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17
Faulttolerant spanners: Better and simpler
 In PODC
, 2011
"... A natural requirement for many distributed structures is faulttolerance: after some failures in the underlying network, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to verte ..."
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A natural requirement for many distributed structures is faulttolerance: after some failures in the underlying network, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults r for all stretch bounds. For stretch k ≥ 3 we design a simple transformation that converts every kspanner construction with at most f(n) edges into an rfaulttolerant kspanner construction with at most O(r 3 log n) · f(2n/r) edges. Applying this to standard greedy spanner constructions gives rfault tolerant kspanners with Õ(r2 1+ 2 n k+1) edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [CLPR09] depends similarly on n but exponentially on r (approximately like k r). For the case of k = 2 and unit edgelengths, an O(r log n)approximation is known from recent work of Dinitz and Krauthgamer [DK11], in which several spanner results are obtained using a common approach of rounding a natural flowbased linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to O(log n), which is, notably, independent of the number of faults r. We further strengthen this bound in terms of the maximum degree by using the Lovász Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.
Directed Spanners via FlowBased Linear Programs
, 2011
"... We examine directed spanners through flowbased linear programming relaxations. We design an Õ(n2/3)approximation algorithm for the directed kspanner problem that works for all k ≥ 1, which is the first sublinear approximation for arbitrary edgelengths. Even in the more restricted setting of unit ..."
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Cited by 8 (3 self)
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We examine directed spanners through flowbased linear programming relaxations. We design an Õ(n2/3)approximation algorithm for the directed kspanner problem that works for all k ≥ 1, which is the first sublinear approximation for arbitrary edgelengths. Even in the more restricted setting of unit edgelengths, our algorithm improves over the previous Õ(n1−1/k) approximation [BGJ + 09] when k ≥ 4. For the special case of k = 3 we design a different algorithm achieving an Õ(√n)approximation, improving the previous Õ(n 2/3) [EP05, BGJ + 09] (independently of our work, an Õ(n 1−1/⌈k/2 ⌉ ) was recently devised [BRR10]). Both of our algorithms easily extend to the faulttolerant setting, which has recently attracted attention but not from an approximation viewpoint. We also prove a nearly matching integrality gap of ˜ Ω(n 1/3−ɛ) for every constant ɛ> 0. A virtue of all our algorithms is that they are relatively simple. Technically, we introduce a new yet natural flowbased relaxation, and show how to approximately solve it even when its size is not polynomial. The main challenge is to design a rounding scheme that “coordinates ” the choices of flowpaths between the many demand pairs while using few edges overall. We achieve this, roughly speaking, by randomization at the level of vertices.
Near Optimal Multicriteria Spanner Constructions in Wireless AdHoc Networks
, 2010
"... In this paper we study asymmetric power assignments which induce a low energy kstrongly connected communication graph with spanner properties. We address two spanner models: energy and distance. The former serves as an indicator for the energy consumed in a message propagation between two nodes, wh ..."
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Cited by 7 (3 self)
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In this paper we study asymmetric power assignments which induce a low energy kstrongly connected communication graph with spanner properties. We address two spanner models: energy and distance. The former serves as an indicator for the energy consumed in a message propagation between two nodes, while the latter reflects the geographic properties of routing in the induced communication graph. We consider a random wireless adhoc network with V = n nodes distributed uniformly and independently in a unit square. For k ∈ {1, 2} we propose several power assignments which obtain a good bicriteria approximation on the total cost and stretch factor under the two models. For k> 2 we analyze a power assignment developed in [1], and derive some interesting bounds on the stretch factor for both models as well. We also describe how to compute all the power assignments distributively, and provide simulation results. To the best of our knowledge, these are the first provable theoretical bounds for low cost spanners in wireless adhoc networks.
Approximate shortest paths avoiding a failed vertex : optimal data structures for unweighted graphs
"... Abstract. Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be tapproximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data struct ..."
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Cited by 6 (0 self)
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Abstract. Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be tapproximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data structure for a given graph G which is capable of answering the following query for any u, v, z ∈ V and t> 1. report tapproximate shortest path between u and v when vertex z fails We present data structures for the single source as well allpairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size nearly match the size of their best static counterparts. 1.
