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56
Approximate distance oracles
 J. ACM
"... 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 279 (10 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. 1
Transitiveclosure spanners
, 2008
"... We define the notion of a transitiveclosure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure as G and (2) diameter at most k. These spanner ..."
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Cited by 38 (11 self)
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We define the notion of a transitiveclosure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure 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 TCspanners. 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 kTCspanner for a given digraph. Our technical contributions fall into three categories: algorithms for general digraphs,
Faster algorithms for approximate distance oracles and allpairs small stretch paths
 In 47th Annual IEEE Symposium 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 28 (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 wellapproximated 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
"... An additive spanner of an unweighted undirected graph G with distortion d is a subgraph H such that for any two vertices u, v ∈ G, we have δH(u, v) ≤ δG(u, v) + d. For ln n every k = O (), we construct a graph G on n vertices ln ln n for which any additive spanner of G with distortion 2k − 1 has Ω ..."
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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 ∈ G, we have δH(u, v) ≤ δG(u, v) + d. For ln n every k = O (), we construct a graph G on n vertices ln ln n for which any additive spanner of G with distortion 2k − 1 has Ω ( 1 k n1+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 emulator of an unweighted undirected graph G with distortion d is like an additive spanner except H may be an arbitrary weighted graph such that δG(u, v) ≤ δH(u, v) ≤ δG(u, v) + d. We show a lower bound of Ω ( 1 k 2 n 1+1/k) edges for distortion(2k − 1) emulators. These are the first nontrivial bounds for k> 3. Second, we parameterize our bounds in terms of the minimum degree of the graph. Namely, for minimum degree n 1/k+c for any c ≥ 0, we prove a bound of Ω ( 1 k n1+1/k−c(1+2/(k−1)) ) for additive spanners and Ω ( 1 k 2 n 1+1/k−c(1+2/(k−1)) ) for emulators. For k = 2 these can be improved to Ω(n 3/2−c). This partially answers a question of Baswana et al (SODA, 2005) for additive spanners. Finally, we continue the study of pairwise and sourcewise 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
"... Abstract — 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 answe ..."
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Cited by 26 (4 self)
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Abstract — 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 cellprobe model. This lower bound holds even for sparse (polylog(n)degree) graphs, and it is not an “incompressibility ” bound: we prove a threeway 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 highgirth 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 highgirth regular graphs and highgirth regular expanders, which might be of independent interest. Keywordsdistance oracle; data structures; lower bounds; cellprobe model; lopsided set disjointness 1.
Efficient algorithms for constructing (1 + ɛ, β)spanners in the distributed and streaming models
 Distributed Computing
, 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 nvertex 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(En 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 allpairsalmostshortestpaths 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
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
Lower bounds for local monotonicity reconstruction from transitiveclosure spanners
, 2010
"... Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure as G and (2) diameter at most k. Transitiveclosure spanners are a common abstraction for applications in access contr ..."
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Cited by 16 (7 self)
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Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure as G and (2) diameter at most k. Transitiveclosure spanners are a common abstraction for applications in access control, property testing and data structures. We show a connection between 2TCspanners and local monotonicity reconstructors. A local monotonicity reconstructor, introduced by Saks and Seshadhri (SIAM Journal on Computing, 2010), is a randomized algorithm that, given access to an oracle for an almost monotone function f: [m] d → R, can quickly evaluate a related function g: [m] d → R which is guaranteed to be monotone. Furthermore, the reconstructor can be implemented in a distributed manner. We show that an efficient local monotonicity reconstructor implies a sparse 2TCspanner of the directed hypergrid (hypercube), providing a new technique for proving lower bounds for local monotonicity reconstructors. Our connection is,
Approximate distance oracles for geometric spanners
 Submitted
, 2002
"... Given an arbitrary real constant ε> 0, and a geometric graph G in ddimensional Euclidean space with n points, O(n) edges, and constant dilation, our main result is a data structure that answers (1 + ε)approximate shortest path length queries in constant time. The data structure can be construct ..."
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Cited by 13 (2 self)
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Given an arbitrary real constant ε> 0, and a geometric graph G in ddimensional Euclidean space with n points, O(n) edges, and constant dilation, our main result is a data structure that answers (1 + ε)approximate shortest path length queries in constant time. The data structure can be constructed in O(n log n) time using O(n log n) space. This represents the first data structure that answers (1 + ε)approximate shortest path queries in constant time, and hence functions as an approximate distance oracle. The data structure is also applied to several other problems. In particular, we also show that approximate shortest path queries between vertices in a planar polygonal domain with “rounded ” obstacles can be answered in constant time. Other applications include query versions of closest pair problems, and the efficient computation of the approximate dilations of geometric graphs. Finally, we show how to extend the main result to answer (1 + ε)approximate shortest path length queries in constant time for geometric spanner graphs with m = ω(n) edges. The resulting data structure can be constructed in O(m + n log n) time using O(n log n) space.