Fault-Tolerant Approximate Shortest-Path Trees

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

doi:10.1007/s00453-017-0396-z

Cited by 8 publications

(7 citation statements)

References 25 publications

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“…The problem of constructing multiplicative approximation replacement paths P s,t,v (i.e., such that | P s,t,v | ≤ α · |P s,t,v |) has been studied in [3,6,10]. In particular its single source variant has been studied in [4,8,21]. In this paper, we further explore this approach.…”

Section: Introductionmentioning

confidence: 99%

Vertex fault tolerant additive spanners

Parter

2015

Distrib. Comput.

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A fault-tolerant structure for a network is required to continue functioning following the failure of some of the network's edges or vertices. In this paper, we address the problem of designing a fault-tolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failure of a single vertex, the surviving part of H still contains an additive spanner for (the surviving partthe problem of constructing fault-tolerant additive spanners resilient to the failure of up to f edges has been considered (Braunschvig et al. Proceedings of the WG, pp 206-214, 2012). The problem of handling vertex failures was left open therein. In this paper we develop new techniques for constructing additive FTspanners overcoming the failure of a single vertex in the graph.Our first result is an FT-spanner with additive stretch 2 and O(n 5/3 ) edges. Our second result is an FT-spanner with additive stretch 6 and O(n 3/2 ) edges. The construction algorithm consists of two main components: (a) constructing an FT-clustering graph and (b) applying a modified path-buying procedure suitably adapted to failure prone settings. Finally, we also describe two constructions for fault-tolerant multisource additive spanners, aiming to guarantee a bounded additive stretch following a vertex failure, for every pair of vertices in S × V for a given subset of sources S ⊆ V . The additive stretch bounds of our constructions are 4 and 8 (using a different number of edges).

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Section: Introductionmentioning

confidence: 99%

Vertex fault tolerant additive spanners

Parter

2015

Distrib. Comput.

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“…Parter and Peleg [19] improved this result by showing that such a subgraph exists with 3n edges. Bilò, Gualà, Leucci, and Proietti [5] showed that any weighted undirected graph has a fault tolerant (1+ )-shortest path subgraph of size O((1/ 2 )n log n). Parter and Peleg [18] showed that any unweighted (un)directed graph has a fault tolerant exact shortest path subgraph with O(n 3/2 ) edges.…”

Section: Related Workmentioning

confidence: 99%

“…For the case of multiple edge faults, Bilò et al [6] obtain a k-fault tolerant (2k+1)-shortest path subgraph with O(kn) edges again only for undirected graphs. They also showed that there is a data structure of size O(kn log 2 n) that reports the (2k + 1)-approximate distance from s in O(k 2 log 2 n) time.…”

Section: Related Workmentioning

confidence: 99%

“…Add last √ n non-tree edges of HD α (v) to H; 4 S ← A uniformly random set of 4 √ n log e n vertices; 5 foreach v ∈ S and α ∈ powers(1 + ) do 6 Add all non-tree edges of HD α (v) to H;…”

Section: Sparse Subgraphmentioning

confidence: 99%

“…Parter and Peleg [18] showed that even in the case of unweighted undirected graphs there are graphs such that the size of their fault tolerant exact shortest path subgraph is at least Ω(n 1.5 ). Bilò, Gualà, Leucci, and Proietti [5] showed that any weighted undirected graph has a fault tolerant (1 + )-shortest path subgraph of size O((1/ 2 )n log n). Their result, however, uses techniques that rely heavily on the undirectedness of the graph and therefore cannot be extended to weighted directed graphs.…”

Section: Introductionmentioning

confidence: 99%

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Approximate Single Source Fault Tolerant Shortest Path

Baswana

Choudhary

Hussain³

et al. 2018

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

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Let G = (V, E) be an n-vertices m-edges directed graph with edge weights in the range [1, W ] and L = log(W ). Let s ∈ V be a designated source. In this paper we address several variants of the problem of maintaining the (1 + )-approximate shortest path from s to each v ∈ V \ {s} in the presence of a failure of an edge or a vertex. From the graph theory perspective we show that G has a subgraph H with O(nL/ ) edges such that for any x, v ∈ V , the graph H \ x contains a path whose length is a (1 + )-approximation of the length of the shortest path from s to v in G \ x. We show that the size of the subgraph H is optimal (up to logarithmic factors) by proving a lower bound of Ω(nL/ ) edges. Demetrescu, Thorup, Chowdhury and Ramachandran [12] showed that the size of a fault tolerant exact shortest path subgraph in weighted directed/undirected graphs is Ω(m). Parter and Peleg [18] showed that even in the restricted case of unweighted undirected graphs the size of any subgraph for the exact shortest path is at least Ω(n 1.5 ). Therefore, a (1 + )-approximation is the best one can hope for. We consider also the data structure problem and show that there exists an O(nL/ ) size oracle that for any v ∈ V reports a (1 + )-approximate distance of v from s on a failure of any x ∈ V in O(log log 1+ (nW )) time. We show that the size of the oracle is optimal (up to logarithmic factors) by proving a lower bound of Ω(nL/ log n). Finally, we present two distributed algorithms. We present a single source routing scheme that can route on a (1 + )-approximation of the shortest path from a fixed source s to any destination t in the presence of a fault. Each vertex has a label and a routing table of O(L/ ) bits. We present also a labeling scheme that assigns each vertex a label of O(L/ ) bits. For any two vertices x, v ∈ V the labeling scheme outputs a (1 + )-approximation of the distance from s to v in G \ x using only the labels of x and v.

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