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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 210 (8 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
Compact Routing with Minimum Stretch
 Journal of Algorithms
"... We present the first universal compact routing algorithm with maximum stretch bounded by 3 that uses sublinear space at every vertex. The algorithm uses local routing tables of size O(n 2=3 log 4=3 n) and achieves paths that are most 3 times the length of the shortest path distances for all node ..."
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Cited by 112 (5 self)
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We present the first universal compact routing algorithm with maximum stretch bounded by 3 that uses sublinear space at every vertex. The algorithm uses local routing tables of size O(n 2=3 log 4=3 n) and achieves paths that are most 3 times the length of the shortest path distances for all nodes in an arbitrary weighted undirected network. This answers an open question of Gavoille and Gengler who showed that any universal compact routing algorithm with maximum stretch strictly less than 3 must use\Omega\Gamma n) local space at some vertex. 1 Introduction Let G = (V; E) with jV j = n be a labeled undirected network. Assuming that a positive cost, or distance is assigned with each edge, the stretch of path p(u; v) from node u to node v is defined as jp(u;v)j jd(u;v)j , where jd(u; v)j is the length of the shortest u \Gamma v path. The approximate allpairs shortest path problem involves a tradeoff of stretch against time short paths with stretch bounded by a constant are com...
All Pairs Almost Shortest Paths
 SIAM Journal on Computing
, 1996
"... Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive onesided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe g) time ..."
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Cited by 83 (8 self)
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Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive onesided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe g) time algorithm APASP 2 for computing all distances in G with an additive onesided error of at most 2. The algorithm APASP 2 is simple, easy to implement, and faster than the fastest known matrix multiplication algorithm. Furthermore, for every even k ? 2, we describe an g) time algorithm APASP k for computing all distances in G with an additive onesided error of at most k.
On graph problems in a semistreaming model
 In 31st International Colloquium on Automata, Languages and Programming
, 2004
"... Abstract. We formalize a potentially rich new streaming model, the semistreaming model, that we believe is necessary for the fruitful study of efficient algorithms for solving problems on massive graphs whose edge sets cannot be stored in memory. In this model, the input graph, G = (V, E), is prese ..."
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Cited by 60 (12 self)
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Abstract. We formalize a potentially rich new streaming model, the semistreaming model, that we believe is necessary for the fruitful study of efficient algorithms for solving problems on massive graphs whose edge sets cannot be stored in memory. In this model, the input graph, G = (V, E), is presented as a stream of edges (in adversarial order), and the storage space of an algorithm is bounded by O(n · polylog n), where n = V . We are particularly interested in algorithms that use only one pass over the input, but, for problems where this is provably insufficient, we also look at algorithms using constant or, in some cases, logarithmically many passes. In the course of this general study, we give semistreaming constant approximation algorithms for the unweighted and weighted matching problems, along with a further algorithm improvement for the bipartite case. We also exhibit log n / log log n semistreaming approximations to the diameter and the problem of computing the distance between specified vertices in a weighted graph. These are complemented by Ω(log (1−ɛ) n) lower bounds. 1
Exact and Approximate Distances in Graphs  a survey
 In ESA
, 2001
"... We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems. ..."
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Cited by 57 (0 self)
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We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.
Designing Networks with Bounded Pairwise Distance
"... We study the following network design problem: Given a communication network, find a minimum cost subset of missing links such that adding these links to the network makes every pair of points within distance at most d from each other. Theproblemhasbeenstudied earlier[17]undertheassumptionthatallli ..."
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Cited by 49 (0 self)
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We study the following network design problem: Given a communication network, find a minimum cost subset of missing links such that adding these links to the network makes every pair of points within distance at most d from each other. Theproblemhasbeenstudied earlier[17]undertheassumptionthatalllinkcostsas wellaslinklengthsareidentical,andwasshowntobe (logn)hardforeveryd4. Wepresentanovellinearprogrammingbasedapproachtoobtainan O(lognlogd)approximationalgorithmforthecaseofuniform linklengthsandcosts. We alsoextendthe(logn)hardnesstod2f2;3g. On the otherhand,iflinkcostscanvary, weshowthattheproblemis(2log1�n)hardford3. Thisversionofour problemcanbeviewedasaspecialcaseoftheminimum cost dspannerproblemandthusourhardnessresultappliesthereaswell. Ford=2,however,weshowthatthe problemcontinuestobeO(logn)approximablebygivingan O(logn)approximationtothemoregeneralminimumcost2spannerproblem.An(2log1�n)hardness resultalsoholdswhenalllinkcostsareidenticalbutlink lengthsmayvary(appliesevenwhenalllengthsare1or
Approximating the stretch factor of Euclidean paths, cycles and trees
 SIAM J. Comput
, 1999
"... Given a set S of n points in R d , and a graph G having the points of S as its vertices, the stretch factor t of G is dened as the maximal value jpqj G =jpqj, where p; q 2 S, p 6= q, jpqj G is the length of a shortest path in G between p and q, and jpqj is the Euclidean distance between p and ..."
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Cited by 48 (7 self)
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Given a set S of n points in R d , and a graph G having the points of S as its vertices, the stretch factor t of G is dened as the maximal value jpqj G =jpqj, where p; q 2 S, p 6= q, jpqj G is the length of a shortest path in G between p and q, and jpqj is the Euclidean distance between p and q. We consider the problem of designing algorithms that, for an arbitrary constant > 0, compute an approximation to this stretch factor, i.e., a value t such that t t (1 + )t. We give eÆcient solutions for the cases when G is a path, cycle, or tree. The main idea used in all the algorithms is to use wellseparated pair decompositions to speed up the computations. 1 Introduction Let S be a set of n points in R d , where d 1 is a small constant, and let G be an undirected connected graph having the points of S as its vertices. The length of any edge (p; q) of G is dened as the Euclidean distance jpqj between the two vertices p and q. The length of a path in G is dened a...
Nearlinear time construction of sparse neighborhood covers
 SIAM Journal on Computing
, 1998
"... Abstract. This paper introduces a nearlinear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to nearlinear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, ..."
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Cited by 43 (4 self)
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Abstract. This paper introduces a nearlinear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to nearlinear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, approximate shortest paths, and weight and distancepreserving graph spanners. In particular, an O(log n) approximation to the kshortest paths problem on an nvertex, Eedge graph is obtained that runs in Õ (n + E + k) time.
A Faster Distributed Protocol for Constructing a Minimum Spanning Tree
 In Proc. of the 15 th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2004
"... Abstract This paper studies the problem of constructing a minimumweight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art to ..."
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Cited by 37 (2 self)
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Abstract This paper studies the problem of constructing a minimumweight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art today isa protocol with running time O(\Lambda (G) + pn * log * n) due to Kutten and Peleg [22], where \Lambda (G) denotesthe diameter of the graph G. Peleg and Rubinovich [29] have shown that ~\Omega (pn) time is required forconstructing an MST even on graphs of small diameter, and claimed that their result "establishes the asymptotic nearoptimality " of the protocol of [22].In this paper we refine this claim, and devise a protocol that constructs the MST in ~ O(u(G,!)+pn)rounds, where u(G,!) is the MSTradius of the graph. The ratio between the diameter and the MSTradius may be as large as \Theta ( n), and, consequently, on some inputs our protocol is faster than theprotocol of [22] by a factor of ~\Omega (p n). Also, on every input, the running time of our protocol is nevergreater than twice the running time of the protocol of [22].