Local Computation of Nearly Additive Spanners
"... An (α, β)spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β) ..."
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Cited by 6 (3 self)
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An (α, β)spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β)spanner H for a given graph G and distortion parameters α and β. It first presents a generic distributed algorithm that in constant number of rounds constructs, for every nnode graph and integer k ≥ 1, an (α, β)spanner of O(βn 1+1/k) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k − 1, 0)spanner of at most kn 1+1/k edges in k rounds, matching the performances of the best known distributed algorithm by Derbel et al. (PODC ’08). For k = 2 and constant ε> 0, it can also produce a (1+ε,2−ε)spanner of O(n 3/2) edges in constant time. More interestingly, for every integer k> 1, it can construct in constant time a (1 + ε, O(1/ε) k−2)spanner of O(ε −k+1 n 1+1/k) edges. Such deterministic
Sparse faulttolerant spanners for doubling metrics with bounded hopdiameter or degree
 IN ICALP
, 2012
"... We study faulttolerant spanners in doubling metrics. A subgraph H for a metric space X is called a kvertexfaulttolerant tspanner ((k, t)VFTS or simply kVFTS), if for any subset S â X with S  â¤ k, it holds that dH\S(x, y) â¤ t Â· d(x, y), for any pair of x, y â X \ S. For any doubl ..."
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Cited by 4 (2 self)
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We study faulttolerant spanners in doubling metrics. A subgraph H for a metric space X is called a kvertexfaulttolerant tspanner ((k, t)VFTS or simply kVFTS), if for any subset S â X with S  â¤ k, it holds that dH\S(x, y) â¤ t Â· d(x, y), for any pair of x, y â X \ S. For any doubling metric, we give a basic construction of kVFTS with stretch arbitrarily close to 1 that has optimal O(kn) edges. In addition, we also consider bounded hopdiameter, which is studied in the context of faulttolerance for the first time even for Euclidean spanners. We provide a construction of kVFTS with bounded hopdiameter: for m â¥ 2n, we can reduce the hopdiameter of the above kVFTS to O(Î±(m, n)) by adding O(km) edges, where Î± is a functional inverse of the Ackermannâs function. Finally, we construct a faulttolerant singlesink spanner with bounded maximum degree, and use it to reduce the maximum degree of our basic kVFTS. As a result, we get a kVFTS with O(k 2 n) edges and maximum degree O(kÂ²).
Sparse faulttolerant BFS trees
 In ESA
, 2013
"... A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This paper considers breadthfirst search (BFS) spanning trees, and addresses the problem of designing a sparse faulttolerant BFS tree, or FTBFS tree for ..."
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Cited by 4 (2 self)
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A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This paper considers breadthfirst search (BFS) spanning trees, and addresses the problem of designing a sparse faulttolerant BFS tree, or FTBFS tree for short, namely, a sparse subgraph T of the given network G such that subsequent to the failure of a single edge or vertex, the surviving part T ′ of T still contains a BFS spanning tree for (the surviving part of) G. For a source node s, a target node t and an edge e ∈ G, the shortest s − t path Ps,t,e that does not go through e is known as a replacement path. Thus, our FTBFS tree contains the collection of all replacement paths Ps,t,e for every t ∈ V (G) and every failed edge e ∈ E(G). Our main results are as follows. We present an algorithm that for every nvertex graph G and source node s constructs a (single edge failure) FTBFS tree rooted at s with O(n · min{Depth(s),√n}) edges, where Depth(s) is the depth of the BFS tree rooted at s. This result is complemented by a matching lower bound, showing that there exist nvertex graphs with a source node s for which any edge (or vertex) FTBFS tree rooted at s has Ω(n3/2) edges. We then consider faulttolerant multisource BFS trees, or FTMBFS trees for short, aiming to provide (following a failure) a BFS tree rooted at each source s ∈ S for some subset of sources S ⊆ V. Again, tight bounds are provided, showing that there exists a polytime algorithm that for every nvertex graph and source set S ⊆ V of size σ constructs a (single failure) FTMBFS tree T ∗(S) from each source si ∈ S, with O( σ · n3/2) edges, and on the other hand there exist nvertex graphs with source sets S ⊆ V of cardinality σ, on which any FTMBFS tree from S has Ω( σ · n3/2) edges.
