ω(f 1/2−1/(2k) · n 1+1/k ) for k ≥ 3. We leave it as an open problem to close this gap. Introduction A spanner
more » ... [45, 46]) of a graph is a subgraph that approximately preserves its shortest path metric. More formally, a subgraph H = called the stretch of the spanner). Spanners were introduced by Peleg and Ullman [46] and Peleg and Schäffer [45], and have a wide range of applications in routing [47], synchronizers [6], broadcasting [5, 43], distance oracles [49], graph sparsifiers [32], and even preconditioning of linear systems [28]. The most common objective in spanners research is to achieve the best possible existential size-stretch trade-off. Most notably, a landmark result of Althöfer et al. [3] proved that for any integer k ≥ 1, every graph G = (V, E) has a (2k − 1)-spanner H ⊆ G with O(n 1+1/k ) edges, and moreover, there exist graphs for which this size-stretch tradeoff cannot be improved (if we assume the girth conjecture of Erdős [29]). In fact, their existentially optimal upper bound was obtained via an extremely simple and natural greedy construction algorithm: consider the edges of G in non-decreasing order of their weight and add an edge {u, v} to the current spanner H if and only if dist H (u, v) > (2k − 1)w (u, v). It is easy to verify that this algorithm never creates cycles of length 2k
doi:10.1137/1.9781611975031.123 dblp:conf/soda/BodwinDPW18 fatcat:t5sr4ib7freujfdjufj4sidzqe