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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.
Toward More Localized Local Algorithms: Removing Assumptions Concerning Global Knowledge
"... Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and ( ∆ + 1)coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the O(∆2)coloring algorithm by Linial [2 ..."
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Cited by 14 (9 self)
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Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and ( ∆ + 1)coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the O(∆2)coloring algorithm by Linial [28]. Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are nonuniform, that is, they assume that all nodes know good estimations of one or more global parameters of the network, e.g., the maximum degree ∆ or the number of nodes n. This paper provides a rather general method for transforming a nonuniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original nonuniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all of the state of the art nonuniform algorithms regarding MIS and Maximal Matching, as well as to many results concerning the coloring problem. (In particular, it applies to all aforementioned algorithms.) To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.
Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance
, 2012
"... In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each r ..."
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In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In the LOCAL model, this is quite simple: each node broadcasts all of its information in each round, and the number of rounds required will be equal to the diameter of the underlying communication graph. In the GOSSIP model, each node must independently choose a single neighbor to contact, and the lack of global information makes it difficult to make any sort of principled choice. As such, researchers have focused on the uniform gossip algorithm, in which each node independently selects a neighbor uniformly at random. When the graph is wellconnected, this works quite well. In a string of beautiful papers, researchers proved a sequence of successively stronger bounds on the number of rounds required in terms of the conductance φ and graph size n, culminating in a bound of O(φ −1 log n). In this paper, we show that a fairly simple modification of the protocol gives an algorithm that solves the information dissemination problem in at most O(D + polylog(n)) rounds in a network of diameter D, with no dependence on the conductance. This is
Distributed Algorithms for Ultrasparse Spanners and Linear Size Skeletons
"... We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages ..."
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We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2 log ∗ n log n)spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n) φ) where φ = 1+ √ 5 2 is the golden ratio. As the distance increases the Fibonacci spanner’s multiplicative distortion passes through four discrete stages, moving from logarithmic to loglogarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.
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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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
Fully Dynamic Randomized Algorithms for Graph Spanners
, 2008
"... Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. ..."
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Cited by 4 (0 self)
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Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a tspanner of itself, the research as well as applications of spanners invariably deal with a tspanner which has as small number of edges as possible. We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner. Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a tspanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a tspanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.
Remotespanners: What to know beyond neighbors
, 2008
"... Motivated by the fact that neighbors are generally known in practical routing algorithms, we introduce the notion of remotespanner. Given an unweighted graph G, a subgraph H with vertex set V (H) = V (G) is an (α, β)remotespanner if for each pair of points u and v the distance between u and v i ..."
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Cited by 3 (1 self)
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Motivated by the fact that neighbors are generally known in practical routing algorithms, we introduce the notion of remotespanner. Given an unweighted graph G, a subgraph H with vertex set V (H) = V (G) is an (α, β)remotespanner if for each pair of points u and v the distance between u and v in Hu, the graph H augmented by all the edges between u and its neighbors in G, is at most α times the distance between u and v in G plus β. We extend this definition to kconnected graphs by considering the minimum length sum over k disjoint paths as a distance. We then say that an (α, β)remotespanner is kconnecting. In this paper, we give distributed algorithms for computing (1 + ε, 1 − 2ε)remotespanners for any ε> 0, kconnecting (1, 0)remotespanners for any k ≥ 1 (yielding (1, 0)remotespanners for k = 1) and 2connecting (2, −1)remotespanners. All these algorithms run in constant time for any unweighted input graph. The number of edges obtained for kconnecting (1, 0)remotespanner is within a logarithmic factor from optimal (compared to the best kconnecting (1, 0)remotespanner of the input graph). Interestingly, sparse (1, 0)remotespanners (i.e. preserving exact distances) with O(n 4/3) edges exist in random unit disk graphs. The number of edges obtained for (1+ε, 1−2ε)remotespanners and 2connecting (2, −1)remotespanners is linear if the input graph is the unit ball graph of a doubling metric (even if distances between nodes are unknown). Our methodology consists in characterizing remotespanners as subgraphs containing the union of small depth tree subgraphs dominating nearby nodes. This leads to simple local distributed algorithms. 1.
What Can Be Observed Locally? RoundBased Models for Quantum Distributed Computing
 COMPUTING, IN "23RD INTERNATIONAL SYMPOSIUM ON DISTRIBUTED COMPUTING (DISC) DISC, ESPAGNE ELCHE/ELX
, 2009
"... It is a wellknown fact that, by resorting to quantum processing in addition to manipulating classical information, it is possible to reduce the time complexity of some centralized algorithms, and also to decrease the bit size of messages exchanged in tasks requiring communication among several agen ..."
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It is a wellknown fact that, by resorting to quantum processing in addition to manipulating classical information, it is possible to reduce the time complexity of some centralized algorithms, and also to decrease the bit size of messages exchanged in tasks requiring communication among several agents. Recently, several claims have been made that certain fundamental problems of distributed computing, including Leader Election and Distributed Consensus, begin to admit feasible and efficient solutions when the model of distributed computation is extended so as to apply quantum processing. This has been achieved in one of two distinct ways: (1) by initializing the system in a quantum entangled state, and/or (2) by applying quantum communication channels. In this paper, we explain why some of these prior claims are misleading, in the sense that they rely on changes to the model unrelated to quantum processing. On the positive side, we consider the aforementioned quantum extensions when applied to Linial’s wellestablished LOCAL model of distributed computing. For both types of extensions, we put forward valid proofofconcept examples of distributed problems whose round complexity
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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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.
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.