Multipath Spanners
"... This paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with n vertices. Classically, a spanner H of stretch s for a graph G is a spanning subgraph such that the distance in H between any two vertices is at most s times the distance in G. We s ..."
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Cited by 3 (3 self)
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This paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with n vertices. Classically, a spanner H of stretch s for a graph G is a spanning subgraph such that the distance in H between any two vertices is at most s times the distance in G. We study in this paper spanners that approximate short cycles, and more generally p edgedisjoint paths with p> 1, between any pair of vertices. For every unweighted graph G, we construct a 2multipath 3spanner of O(n 3/2) edges. In other words, for any two vertices u, v of G, the length of the shortest cycle (with no edge replication) traversing u, v in the spanner is at most thrice the length of the shortest one in G. This construction is shown to be optimal in term of stretch and of size. In a second construction, we produce a 2multipath (2, 8)spanner of O(n 3/2) edges, i.e., the length of the shortest cycle traversing any two vertices have length at most twice the shortest length in G plus eight. For arbitrary p, we observe that, for each integer k � 1, every weighted graph has a pmultipath p(2k−1)spanner with O(pn 1+1/k) edges, leaving open the question whether, with similar size, the stretch of the spanner can be reduced to 2k − 1 for all p> 1.
Fault tolerant approximate bfs structures
 In SODA
, 2014
"... A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This paper addresses the problem of designing a faulttolerant (α, β) approximate BFS structure (or FTABFS structure for short), namely, a subgraph H of th ..."
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Cited by 3 (1 self)
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A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This paper addresses the problem of designing a faulttolerant (α, β) approximate BFS structure (or FTABFS structure for short), namely, a subgraph H of the network G such that subsequent to the failure of some subset F of edges or vertices, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, satisfying dist(s, v,H \ F) ≤ α · dist(s, v,G \ F) + β for every v ∈ V. We first consider multiplicative (α, 0) FTABFS structures resilient to a failure of a single edge and present an algorithm that given an nvertex unweighted undirected graph G and a source s constructs a (3, 0) FTABFS structure rooted at s with at most 4n edges (improving by an O(log n) factor on the neartight result of [3] for the special case of edge failures). Assuming at most f edge failures, for constant integer f> 1, we prove that there exists a (polytime constructible) (3(f + 1), (f + 1) log n) FTABFS structure with O(fn) edges. We then consider additive (1, β) FTABFS structures. In contrast to the linear size of (α, 0) FTABFS structures, we show that for every β ∈ [1, O(log n)] there exists an nvertex graph G with a source s for which any (1, β) FTABFS structure rooted at s has Ω(n1+(β)) edges, for some function (β) ∈ (0, 1). In particular, (1, 3) FTABFS structures admit a lower bound of Ω(n5/4) edges. These lower bounds demonstrate an interesting dichotomy between multiplicative and additive
NodeDisjoint Multipath Spanners and their Relationship with FaultTolerant Spanners
, 2011
"... Motivated by multipath routing, we introduce a multiconnected variant of spanners. For that purpose we introduce the pmultipath cost between two nodes u and v as the minimum weight of a collection of p internally vertexdisjoint paths between u and v. Given a weighted graph G, a subgraph H is a p ..."
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Motivated by multipath routing, we introduce a multiconnected variant of spanners. For that purpose we introduce the pmultipath cost between two nodes u and v as the minimum weight of a collection of p internally vertexdisjoint paths between u and v. Given a weighted graph G, a subgraph H is a pmultipath sspanner if for all u, v, the pmultipath cost between u and v in H is at most s times the pmultipath cost in G. The s factor is called the stretch. Building upon recent results on faulttolerant spanners, we show how to build pmultipath spanners of constant stretch and of Õ(n1+1/k) edges 1, for fixed parameters p and k, n being the number of nodes of the graph. Such spanners can be constructed by a distributed algorithm running in O(k) rounds. Additionally, we give an improved construction for the case p = k = 2. Our spanner H has O(n 3/2) edges and the pmultipath cost in H between any two node is at most twice the corresponding one in G plus O(W), W being the maximum edge weight.