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Efficient and Simple Algorithms for Fault-Tolerant Spanners

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... There has been a fruitful line of work tackling this question for spanners and emulators (e.g., [6][7][8][9][10][11][12][13]16]). This has produced optimal bounds on the price of fault tolerance in some settings, distinctions between edge and vertex faults, and a suite of algorithmic techniques that allow for efficient computation of sparse fault tolerant spanners/emulators (including in distributed and parallel models of computation). ...
... After significant work following [11] (see in particular [7][8][9][10]13]), we now have a generally good understanding of both vertex-and edge-fault tolerant spanners. Precise bounds are given in Table 1, but the high-level view is that the price of fault-tolerance is f 1−1/k for vertex fault tolerant spanners, and even smaller for edge fault tolerant spanners (approximately f 1/2 , although the exact bounds remain open). ...
... While the "main" algorithm we analyze does not run in polynomial time (the obvious implementation of it would take time at least Ω(n f )), in Section 7 we show a variant that runs in polynomial time, paying an additional O(k) factor in emulator size. This algorithm is based on ideas from [13]. ...
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A $t$-emulator of a graph $G$ is a graph $H$ that approximates its pairwise shortest path distances up to multiplicative $t$ error. We study fault tolerant $t$-emulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhibit surprisingly different behavior. In particular, our main result is that, for $(2k-1)$-emulators with $k$ odd, we can tolerate a polynomial number of edge faults for free. For example: for any $n$-node input graph, we construct a $5$-emulator ($k=3$) on $O(n^{4/3})$ edges that is robust to $f = O(n^{2/9})$ edge faults. It is well known that $\Omega(n^{4/3})$ edges are necessary even if the $5$-emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd $k$, and whether a similar phenomenon can be proved for even $k$.
... Unlike the (fault-free) greedy algorithm of Althöfer et al. [ADD + 93], the naive implementation of the FT-greedy algorithms of [BDPW18,BP19] requires exponential time, in the number of faults f . Dinitz and Robelle [DR20] presented an elegant implementation of these greedy algorithms to run in O(k · f 2−1/k n 1+1/k · m) time, and with nearly optimal sparsity of O(k f 1−1/k n 1+1/k ) edges. In a more recent work, Bodwin, Dinitz and Robelle [BDR21a] obtained truly optimal spanners in time O( f 1−1/k · n 2+1/k + m f 2 ). ...
... Indeed, despite the fact that the key motivation for fault tolerant spanners comes from distributed networks, currently we are lacking time-efficient algorithms for optimal FT spanners. The known algorithms by Dinitz and Krauthgamer [DK11] and Dinitz and Robelle [DR20] provide FT-spanners with sub-optimal size of O( f 2−1/k n 1+1/k ) edges, and using O( f 2−1/k ) congest rounds [Pel00a]. We note that even for the simpler setting of edge-FT spanners (resilient to f edge faults), currently there are no local solutions, i.e., with O(1) congest rounds, even when settling for spanners with sub-optimal sparsity 2 . ...
... This connection immediately lead to O(λ)-round distributed algorithms for nearly sparse λ-edge certificates. Using the algorithm of [DR20], one can obtain λ-vertex certificates with O(λ 2 n) edges using O(λ) rounds. In this paper, we will use this connection to provide nearly-sparse vertex-certificates, i.e., with O(λn) edges, and in nearly optimal parallel and distributed runtime (e.g., in O(1) congest rounds). ...
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We (nearly) settle the time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT spanners are sparse subgraphs that preserve distance information, up to a small multiplicative stretch, in the presence of vertex failures. These structures were introduced by [Chechik et al., STOC 2009] and have received a lot of attention since then. We provide algorithms for computing nearly optimal $f$-VFT spanners for any $n$-vertex $m$-edge graph, with near optimal running time in several computational models: - A randomized sequential algorithm with a runtime of $\widetilde{O}(m)$ (i.e., independent in the number of faults $f$). The state-of-the-art time bound is $\widetilde{O}(f^{1-1/k}\cdot n^{2+1/k}+f^2 m)$ by [Bodwin, Dinitz and Robelle, SODA 2021]. - A distributed congest algorithm of $\widetilde{O}(1)$ rounds. Improving upon [Dinitz and Robelle, PODC 2020] that obtained FT spanners with near-optimal sparsity in $\widetilde{O}(f^{2})$ rounds. - A PRAM (CRCW) algorithm with $\widetilde{O}(m)$ work and $\widetilde{O}(1)$ depth. Prior bounds implied by [Dinitz and Krauthgamer, PODC 2011] obtained sub-optimal FT spanners using $\widetilde{O}(f^3m)$ work and $\widetilde{O}(f^3)$ depth. An immediate corollary provides the first nearly-optimal PRAM algorithm for computing nearly optimal $\lambda$-\emph{vertex} connectivity certificates using polylogarithmic depth and near-linear work. This improves the state-of-the-art parallel bounds of $\widetilde{O}(1)$ depth and $O(\lambda m)$ work, by [Karger and Motwani, STOC'93].
