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Low Distortion Spanners
"... A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an f-spanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tr ..."
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A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an f-spanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tradeoff between the sparsity of H and the distortion function f, though the nature of this tradeoff is still poorly understood. In this paper we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2- and 6-spanners of Aingworth et al. and Baswana et al. and improve on the (1+ɛ, β)-spanners of Elkin and Peleg, the sublinear additive spanners of Thorup and Zwick, and the (non constant) additive spanners of Baswana et al. Our constructions rival the simplicity of all comparable algorithms and provide substantially better spanners, in some cases reducing the density doubly exponentially.
On the locality of distributed sparse spanner construction
- In ACM Press, editor, 27th Annual ACM Symp. on Principles of Distributed Computing (PODC
, 2008
"... The paper presents a deterministic distributed algorithm that, given k � 1, constructs in k rounds a (2k−1, 0)-spanner of O(kn 1+1/k)edgesforeveryn-node unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)-spanner with O ..."
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Cited by 17 (6 self)
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The paper presents a deterministic distributed algorithm that, given k � 1, constructs in k rounds a (2k−1, 0)-spanner of O(kn 1+1/k)edgesforeveryn-node unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)-spanner with O(kn 1+1/k) edges.) Previous distributed solutions achieving such optimal stretch-size trade-off either make use of randomization providing performance guarantees in expectation only, or perform in log Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every ɛ>0, constructs a (1 + ɛ, 2)-spanner of O(ɛ −1 n 3/2)edgesin O(ɛ −1) rounds, without any prior knowledge on the graph. Our algorithms are complemented with lower bounds, which hold even under the assumption that n is known to the nodes. It is shown that any (randomized) distributed algorithm requires k rounds in expectation to compute a (2k − 1, 0)-spanner of o(n 1+1/(k−1))edgesfork ∈{2, 3, 5}. It is also shown that for every k>1, any (randomized) distributed algorithm that constructs a spanner with fewer than n 1+1/k+ɛ edges in at most n ɛ expected rounds must stretch some distances by an additive factor of n Ω(ɛ).Inotherwords, while additive stretched spanners with O(n 1+1/k) edges may exist, e.g., for k =2, 3, they cannot be computed distributively in a sub-polynomial number of rounds in expectation. Supported by the équipe-projet INRIA “DOLPHIN”. Supported by the ANR-project “ALADDIN”, and the
Fault-tolerant spanners for general graphs
- in STOC’09, 2009
"... The paper concerns graph spanners that are resistant to ver-tex or edge failures. Given a weighted undirected n-vertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a k-spanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u ..."
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The paper concerns graph spanners that are resistant to ver-tex or edge failures. Given a weighted undirected n-vertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a k-spanner) 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 intro-duction over two decades ago. It is known how to efficiently construct a (2k−1)-spanner of size O(n1+1/k), and this size-stretch 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 f-vertex fault tolerant k-spanner 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 f-vertex 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 f-vertex fault tolerant (2k−1)-spanner of size
Additive Spanners and (α, β)-Spanners
"... An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k) and an (additive) (1, 2)-spanner of size O(n 3/2). How ..."
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Cited by 14 (3 self)
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An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k) and an (additive) (1, 2)-spanner of size O(n 3/2). However no other additive spanners are known to exist. In this paper we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1, 6)-spanner of size O(n 4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well-approximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that for any k, a (k, k − 1)-spanner with size O(kn 1+1/k) can be found in linear time, and further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.
All-pairs nearly 2-approximate shortest-paths in O(n² polylog n) time
- IN PROCEEDINGS OF 22ND ANNUAL SYMPOSIUM ON THEORETICAL ASPECT OF COMPUTER SCIENCE, VOLUME 3404 OF LNCS
, 2005
"... Let G(V, E) be an unweighted undirected graph on |V| = n vertices. Let δ(u, v) denote the shortest distance between vertices u, v ∈ V. An algorithm is said to compute all-pairs t-approximate shortest-paths/distances, for some t ≥ 1, if for each pair of vertices u, v ∈ V, the path/distance reported ..."
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Cited by 13 (6 self)
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Let G(V, E) be an unweighted undirected graph on |V| = n vertices. Let δ(u, v) denote the shortest distance between vertices u, v ∈ V. An algorithm is said to compute all-pairs t-approximate shortest-paths/distances, for some t ≥ 1, if for each pair of vertices u, v ∈ V, the path/distance reported by the algorithm is not longer/greater than t · δ(u, v). This paper presents two randomized algorithms for computing all-pairs nearly 2-approximate distances. The first algorithm takes expected O(m 2/3 n log n+n²) time, and for any u, v ∈ V reports distance no greater than 2δ(u, v) + 1. Our second algorithm requires expected O(n² log 3/2) time, and for any u, v ∈ V reports distance bounded by 2δ(u, v) + 3. This paper also presents the first expected O(n 2) time algorithm to compute all-pairs 3-approximate distances.
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 non-trivial 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 non-trivial 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 log-logarithmic, 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.
Distance oracles for unweighted graphs: breaking the quadratic barrier with constant additive error
, 2008
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Faster approximation of distances in graphs
- IN PROC. WADS
, 2007
"... Let G = (V, E) be an weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating allpairs shortest paths in G. Our first algorithm runs in Õ(n2) time, and for any u, v ∈ V reports distance no greater th ..."
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Cited by 7 (0 self)
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Let G = (V, E) be an weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating allpairs shortest paths in G. Our first algorithm runs in Õ(n2) time, and for any u, v ∈ V reports distance no greater than 2dG(u, v)+h(u, v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ∈ V reports distance no greater than (1+ǫ)dG(u, v)+2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n 2.24+o(1) ǫ −3 log(nǫ −1)) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in Õ(m √ n + n 2) time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for diameter approximation. Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.
Distance Oracles for Vertex-Labeled Graphs
"... Abstract. Given a graph G = (V, E) with non-negative edge lengths whose vertices are assigned a label from L = {λ1,..., λℓ}, we construct a compact distance oracle that answers queries of the form: “What is δ(v, λ)?”, where v ∈ V is a vertex in the graph, λ ∈ L a vertex label, and δ(v, λ) is the dis ..."
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Abstract. Given a graph G = (V, E) with non-negative edge lengths whose vertices are assigned a label from L = {λ1,..., λℓ}, we construct a compact distance oracle that answers queries of the form: “What is δ(v, λ)?”, where v ∈ V is a vertex in the graph, λ ∈ L a vertex label, and δ(v, λ) is the distance (length of a shortest path) between v and the closest vertex labeled λ in G. We formalize this natural problem and provide a hierarchy of approximate distance oracles that require subquadratic space and return a distance of constant stretch. We also extend our solution to dynamic oracles that handle label changes in sublinear time. 1
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 t-approximate 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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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 t-approximate 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 t-approximate shortest path between u and v when vertex z fails We present data structures for the single source as well all-pairs 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.
