Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

Bilò, Davide; Gualà, Luciano; Leucci, Stefano; Proietti, Guido

doi:10.1007/s00453-021-00879-8

Cited by 6 publications

(5 citation statements)

References 48 publications

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“…SND is well-studied, see [26,23,16,22] for a sample; notably, Jain gives a 2-approximation algorithm for the problem in a seminal paper [23]. Beyond SND, edge faulttolerance has been studied in many particular cases and related problems, among them k-Edge Connected Spanning Subgraph, Fault-Tolerant Group Steiner Tree, Fault-Tolerant Spanner, and Fault-Tolerant Shortest Paths problems; see [16,24,20,4] for examples. These and other classical fault-tolerance problems, including SND, are absolute fault-tolerance problems -if up to k objects fail, the remaining graph should function as desired.…”

Section: Survivable Network Design (Snd)mentioning

confidence: 99%

Approximation Algorithms for Relative Survivable Network Design Problems

Dinitz,

Koranteng,

Kortsarz

et al. 2024

Preprint

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In the {\sc Survivable Network Design} ({\SND}) problem we seek a min-cost subgraph that satisfies pairwise edge-connectivity demands. This encompasses the {\sc $k$-Edge-Connected Spanning Subgraph} ({\kECSS}) problem (the case of uniform demands), and the case of a single demand is essentially the {\sc Min-cost $k$-Flow} problem with unit capacities. A weakness of these formulations is that we cannot ask for fault-tolerance larger than the connectivity. Inspired by recent definitions and progress in graph spanners, we study new variants of these problems under a notion of \emph{relative} fault-tolerance. Informally, we require not that two nodes are connected if there are $k$ faults (as in the classical setting), but that two nodes are connected if there are $k$ faults \emph{and the two nodes are connected in the underlying graph post-faults}. That is, the subgraph must ''behave'' identically to the underlying graph with respect to connectivity after $k$ faults. We define and introduce these ''relative'' problems, and provide approximation algorithms: a $2$-approximation for {\sc Relative}{\kECSS} ({\RkECSS}), a $(1+4/k)$-approximation for unweighted {\RkECSS}, a $2$-approximation for {\sc Relative SND} with demands $\leq 3$ ({\RtSND}), and a $2^{O(k^2)}$-approximation for {\RSND} with a single demand of $k$.

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Section: Survivable Network Design (Snd)mentioning

confidence: 99%

Approximation Algorithms for Relative Survivable Network Design Problems

Dinitz,

Koranteng,

Kortsarz

et al. 2024

Preprint

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show abstract

“…Besides FT spanners, there are many other kinds of fault-tolerant structures, e.g. [14,15,17,47,[49][50][51][52]. We refer the reader to the excellent survey of [48].…”

Section: More Related Workmentioning

confidence: 99%

“…For undirected graphs, we can answer connectivity queries under d edge failures 1 [33,34,53] and d vertex failures [33,34] in poly(d, log n) time. Chechik et al [24] designed a data structure that maintains O(d)-approximate shortest paths under d edge failures in an undirected graph, and Bilò et al [14] improved the approximation ratio to 2d + 1. For any > 0, Chechik et al [23] designed a data structure that (1 + )-approximates shortest paths under d edge failures in an undirected graph, but the space complexity of their data structure is exponential in d, and becomes super-polynomial when d = ω(log n/ log log n).…”

Section: Introductionmentioning

confidence: 99%

Approximate Distance Oracles Subject to Multiple Vertex Failures

Duan

Ren

2020

Preprint

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Given an undirected graph G = (V, E) of n vertices and m edges with weights in [1, W ], we construct vertex sensitive distance oracles (VSDO), which are data structures that preprocess the graph, and answer the following kind of queries: Given a source vertex u, a target vertex v, and a batch of d failed vertices D, output (an approximation of) the distance between u and v in G − D (that is, the graph G with vertices in D removed). An oracle has stretchis the actual distance between u and v in G − D, and δ(u, v) is the distance reported by the oracle.In this paper we construct efficient VSDOs for any number d of failures. For any constant c ≥ 1, we propose two oracles:• The first oracle has size n 2+1/c (log n/ ) O(d) •log W , answers a query in poly(log n, d c , log log W, −1 ) time, and has stretch 1 + , for any constant > 0. • The second oracle has size n 2+1/c poly(log(nW ), d), answers a query in poly(log n, d c , log log W ) time, and has stretch poly(log n, d).

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“…Recently Bilò et al [6] built the SDO(1) (described in [5]) in O(m n + n 2 ) time. Many different aspects of distance oracles have been studied in literature [3,[7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path problem

Dey,

Gupta

2023

Preprint

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In a graph G with fixed source s, we design a distance oracle that can answer the following query: QU(s,t,e) -- find the length of the shortest path from the source s to any destination vertex t while avoiding any edge e. We design a deterministic algorithm that builds such an oracle in $\Tilde{O}(m\sqrt n)$ time\footnote{$\Tilde{O}$ hides poly$\log n$ factor} \footnote{A preliminary version of this article appeared in the Proceedings of 30th Annual European Symposium on Algorithms (ESA 2022), 5-9 September 2022, Potsdam, Germany \cite{DeyGupta} }. Our oracle uses $\Tilde{O}(n\sqrt n)$ space and can answer queries in $\Tilde{O}(1)$ time. Our oracle is an improvement of the work of Bil\'{o} et al. (ESA 2021) in the preprocessing time, which constructs the first deterministic oracle for this problem in $\Tilde{O}(m\sqrt n+n^2)$ time. Using our distance oracle, we also solve the single source replacement path problem (SSRP problem).Chechik and Cohen (SODA 2019) designed a randomised combinatorial algorithm to solve the $\SSR$ problem. The running time of their algorithm is $\Tilde{O}(m\sqrt n + |R|)$ time, where R is the output set of the SSRP problem in G. Our SSRP algorithm is optimal (upto polylogarithmic factor) as there is a conditional lower bound of $\Omega(m\sqrt n)$ for any combinatorial algorithm that solves this problem.

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Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

Cited by 6 publications

References 48 publications

Approximation Algorithms for Relative Survivable Network Design Problems

Approximation Algorithms for Relative Survivable Network Design Problems

Approximate Distance Oracles Subject to Multiple Vertex Failures

Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path problem

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