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24
Faulttolerant spanners for general graphs
 in STOC’09, 2009
"... The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected nvertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a kspanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u ..."
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Cited by 17 (4 self)
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The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected nvertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a kspanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u, v) denotes the distance between u and v in G Graph spanners were extensively studied since their introduction over two decades ago. It is known how to efficiently construct a (2k−1)spanner of size O(n1+1/k), and this sizestretch tradeoff is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [Levcopoulos et al., STOC’98]. A subgraph H is an fvertex fault tolerant kspanner of the graph G if for any set F ⊆ V of size at most f and any pair of vertices u, v ∈ V \ F, the distances in H satisfy δH\F (u, v) ≤ k · δG\F (u, v). Levcopoulos et al. presented an efficient algorithm that given a set S of n points in Rd, constructs an fvertex fault tolerant geometric (1+)spanner for S, that is, a sparse graph H such that for every set F ⊆ S of size f and any pair of points u, v ∈ S \F, δH\F (u, v) ≤ (1+)uv, where uv  is the Euclidean distance between u and v. A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [Czumaj & Zhao, SoCG’03]. This paper also raised as an open problem the question whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph. The current paper answers this question in the affirmative, presenting an fvertex fault tolerant (2k−1)spanner of size
Local Approximation Schemes for Topology Control
 IN: PROC. OF THE 25 TH ANNUAL ACM SYMPOSIUM ON PRINCIPLES OF DISTRIBUTED COMPUTING (PODC). (2006
, 2006
"... This paper presents a distributed algorithm on wireless adhoc networks that runs in polylogarithmic number of rounds in the size of the network and constructs a linear size, lightweight, (1 + ε)spanner for any given ε> 0. A wireless network is modeled by a ddimensional αquasi unit ball graph ( ..."
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Cited by 15 (1 self)
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This paper presents a distributed algorithm on wireless adhoc networks that runs in polylogarithmic number of rounds in the size of the network and constructs a linear size, lightweight, (1 + ε)spanner for any given ε> 0. A wireless network is modeled by a ddimensional αquasi unit ball graph (αUBG), which is a higher dimensional generalization of the standard unit disk graph (UDG) model. The ddimensional αUBG model goes beyond the unrealistic “flat world ” assumption of UDGs and also takes into account transmission errors, fading signal strength, and physical obstructions. The main result in the paper is this: for any fixed ε> 0, 0 < α ≤ 1, and d ≥ 2 there is a distributed algorithm running in O(log n·log ∗ n) communication rounds on an nnode, ddimensional αUBG G that computes a (1+ε)spanner G ′ of G with maximum degree ∆(G ′ ) = O(1) and total weight w(G ′ ) = O(w(MST (G)). This result is motivated by the topology control problem in wireless adhoc networks and improves on existing topology control algorithms along several dimensions. The technical contributions of the paper include a new, sequential, greedy algorithm with relaxed edge ordering and lazy updating, and clustering techniques for filtering out unnecessary edges.
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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Cited by 14 (3 self)
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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.
Distributed Spanner Construction in Doubling Metric Spaces
"... This paper presents a distributed algorithm that runs on an nnode unit ball graph (UBG) G residing in a metric space of constant doubling dimension, and constructs, for any ε> 0, a (1 + ε)spanner H of G with maximum degree bounded above by a constant. In addition, we show that H is “lightweight ..."
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This paper presents a distributed algorithm that runs on an nnode unit ball graph (UBG) G residing in a metric space of constant doubling dimension, and constructs, for any ε> 0, a (1 + ε)spanner H of G with maximum degree bounded above by a constant. In addition, we show that H is “lightweight”, in the following sense. Let ∆ denote the aspect ratio of G, that is, the ratio of the length of a longest edge in G to the length of a shortest edge in G. The total weight of H is bounded above by O(log ∆) · wt(MST), where MST denotes a minimum spanning tree of the metric space. Finally, we show that H satisfies the so called leapfrog property, an immediate implication being that, for the special case of Euclidean metric spaces with fixed dimension, the weight of H is bounded above by O(wt(MST)). Thus, the current result subsumes the results of the authors in PODC 2006 that apply to Euclidean metric spaces, and extends these results to metric spaces with constant doubling dimension.
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.
Incubators vs Zombies: FaultTolerant, Short, Thin and Lanky Spanners for Doubling Metrics
, 2012
"... Recently Elkin and Solomon gave a construction of spanners for doubling metrics that has constant maximum degree, hopdiameter O(log n) and lightness O(log n) (i.e., weight O(log n)·w(MST)). This resolves a long standing conjecture proposed by Arya et al. in a seminal STOC 1995 paper. However, Elkin ..."
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Cited by 5 (1 self)
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Recently Elkin and Solomon gave a construction of spanners for doubling metrics that has constant maximum degree, hopdiameter O(log n) and lightness O(log n) (i.e., weight O(log n)·w(MST)). This resolves a long standing conjecture proposed by Arya et al. in a seminal STOC 1995 paper. However, Elkin and Solomon’s spanner construction is extremely complicated; we offer a simple alternative construction that is very intuitive and is based on the standard technique of net tree with cross edges. Indeed, our approach can be readily applied to our previous construction of kfault tolerant spanners (ICALP 2012) to achieve kfault tolerance, maximum degree O(k²), hopdiameter O(log n) and lightness O(k³ log n).
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).
Minimum power assignment in wireless ad hoc networks with spanner property
 Journal of Combinatorial Optimization
, 2006
"... Abstract — Power assignment for wireless networks is to assign a power for each wireless node such that the induced communication graph has some required properties. In this paper, we study the power assignment such that the induced communication graph is a spanner for the original communication gra ..."
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Abstract — Power assignment for wireless networks is to assign a power for each wireless node such that the induced communication graph has some required properties. In this paper, we study the power assignment such that the induced communication graph is a spanner for the original communication graph when all nodes have the maximum power. Polynomial time algorithm is given to minimize the maximum assigned power. Then we propose a new polynomial time approximation method to minimize the total transmission radius of all nodes. We also give two heuristics and conduct extensive simulations to study their performance when we want to minimize the total assigned power of all nodes. Our simulations validate our theoretical claims. Keywords—Power assignment, spanner, wireless ad hoc networks. I.
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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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.
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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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.