Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

Abstract

A k-spanner of a graph G is a sparse subgraph H whose shortest path distances match those of G up to a multiplicative error k. In this paper we study spanners that are resistant to faults. A subgraph H ⊆ G is an f vertex fault tolerant (VFT) k-spanner if H \ F is a k-spanner of G \ F for any small set F of f vertices that might "fail." One of the main questions in the area is: what is the minimum size of an f fault tolerant k-spanner that holds for all n node graphs (as a function of f , k and n)? In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor k is fixed. Specifically, we prove that every (undirected, possibly weighted) n-node graph G has a (2k − 1)-spanner resilient to f vertex faults with O k (f 1−1/k n 1+1/k ) edges, and this is fully optimal (unless the famous Erdös Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating dist G\F (s, t) similarly can beat the space usage of our spanner in the worst case. To the best of our knowledge, this is the first instance in fault tolerant network design in which introducing fault tolerance to the structure increases the size of the (non-FT) structure by a sublinear factor in f . Another advantage of this result is that our spanners are constructed by a very natural and simple greedy algorithm, which is the obvious extension of the standard * Supported in part by NSF awards 1464239 and 1535887. Supported in part by NSF grants CCF-1417238, CCF-1528078 and CCF-1514339, and BSF grant BSF:2012338. 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.

A Trivial Yet Optimal Solution to Vertex Fault Tolerant Spanners

“…This corollary improves over the previous best upper bound in [9] by a factor of exp(k). The famous Erdös girth conjecture [20] posits that the Moore Bounds are tight, which would then imply that this corollary is best possible, at least for VFT spanners.…”

“…Correctness is again obvious, but unfortunately the analyses used in the non-faulty setting all seem to break for the FT greedy algorithm. Most prior work on FT spanners has thus abandoned the greedy approach in favor of more involved constructions [23,14,13,16,5] (see also [24,25,26,10,12,15,7,22,30,17,8]); an analysis of the FT greedy algorithm was only obtained recently via fairly complex arguments [9]. In contrast, our main result is a simple analysis of the FT greedy algorithm which improves on all of these previous bounds.…”

“…However, no such improvement is possible: for any k we can show a graph H on Ω(f 2 · b(n/f, k + 1)) edges that has an edge (k + 1)-blocking set of size ≤ f |E(H)|, and so our analog of Lemma 3 alone is not powerful enough to get improved upper bounds in the EFT setting. Specifically, this H is the same as the VFT lower bound graph of [9]: it is the Cartesian product of an arbitrary graph of girth > k + 1 with a biclique on ⌊f /2⌋ nodes; the blocking set is then all pairs of edges that share an endpoint in the product graph and which correspond to the same edge in the initial high-girth graph. Hence any improvement to our EFT upper bounds (if possible) will need to exploit stronger properties of H than just the existence of a small edge blocking set.…”

“…This upper bound is best possible in the VFT setting, for any construction algorithm, due to a simple lower bound construction in [9] (meaning that any asymptotic tradeoff between n, f, k, |E(H)| not promised by this theorem does not exist in general). In the EFT setting, the bound of Theorem 1 was already known to be the best possible tradeoff when k < 5 [9], but for larger k it is still conceivable to improve the upper bound as far as…”

Partially Optimal Edge Fault-Tolerant Spanners

“…At this point, it appeared that the EFT setting might be substantially easier than the VFT setting, in the sense that it allowed for a smaller dependence on f in spanner size. However, a recent series of papers has developed a set of techniques that apply equally well to both settings, yielding the same improved bounds for each [5][6][7]13]. This has culminated in the following theorem: Theorem 1.1.…”