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37
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 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.
A faster algorithm for Minimum Cycle Basis of graphs
 In Proc. of ICALP, LNCS 3142
, 2004
"... Abstract. In this paper we consider the problem of computing a minimum cycle basis in a graph G with m edges and n vertices. The edges of G have nonnegative weights on them. The previous best result for this problem was an O(mωn) algorithm, where ω is the best exponent of matrix multiplication. It ..."
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Cited by 22 (10 self)
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Abstract. In this paper we consider the problem of computing a minimum cycle basis in a graph G with m edges and n vertices. The edges of G have nonnegative weights on them. The previous best result for this problem was an O(mωn) algorithm, where ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. We obtain an O(m2n + mn2 logn) algorithm for this problem. Our algorithm also uses fast matrix multiplication. When the edge weights are integers, we have an O(m2n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(mω) time. For any > 0, we also design a 1 + approximation algorithm to compute a cycle basis which is at most 1+ times the weight of a minimum cycle basis. The running time of this algorithm is O(m ω log(W/)) for reasonably dense graphs, where W is the largest edge weight.
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 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
Finding Least Common Ancestors in Directed Acyclic Graphs
 PROC. 12TH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA’01
, 2001
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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 with improved query time
 MATHEMATICS AND COMPUTER SCIENCE DEPARTMENT, OPEN UNIVERSITY OF ISRAEL
, 2011
"... Given an undirected graph G with m edges, n vertices, and nonnegative edge weights, and given an integer k ≥ 1, we show that for some universal constant c, a (2k − 1)approximate distance oracle for G of size O(kn 1+1/k) can be constructed in O ( √ km+kn 1+c/ √ k) time and can answer queries in O(k ..."
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Cited by 12 (1 self)
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Given an undirected graph G with m edges, n vertices, and nonnegative edge weights, and given an integer k ≥ 1, we show that for some universal constant c, a (2k − 1)approximate distance oracle for G of size O(kn 1+1/k) can be constructed in O ( √ km+kn 1+c/ √ k) time and can answer queries in O(k) time. We also give an oracle which is faster for smaller k. Our results break the quadratic preprocessing time bound of Baswana and Kavitha for all k ≥ 6 and improve the O(kmn 1/k) time bound of Thorup and Zwick except for very sparse graphs and small k. When m = Ω(n 1+c/ √ k) and k = O(1), our oracle is optimal w.r.t. both stretch, size, preprocessing time, and query time, assuming a widely believed girth conjecture by Erdős.
Distance oracles for unweighted graphs: breaking the quadratic barrier with constant additive error
, 2008
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A Symbolic Approach to the AllPairs ShortestPaths Problem
 In WG 2004, LNCS 3353
, 2004
"... Abstract. Graphs can be represented symbolically by the Ordered Binary Decision Diagram (OBDD) of their characteristic function. To solve problems in such implicitly given graphs, specialized symbolic algorithms are needed which are restricted to the use of functional operations offered by the OBDD ..."
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Cited by 9 (5 self)
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Abstract. Graphs can be represented symbolically by the Ordered Binary Decision Diagram (OBDD) of their characteristic function. To solve problems in such implicitly given graphs, specialized symbolic algorithms are needed which are restricted to the use of functional operations offered by the OBDD data structure. In this paper, a symbolic algorithm for the allpairs shortestpaths (APSP) problem in loopless directed graphs with strictly positive integral edge weights is presented. It requires Θ ( log 2 (NB) ) OBDDoperations to obtain the lengths and edges of all shortest paths in graphs with N nodes and maximum edge weight B. It is proved that runtime and space usage are polylogarithmic w. r. t. N and B on graph sequences with characteristic boundedwidth functions. This convenient property is closed under certain graph composition operations. Moreover, an alternative symbolic approach for general integral edge weights is sketched which does not behave efficiently on general graph sequences with boundedwidth functions. Finally, two variants of theAPSPproblemarebrieflydiscussed. 1