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53
Approximate distance oracles
, 2004
"... Let G = (V, E) be an undirected weighted graph with |V | = n and |E | = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in ..."
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Cited by 273 (9 self)
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Let G = (V, E) be an undirected weighted graph with |V | = n and |E | = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdős, implies that Ω(n 1+1/k) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name “oracle”. Previously, data structures that used only O(n 1+1/k) space had a query time of Ω(n 1/k). Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
Transitive-closure spanners
, 2008
"... We define the notion of a transitive-closure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanner ..."
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Cited by 35 (11 self)
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We define the notion of a transitive-closure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were studied implicitly in access control, property testing, and data structures, and properties of these spanners have been rediscovered over the span of 20 years. We bring these areas under the unifying framework of TC-spanners. We abstract the common task implicitly tackled in these diverse applications as the problem of constructing sparse TCspanners. We study the approximability of the size of the sparsest k-TC-spanner for a given digraph. Our technical contributions fall into three categories: algorithms for general digraphs,
A simple and linear time randomized algorithm for computing sparse . . .
"... Let G = (V, E) be an undirected weighted graph on |V | = n vertices, and |E | = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V, ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Comput ..."
Abstract
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Cited by 34 (5 self)
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Let G = (V, E) be an undirected weighted graph on |V | = n vertices, and |E | = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V, ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Computing a t-spanner of minimum size (number of edges) has been a widely studied and well motivated problem in computer science. In this paper we present the first linear time randomized algorithm that computes a t-spanner of a given weighted graph. Moreover, the size of the t-spanner computed essentially matches the worst case lower bound implied by a 43 years old girth conjecture made independently by Erdős [26], Bollobás [19], and Bondy & Simonovits [21]. Our algorithm uses a novel clustering approach that avoids any distance computation altogether. This feature is somewhat surprising since all the previously existing algorithms employ computation of some sort of local or global distance information which involves growing either breadth first search trees up to θ(t)-levels or full shortest path trees on a large fraction of vertices. The truly local approach of our algorithm also leads to equally simple and efficient algorithms for computing spanners in other important computational environments like distributed, parallel, and external memory.
Faster algorithms for approximate distance oracles and all-pairs small stretch paths
- In 47th Annual IEEE Symp. on Foundations of Computer Science (FOCS
, 2006
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New Constructions of (α, β)-Spanners and Purely Additive Spanners
, 2005
"... An ¦ α § β ¨-spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u § v: δH ¦ u § v¨� © αδG ¦ u § v¨� � β, where δG is the distance w.r.t. G. It is well known that there exist (multiplicative) ¦ 2k � 1 § 0 ¨-spanners of size ..."
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Cited by 27 (6 self)
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An ¦ α § β ¨-spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u § v: δH ¦ u § v¨� © αδG ¦ u § v¨� � β, where δG is the distance w.r.t. G. It is well known that there exist (multiplicative) ¦ 2k � 1 § 0 ¨-spanners of size O ¦ n 1 � 1 � k ¨ and that there exist (purely additive) ¦ 1 § 2 ¨-spanners of size O ¦ n 3 � 2 ¨. However no other ¦ 1 § O ¦ 1¨� ¨-spanners are known to exist. In this paper we develop a couple new techniques for constructing ¦ α § β ¨-spanners. The first result is a purely additive ¦ 1 § 6 ¨-spanner of size O ¦ n 4 � 3 ¨. Our 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. This general approach should lead to new spanner constructions. The second result is a truly simple linear time construction of ¦ k § k � 1 ¨-spanners with size O ¦ n 1 � 1 � k ¨. In a distributed network the algorithm terminates in a constant number of rounds and has expected size O ¦ n 1 � 1 � k ¨. The new idea here is primarily in the analysis of the construction. We show that a few simple and local rules for picking spanner edges induce seemingly coordinated global behavior.
Lower Bounds for Additive Spanners, Emulators, and More
, 2006
"... An additive spanner of an unweighted undirected graph G with distortion d is a subgraph H such that for any two vertices u, v 2 G, we have ffiH(u, v) < = ffiG(u, v) + d. Forevery k = O ( ln nln ln n), we construct a graph G on n vertices for which any additive spanner of G with distortion 2k-1 h ..."
