Improved Distance Sensitivity Oracles via Tree Partitioning

Duan, Ran; Zhang, Tianyi

doi:10.1007/978-3-319-62127-2_30

Cited by 12 publications

(9 citation statements)

References 23 publications

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“…A natural data structure analog of fault-tolerant subgraphs are distance sensitivity oracles, which are small data structure that are used to answer queries of the form: "what is the distance between s and t in G when a set of F edges fail"? There is a long line of literature on constructing distance sensitivity oracles [23,8,9,30,7,49,26,12] and related structures such a fault tolerant routing schemes [29,19] and labeling schemes [2]. Recently, [20] provided an efficient construction of distance sensitivity oracles that support f = O(log n/ log log n) many faults with polylogarithmic query time.…”

Section: Introductionmentioning

confidence: 99%

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

Bodwin¹,

Dinitz

Parter³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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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.

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Section: Introductionmentioning

confidence: 99%

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

Bodwin¹,

Dinitz

Parter³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In this sense, the Õ(mn) time bound of [BK09] is optimal. Duan and Zhang [DZ17] improved the space complexity of the DSO to O(n 2 ), eliminating the last log n factor, while preserving constant query time and Õ(mn) preprocessing time.…”

Section: Related Workmentioning

confidence: 99%

Constructing a Distance Sensitivity Oracle in $O(n^{2.5794}M)$ Time

Gu,

Ren

2021

Preprint

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We continue the study of distance sensitivity oracles (DSOs). Given a directed graph G with n vertices and edge weights in {1, 2, . . . , M }, we want to build a data structure such that given any source vertex u, any target vertex v, and any failure f (which is either a vertex or an edge), it outputs the length of the shortest path from u to v not going through f . Our main result is a DSO with preprocessing time O(n 2.5794 M ) and constant query time. Previously, the best preprocessing time of DSOs for directed graphs is O(n 2.7233 M ), and even in the easier case of undirected graphs, the best preprocessing time is O(n 2.6865 M ) [Ren, ESA 2020]. One drawback of our DSOs, though, is that it only supports distance queries but not path queries.Our main technical ingredient is an algorithm that computes the inverse of a degree-d polynomial matrix (i.e. a matrix whose entries are degree-d univariate polynomials) modulo x r . The algorithm is adapted from [Zhou, Labahn and Storjohann, Journal of Complexity, 2015], and we replace some of its intermediate steps with faster rectangular matrix multiplication algorithms.We also show how to compute unique shortest paths in a directed graph with edge weights in {1, 2, . . . , M }, in O(n 2.5286 M ) time. This algorithm is crucial in the preprocessing algorithm of our DSO. Our solution improves the O(n 2.6865 M ) time bound in [Ren, ESA 2020], and matches the current best time bound for computing all-pairs shortest paths.

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“…They showed that it is possible to preprocess a directed weighted graph in O(mn 2 ) time to compute a data-structure of size O(n 2 log n) capable of answering distance queries in constant time. Bernstein and Karger [8] improved the preprocessing time to O(mn) and Duan and Zhang [23] reduced the space to O(n 2 ), which is asymptotically optimal.…”

Section: Related Workmentioning

confidence: 99%

Space-Efficient Fault-Tolerant Diameter Oracles

Bilò¹,

Cohen²,

Friedrich³

et al. 2021

Preprint

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We design f -edge fault-tolerant diameter oracles (f -FDO, or simply FDO if f = 1). For a given directed or undirected and possibly edge-weighted graph G with n vertices and m edges and a positive integer f , we preprocess the graph and construct a data structure that, when queried with a set F of edges, whereFor the case of a single edge failure (f = 1) in an unweighted directed graph, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch (1 + ε), constant query time, space O(m), and a combinatorial preprocessing time of O(mn + n 1.5 Dm/ε ), where D is the diameter.We present an FDO for directed graphs with the same stretch, query time, and space. It has a preprocessing time of O(mn + n 2 /ε), which is better for constant ε > 0. The preprocessing time nearly matches a conditional lower bound for combinatorial algorithms, also by Henzinger et al. With fast matrix multiplication, we achieve a preprocessing time of O(n 2.5794 + n 2 /ε). We further prove an information-theoretic lower bound showing that any FDO with stretch better than 3/2 requires Ω(m) bits of space. Thus, for constant 0 < ε < 3/2, our combinatorial (1 + ε)-approximate FDO is near-optimal in all parameters.In the case of multiple edge failures (f > 1) in undirected graphs with non-negative edge weights, we give an f -FDO with stretch (f + 2), query time O(f 2 log 2 n), O(f n) space, and preprocessing time O(f m). We complement this with a lower bound excluding any finite stretch in o(f n) space.Many real-world networks have polylogarithmic diameter. We show that for those graphs and up to f = o(log n/ log log n) failures one can swap approximation for query time and space. We present an exact combinatorial f -FDO with preprocessing time mn 1+o(1) , query time n o(1) , and space n 2+o(1) . When using fast matrix multiplication instead, the preprocessing time can be improved to n ω+o(1) , where ω < 2.373 is the matrix multiplication exponent. ACM Subject ClassificationTheory of computation → Shortest paths; Theory of computation → Data structures design and analysis; Theory of computation → Cell probe models and lower bounds; Theory of computation → Pseudorandomness and derandomization

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Improved Distance Sensitivity Oracles via Tree Partitioning

Cited by 12 publications

References 23 publications

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

Constructing a Distance Sensitivity Oracle in $O(n^{2.5794}M)$ Time

Space-Efficient Fault-Tolerant Diameter Oracles

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