... 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. ...
... The spanner construction algorithm that we use to prove Theorem 1.4 is the same greedy algorithm as in [5,7] (adapted for edge fault tolerance), which requires exponential time. However, by combining our new analysis with the ideas used by [13], we can obtain polynomial time at the price of a slightly worse dependence on k: Theorem 1.5. There is a polynomial time algorithm that, given positive integers f, k and an n-node input graph, outputs an f -EFT (2k − 1)-spanner H with ...
... 2. The following analyses [6,7,13] took an alternate view that greedy FT-spanners are structurally similar to high-girth graphs; in particular, they certify sparsity by showing that the output spanners of the FT-greedy algorithm have high-girth subgraphs that keep most of the density of the original graph. This high-girth subgraph must be sparse, since the Moore bounds apply to this subgraph directly, and the way the subgraph is constructed ensures that its density cannot be too far away from that of the original output spanner. ...
... After significant work following [17], we now completely understand the achievable bounds on fault-tolerant spanners: Bodwin and Patel [14] proved that every graph has an f -VFT (2k − 1) spanner with at most O f 1−1/k n 1+1/k edges (and the same bounds were shown to be achievable in polynomial time by [11,18]), and Bodwin, Dinitz, Parter, and Williams [10] gave examples (under the girth conjecture) of graphs on which this bound cannot be improved in any range of parameters. ...
... The algorithm we design to prove Theorem 1.1 starts from the basic greedy VFT spanner algorithm of [10] (and its polytime extension in [18]), where we consider edges in nondecreasing weight order and add an edge if there is a fault set that forces us to add it. To take advantage of the power of emulators, though, we augment this with an extra "path sampling" step: intuitively, when we decide to add a spanner edge, we also flip a biased coin for every k-path that it completes to decide whether to also add an emulator edge between the endpoints of the path. ...
... We begin by proving Theorem 1.1 in the special case k = 3 in Section 2. This introduces the main ideas and approach that we use to prove Theorem 1.1 in general, but it also happens to avoid a few technical details that become necessary only when we move to larger k (allowing us to replace the complicated SALAD paths with simpler "middle-heavy fault-avoiding" paths). We then prove Theorem 1.1 in its full generality: in Section 3 we design an exponential-time algorithm which proves existence of sparse fault-tolerant emulators for all k, and then in Section 4 we show how to use ideas from [18] to make the algorithm polynomial-time without significant loss in emulator sparsity. We then prove our lower bounds (Theorems 1.2 and 1.3) in Section 5, and we conclude with our results on additive spanners in Section 6. ...
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A $k$-spanner of a graph $G$ is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of $k$, and a $k$-emulator is similar but not required to be a subgraph of $G$. A classic theorem by Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant emulators, and then we show a three-way tradeoff between size, stretch, and fault-tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners. We complement our emulator upper bound with a lower bound construction that is essentially tight (within $\log n$ factors of the upper bound) when the stretch is $2k-1$ and $k$ is either a fixed odd integer or $2$. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive 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]11]. This has culminated in the following theorem: Theorem 1.1 (FT upper bounds [6]). ...
... Polytime? Citation O f · n 1+1/k [8] O exp(k)f 1−1/k · n 1+1/k k = 2 only [5] O f 1−1/k · n 1+1/k k = 2 only [7] O kf 1−1/k · n 1+1/k k = 2 only [11] O f 1−1/k · n 1+1/k k = 2 only [6] O k 2 f 1/2 · n 1+1/k + kf n for even k k = 2 only ( * ) this paper ...
... The spanner construction algorithm that we use to prove Theorem 1.4 is the same greedy algorithm as in [5,7] (adapted for edge fault tolerance), which requires exponential time. However, by combining our new analysis with the ideas used by [11], we can obtain polynomial time at the price of a slightly worse dependence on k: Theorem 1.6. There is a polynomial time algorithm that, given positive integers f, k and an n-node input graph, outputs an f -EFT (2k − 1)-spanner H with ...