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Cited by 26 (2 self)
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An additive spanner of an unweighted undirected graph G with distortion d is a subgraph H such that for any two vertices u, v 2 G, we have ffiH(u, v) < = ffiG(u, v) + d. Forevery k = O ( ln nln ln n), we construct a graph G on n vertices for which any additive spanner of G with distortion 2k-1 has \Omega ( 1 k n 1+1/k) edges. This matches the lower bound previously known only to hold under a 1963 conjecture of Erdös. We generalize our lower bound in a number of ways.First, we consider graph emulators introduced by Dor, Halperin, and Zwick (FOCS, 1996), where an emulatorof an unweighted undirected graph G with distortion dis like an additive spanner except H may be an arbi-trary weighted graph such that ffiG(u, v) < = ffiH(u, v) < = ffiG(u, v) + d. We show a lower bound of \Omega ( 1k2 n1+1/k)edges for distortion-(2 k- 1) emulators. These are the first non-trivial bounds for k> 3. Second, we parameterize our bounds in terms of the minimum degree of the graph. Namely, for minimum degree n1/k+c for any c> = 0,we prove a bound of \Omega ( 1 k n1+1/k-c(1+2/(k-1))) for addi-tive spanners and \Omega ( 1 k2 n 1+1/k-c(1+2/(k-1))) for emulators. For k = 2 these can be improved to \Omega (n3/2-c).This partially answers a question of Baswana et al (SODA, 2005) for additive spanners. Finally, we continue the study of pair-wise and source-wise distance preservers defined by Coppersmith and Elkin (SODA, 2005) by considering their approximate variants and their relaxation to emulators. We prove the first lower bounds for such graphs.
Distance oracles for sparse graphs
- IN PROCEEDINGS OF THE 50TH IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
"... Thorup and Zwick, in their seminal work, introduced the approximate distance oracle, which is a data structure that answers distance queries in a graph. For any integer k, they showed an efficient algorithm to construct an approximate distance oracle using space O(kn 1+1/k) that can answer queries i ..."
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Cited by 26 (4 self)
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Thorup and Zwick, in their seminal work, introduced the approximate distance oracle, which is a data structure that answers distance queries in a graph. For any integer k, they showed an efficient algorithm to construct an approximate distance oracle using space O(kn 1+1/k) that can answer queries in time O(k) with a distance estimate that is at most α = 2k − 1 times larger than the actual shortest distance (α is called the stretch). They proved that, under a combinatorial conjecture, their data structure is optimal in terms of space: if a stretch of at most 2k−1 is desired, then the space complexity is at least n 1+1/k. Their proof holds even if infinite query time is allowed: it is essentially an “incompressibility ” result. Also, the proof only holds for dense graphs, and the best bound it can prove only implies that the size of the data structure is lower bounded by the number of edges of the graph. Naturally, the following question arises: what happens for sparse graphs? In this paper we give a new lower bound for approximate distance oracles in the cell-probe model. This lower bound holds even for sparse (polylog(n)-degree) graphs, and it is not an “incompressibility ” bound: we prove a three-way tradeoff between space, stretch and query time. We show that, when the query time is t, and the stretch is α, then the space S must be S ≥ n 1+Ω(1/tα) / lg n. (1) This lower bound follows by a reduction from lopsided set disjointness to distance oracles, based on and motivated by recent work of Pǎtras¸cu. Our results in fact show that for any high-girth regular graph, an approximate distance oracle that supports efficient queries for all subgraphs of G must obey Eq. (1). We also prove some lemmas that count sets of paths in high-girth regular graphs and high-girth regular expanders, which might be of independent interest.
Efficient algorithms for constructing (1 + ɛ, β)-spanners in the distributed and streaming models (Extended Abstract)
- PODC
, 2004
"... For an unweighted undirected graph G = (V, E), and a pair of positive integers α ≥ 1, β ≥ 0, a subgraph G ′ = (V, H), H ⊆ E, is called an (α, β)-spanner of G if for every pair of vertices u, v ∈ V, distG ′(u, v) ≤ α · distG(u, v) + β. It was shown in [20] that for any ɛ> 0, κ = 1, 2,..., there ..."
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Cited by 20 (6 self)
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For an unweighted undirected graph G = (V, E), and a pair of positive integers α ≥ 1, β ≥ 0, a subgraph G ′ = (V, H), H ⊆ E, is called an (α, β)-spanner of G if for every pair of vertices u, v ∈ V, distG ′(u, v) ≤ α · distG(u, v) + β. It was shown in [20] that for any ɛ> 0, κ = 1, 2,..., there exists an integer β = β(ɛ, κ) such that for every n-vertex graph G there exists a (1 + ɛ, β)-spanner G ′ with O(n 1+1/κ) edges. An efficient distributed protocol for constructing (1+ ɛ, β)-spanners was devised in [18]. The running time and the communication complexity of that protocol are O(n 1+ρ) and O(|E|n ρ), respectively, where ρ is an additional control parameter of the protocol that affects only the additive term β. In this paper we devise a protocol with a drastically improved running time (O(n ρ) as opposed to O(n 1+ρ)) for constructing (1 + ɛ, β)-spanners. Our protocol has the same communication complexity as the protocol of [18], and it constructs spanners with essentially the same properties as the spanners that are constructed by the protocol of [18]. We also show that our protocol for constructing (1+ɛ, β)spanners can be adapted to the streaming model, and devise a streaming algorithm that uses a constant number of passes and O(n 1+1/κ · log n) bits of space for computing allpairs-almost-shortest-paths of length at most by a multiplicative factor (1 + ɛ) and an additive term of β greater than the shortest paths. Our algorithm processes each edge in time O(n ρ), for an arbitrarily small ρ> 0. The only
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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Cited by 17 (4 self)
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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.
