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53
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 13 (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,
Allpairs nearly 2approximate shortestpaths 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 allpairs tapproximate shortestpaths/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 allpairs tapproximate shortestpaths/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 allpairs nearly 2approximate 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 allpairs 3approximate distances.
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 12 (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.
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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Cited by 10 (0 self)
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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.
Distance oracles for unweighted graphs: breaking the quadratic barrier with constant additive error
, 2008
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Spanners with slack
 PROCEEDINGS OF THE 14TH EUROPEAN SYMPOSIUM ON ALGORITHMS
, 2006
"... Given a metric (V,d), a spanner is a sparse graph whose shortestpath metric approximates the distance d to within a small multiplicative distortion. In this paper, we study the problem of spanners with slack: e.g., can we find sparse spanners where we are allowed to incur an arbitrarily large dist ..."
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Cited by 9 (2 self)
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Given a metric (V,d), a spanner is a sparse graph whose shortestpath metric approximates the distance d to within a small multiplicative distortion. In this paper, we study the problem of spanners with slack: e.g., can we find sparse spanners where we are allowed to incur an arbitrarily large distortion on a small constant fraction of the distances, but are then required to incur only a constant (independent of n) distortion on the remaining distances? We answer this question in the affirmative, thus complementing similar recent results on embeddings with slack into ℓp spaces. For instance, we show that if we ignore an ɛ fraction of the distances, we can get spanners with O(n) edgesand O(log 1) distortion for the remaining distances. ɛ We also show how to obtain sparse and lowweight spanners with slack from existing constructions of conventional spanners, and these techniques allow us to also obtain the best known results for distance oracles and distance labelings with slack. This paper complements similar results obtained in recent research on slack embeddings into normed metric spaces.
Fast deterministic distributed algorithms for sparse spanners
 IN 13 TH INTERNATIONAL COLLOQUIUM ON STRUCTURAL INFORMATION & COMMUNICATION COMPLEXITY (SIROCCO
, 2006
"... This paper concerns the efficient construction of sparse and low stretch spanners for unweighted arbitrary graphs with n nodes. All previous deterministic distributed algorithms, for constant stretch spanner of o(n²) edges, have a running time Ω(n^ɛ) for some constant ɛ > 0 depending on the stret ..."
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Cited by 9 (5 self)
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This paper concerns the efficient construction of sparse and low stretch spanners for unweighted arbitrary graphs with n nodes. All previous deterministic distributed algorithms, for constant stretch spanner of o(n²) edges, have a running time Ω(n^ɛ) for some constant ɛ > 0 depending on the stretch. Our deterministic distributed algorithms construct constant stretch spanners of o(n²) edges in o(n^ɛ) time for any constant ɛ > 0. More precisely, in the Linial’s free model, we construct in n O(1/ √ log n) time, for every graph, a 5spanner of O(n 3/2) edges. The result is extended to O(k 2.322)spanners with O(n 1+1/k) edges for every parameter k � 1. If the minimum degree of the graph is Ω(√n), then, in the same time complexity, a 9spanner with O(n) edges can be constructed.
FASTER ALGORITHMS FOR ALLPAIRS APPROXIMATE SHORTEST PATHS IN UNDIRECTED GRAPHS
, 2006
"... Let G = (V, E) be a weighted undirected graph having nonnegative edge weights. An estimate ˆ δ(u, v) of the actual distance δ(u, v) between u, v ∈ V is said to be of stretch t iff δ(u, v) ≤ ˆ δ(u, v) ≤ t · δ(u, v). Computing allpairs small stretch distances efficiently (both in terms of time ..."
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Cited by 9 (2 self)
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Let G = (V, E) be a weighted undirected graph having nonnegative edge weights. An estimate ˆ δ(u, v) of the actual distance δ(u, v) between u, v ∈ V is said to be of stretch t iff δ(u, v) ≤ ˆ δ(u, v) ≤ t · δ(u, v). Computing allpairs small stretch distances efficiently (both in terms of time and space) is a wellstudied problem in graph algorithms. We present a simple, novel and generic scheme for allpairs approximate shortest paths. Using this scheme and some new ideas and tools, we design faster algorithms for allpairs tstretch distances for a whole range of stretch t, and also answer an open question posed by Thorup and Zwick in their
TransitiveClosure Spanners: A Survey
"... We survey results on transitiveclosure spanners and their applications. 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 8 (5 self)
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We survey results on transitiveclosure spanners and their applications. 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 different areas of computer science, and properties of these spanners have been rediscovered over the span of 20 years. The common task implicitly tackled in these diverse applications can be abstracted as the problem of constructing sparse TCspanners. In this article, we survey combinatorial bounds on the size of sparsest TCspanners, and algorithms and inapproximability results for the problem of computing the sparsest TCspanner of a given directed graph. We also describe multiple applications of TCspanners, including property testing, property reconstruction, key management in access control hierarchies and data structures.
ISLABEL: an IndependentSet based Labeling Scheme for PointtoPoint Distance Querying
"... We study the problem of computing shortest path or distance between two query vertices in a graph, which has numerous important applications. Quite a number of indexes have been proposed to answer such distance queries. However, all of these indexes can only process graphs of size barely up to 1 mil ..."
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Cited by 7 (2 self)
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We study the problem of computing shortest path or distance between two query vertices in a graph, which has numerous important applications. Quite a number of indexes have been proposed to answer such distance queries. However, all of these indexes can only process graphs of size barely up to 1 million vertices, which is rather small in view of many of the fastgrowing realworld graphs today such as social networks and Web graphs. We propose an efficient index, which is a novel labeling scheme based on the independent set of a graph. We show that our method can handle graphs of size orders of magnitude larger than existing indexes. 1.