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Recent work has established that, for every positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges that is resilient to $f$ edge or vertex faults. For vertex faults, this bound is tight. However, the case of edge faults is not as well understood: the best known lower bound for general $k$ is $\Omega(f^{\frac12 - \frac{1}{2k}} n^{1+1/k} +fn)$. Our main result is to nearly close this gap with an improved upper bound, thus separating the cases of edge and vertex faults. For odd $k$, our new upper bound is $O_k(f^{\frac12 - \frac{1}{2k}} n^{1+1/k} + fn)$, which is tight up to hidden $poly(k)$ factors. For even $k$, our new upper bound is $O_k(f^{1/2} n^{1+1/k} +fn)$, which leaves a gap of $poly(k) f^{1/(2k)}$. Our proof is an analysis of the fault-tolerant greedy algorithm, which requires exponential time, but we also show that there is a polynomial-time algorithm which creates edge fault tolerant spanners that are larger only by factors of $k$.
... The edge test in the FT greedy algorithm, i.e., whether or not there exists a fault set under which (1.2) holds, is an NP-hard problem known as lengthbounded cut [BEH + 06], and hence the algorithm inherently runs in exponential time. Addressing this, a greedy algorithm with slack was recently proposed in [DR20]. This algorithm is an adaptation of the FTgreedy algorithm which replaces the exponential-time edge test with a different subroutine test(u, v), which accepts every edge (u, v) where there exist |F | ≤ f faults under which (1.2) fails, and possibly some other edges too. ...
... This slack maintains correctness and allows one to escape NP-hardness, but it introduces the challenge of bounding the number of additional edges added. The approach in [DR20] is to design an O(k)-approximation algorithm for length-bounded cut and use this in an efficiently computable test subroutine. This gives a polynomial runtime, but pays the approximation ratio of O(k) in spanner size over optimal. ...
... This gives a polynomial runtime, but pays the approximation ratio of O(k) in spanner size over optimal. So the result in [DR20] takes an important step forward (polynomial time) but also a step back (non-optimal size, by a factor of O(k)). ...
... This technique is inspired by the color-coding technique [2], and provides a general recipe for translating a given fault-free algorithm for a given task into a fault-tolerant one while paying a relatively small overhead in terms of computation time and other complexity measures of interest (e.g., space). Indeed this approach has been applied in the context of distance sensitivity oracles [15,15,7], faulttolerant spanners [11,5,12], fault-tolerant reachability preservers [6], distributed minimum-cut computation [20], and resilient distributed computation [24,23,10,17]. The high-level idea of this technique is based on sampling a (relatively) small number of subgraphs G 1 , . . . ...
... Comparison with a recent independent work of [4]. Independent to our work, [4] presented a new slack version of the greedy algorithm from [3,12] to obtain a (vertex) fault-tolerant spanners with optimal size bounds. Their main algorithm is randomized with and the emphasis there is on optimizing the size of the output spanner. ...
... Distributed constructions of fault-tolerant preservers. Distributed constructions of FT preservers attracted attention recently [15,20,16,29]. In the context of exact distance preservers, Ghaffari and Parter [20] presented the first distributed constructions of fault tolerant distance preserving structures. ...
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The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [Dist. Comp. '02] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two original shortest paths. However, the lemma is tiebreaking-sensitive: if one selects a particular canonical shortest path for each node pair, it is no longer guaranteed that one can build replacement paths by concatenating two selected shortest paths. They left as an open problem whether a method of shortest path tiebreaking with this desirable property is generally possible. We settle this question affirmatively with the first general construction of restorable tiebreaking schemes. We then show applications to various problems in fault-tolerant network design. These include a faster algorithm for subset replacement paths, more efficient fault-tolerant (exact) distance labeling schemes, fault-tolerant subset distance preservers and $+4$ additive spanners with improved sparsity, and fast distributed algorithms that construct these objects. For example, an almost immediate corollary of our restorable tiebreaking scheme is the first nontrivial distributed construction of sparse fault-tolerant distance preservers resilient to three faults.
... This technique is inspired by the color-coding technique [AYZ95], and provides a general recipe for translating a given fault-free algorithm for a given task into a fault-tolerant one while paying a relatively small overhead in terms of computation time and other complexity measures of interest (e.g., space). Indeed this approach has been applied in the context of distance sensitivity oracles [GW20, GW20,CC20b], fault-tolerant spanners [DK11,BCPS15,DR20], fault-tolerant reachability preservers [CC20a], distributed minimum-cut computation [Par19a], and resilient distributed computation [PY19b,PY19a,CPT20,HP20]. The high-level idea of this technique is based on sampling a (relatively) small number of subgraphs G 1 , . . . ...
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