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10
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 24 (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.
Faster algorithms for approximate distance oracles and allpairs small stretch paths
 In Proceedings of the 47th Annual IEEE FOCS
, 2006
"... Let G = (V, E) be a weighted undirected graph with V  = n and E  = m. An estimate ˆ δ(u, v) of the 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 distances of small stretch efficiently is a wellstudied problem in graph algorith ..."
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Cited by 21 (6 self)
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Let G = (V, E) be a weighted undirected graph with V  = n and E  = m. An estimate ˆ δ(u, v) of the 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 distances of small stretch efficiently is a wellstudied problem in graph algorithms. The most efficient algorithms known here are the approximate distance oracles of [16] and the three algorithms in [9] to compute allpairs stretch t distances for t = 2, 7/3, and 3. We present faster algorithms for these problems. For any integer k ≥ 1, Thorup and Zwick in [16] gave an O(kmn 1/k) algorithm to construct a data structure of size O(kn 1+1/k) to answer approximate distance queries. For a query (u, v) ∈ V × V, the distance returned is of stretch at most 2k−1. The query answering time is O(k), which is essentially a constant since we are interested in smallstretch distances, or equivalently, small values of k. But for small values of k, the time to construct the oracle is rather high. The case k = 2 is particularly interesting and the ThorupZwick algorithm takes O(m √ n) time, which could be as large as Θ(n 5/2). Here we present an O(n 2 log n) algorithm to construct such a data structure of size O(kn 1+1/k) for all integers k ≥ 2. Our query answering time is O(k) for k> 2 and Θ(log n) for k = 2. We obtain these results with a new generic scheme for allpairs approximate shortest paths. Using this scheme, we also design faster algorithms for allpairs tstretch distances for t = 2, 7/3, 3. 1.
Local Computation of Nearly Additive Spanners
"... Abstract. An (α, β)spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a spar ..."
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Cited by 6 (3 self)
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Abstract. An (α, β)spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β)spanner H for a given graph G and distortion parameters α and β. It first presents a generic distributed algorithm that in constant number of rounds constructs, for every nnode graph and integer k ≥ 1, an (α, β)spanner of O(βn 1+1/k) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k − 1, 0)spanner of at most kn 1+1/k edges in k rounds, matching the performances of the best known distributed algorithm by Derbel et al. (PODC ’08). For k = 2 and constant ε> 0, it can also produce a (1+ε,2−ε)spanner of O(n 3/2) edges in constant time. More interestingly, for every integer k> 1, it can construct in constant time a (1 + ε, O(1/ε) k−2)spanner of O(ε −k+1 n 1+1/k) edges. Such deterministic
Distance oracles for unweighted graphs: breaking the quadratic barrier with constant additive error
, 2008
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FASTER ALGORITHMS FOR ALLPAIRS APPROXIMATE SHORTEST PATHS IN UNDIRECTED GRAPHS ∗
"... Abstract. 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 ..."
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Cited by 4 (1 self)
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Abstract. 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
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 3 (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 allpairs 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 3/2approximation 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 3/2approximation of both the diameter and the radius can be obtained more efficiently. 1
Faster Algorithms for AllPairs Small Stretch Distances in Weighted Graphs
"... Abstract. Let G = (V,E) be a weighted undirected graph, with nonnegative edge weights. We consider the problem of efficiently computing approximate distances between all pairs of vertices in G. While many efficient algorithms are known for this problem in unweighted graphs, not many results are kno ..."
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Abstract. Let G = (V,E) be a weighted undirected graph, with nonnegative edge weights. We consider the problem of efficiently computing approximate distances between all pairs of vertices in G. While many efficient algorithms are known for this problem in unweighted graphs, not many results are known for this problem in weighted graphs. Zwick [14] showed that for any fixed ε> 0, stretch 1 1 + ε distances between all pairs of vertices in a weighted directed graph on n vertices can be computed in Õ(n ω) time, where ω < 2.376 is the exponent of matrix multiplication and n is the number of vertices. It is known that finding distances of stretch less than 2 between all pairs of vertices in G is at least as hard as Boolean matrix multiplication of two n×n matrices. It is also known that allpairs stretch 3 distances can be computed in Õ(n 2) time and allpairs stretch 7/3 distances can be computed in Õ(n 7/3) time. Here we consider efficient algorithms for the problem of computing allpairs stretch (2+ε) distances in G, for any 0 < ε < 1. We show that all pairs stretch (2 + ε) distances for any fixed ε> 0 in G can be computed in expected time O(n 9/4 logn). This algorithm uses a fast rectangular matrix multiplication subroutine. We also present a combinatorial algorithm (that is, it does not use fast matrix multiplication) with expected running time O(n 9/4) for computing allpairs stretch 5/2 distances in G. 1
Fast Approximation Algorithms for the Diameter and Radius of Sparse Graphs
"... The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the allpairs shortest paths problem (APSP) and has a running time of Õ(mn) i ..."
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The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the allpairs shortest paths problem (APSP) and has a running time of Õ(mn) in medge, nnode graphs. In a seminal paper, Aingworth, Chekuri, Indyk and Motwani [SODA’96 and SICOMP’99] presented an algorithm that computes in Õ(m √ n+n 2) time an estimate ˆ D for the diameter D, such that ⌊2/3D ⌋ ≤ ˆ D ≤ D. Their paper spawned a long line of research on approximate APSP. For the specific problem of diameter approximation, however, no improvement has been achieved in over 15 years. Our paper presents the first improvement over the diameter approximation algorithm of Aingworth et al., producing an algorithm with the same estimate but with an expected running time of Õ(m √ n). We thus show that for all sparse enough graphs, the diameter can be 3/2approximated in o(n 2) time. Our algorithm is obtained using a surprisingly simple method of neighborhood depth estimation that is strong enough to also approximate, in the same running time, the radius and more generally, all of the eccentricities, i.e. for every node the distance to its furthest node. Wealsoprovidestrongevidencethatourdiameterapproximation result may be hard to improve. We show that if for some constant ε> 0 there is an O(m 2−ε) time (3/2 − ε)approximation algorithm for the diameter of undirected unweighted graphs, then there is an O ∗ ((2 − δ) n) time algorithm for CNFSAT on n variables for constant δ> 0, and the strong exponential time hypothesis of [Impagliazzo, Paturi, Zane JCSS’01] is false.
An Optimized All pair Shortest Paths Algorithm
, 2010
"... In this paper, we present an algorithm to compute all pairs optimized shortest paths in an unweighted and undirected graph with some additive error of at most 2.This algorithm can be extended for weighted graph also but it will not work for directed graph due to absence of commutative property. The ..."
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In this paper, we present an algorithm to compute all pairs optimized shortest paths in an unweighted and undirected graph with some additive error of at most 2.This algorithm can be extended for weighted graph also but it will not work for directed graph due to absence of commutative property. The algorithm runs in n 5/2) times, where n is the number of vertices in the graph. This algorithm is much simpler than the existing algorithms. A study of upper bounds on the size of a maximal independent set of such graphs has been performed.