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19
More algorithms for allpairs shortest paths in weighted graphs
 In Proceedings of 39th Annual ACM Symposium on Theory of Computing
, 2007
"... In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multipl ..."
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Cited by 54 (3 self)
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In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted ” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4)), where ω < 2.376; in two dimensions, this is O(n 2.922). Our framework greatly extends the previously considered case of smallintegerweighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4) = O(n 2.844) time) for APSP in realvertexweighted graphs, as well as an improved result (near O(n (3+ω)/2) = O(n 2.688) time) for the allpairs lightest shortest path problem for smallintegerweighted graphs. 1
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.
Allpairs nearly 2approximate shortest paths in O(n 2 polylog n) time
 In Proceedings of 22nd Annual Symposium on Theoretical Aspect of Computer Science, volume 3404 of LNCS
, 2005
"... Abstract. 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 ..."
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Cited by 12 (5 self)
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Abstract. 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+n 2) time, and for any u, v ∈ V reports distance no greater than 2δ(u, v) + 1. Our second algorithm requires expected O(n 2 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. 1
A shortest path algorithm for realweighted undirected graphs
 in 13th ACMSIAM Symp. on Discrete Algs
, 1985
"... Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) ti ..."
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Cited by 12 (3 self)
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Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) time, where α = α(m, n) is the very slowly growing inverseAckermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the allpairs and singlesource shortest paths problems. We solve the allpairs problem in O(mnlog α(m, n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n (log n)O(1) , we can solve the singlesource problem in O(m + nlog log n) time. Both these results are theoretical improvements over Dijkstra’s algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchybased approach invented by Thorup. Key words. singlesource shortest paths, allpairs shortest paths, undirected graphs, Dijkstra’s
Networks Cannot Compute Their Diameter in Sublinear Time preliminary version please check for updates
, 2011
"... We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an ˜ Ω(n) lower bound for the number of commun ..."
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Cited by 10 (2 self)
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We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an ˜ Ω(n) lower bound for the number of communication rounds needed, where n denotes the number of nodes in the network. This lower bound is valid even if the diameter of the network is a small constant. We also show that a (3/2 − ε)approximation of the diameter requires ˜ Ω ( √ n) rounds. Furthermore we use our new technique to prove an ˜ Ω ( √ n) lower bound on approximating the girth of a graph by a factor 2 − ε. Contact author:
Oracles for distances avoiding a failed node or link
 SIAM J. Comput
"... Abstract. We consider the problem of preprocessing an edgeweighted directed graph G to answer queries that ask for the length and first hop of a shortest path from any given vertex x to any given vertex y avoiding any given vertex or edge. As a natural application, this problem models routing in ne ..."
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Cited by 7 (0 self)
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Abstract. We consider the problem of preprocessing an edgeweighted directed graph G to answer queries that ask for the length and first hop of a shortest path from any given vertex x to any given vertex y avoiding any given vertex or edge. As a natural application, this problem models routing in networks subject to node or link failures. We describe a deterministic oracle with constant query time for this problem that uses O(n2 log n) space, where n is the number of vertices in G. The construction time for our oracle is O(mn2 + n3 log n). However, if one is willing to settle for Θ(n2.5) space, we can improve the preprocessing time to O(mn1.5 + n2.5 log n) while maintaining the constant query time. Our algorithms can find the shortest path avoiding a failed node or link in time proportional to the length of the path.
Approximate distance oracles with improved query time. Preprint, available at http: //arxiv.org/abs/1202.2336
 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 4 (0 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. 1
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
A MaxentStress Model for Graph Layout
"... In some applications of graph visualization, input edges have associated target lengths. Dealing with these lengths is a challenge, especially for large graphs. Stress models are often employed in this situation. However, the traditional full stress model is not scalable due to its reliance on an in ..."
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Cited by 3 (2 self)
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In some applications of graph visualization, input edges have associated target lengths. Dealing with these lengths is a challenge, especially for large graphs. Stress models are often employed in this situation. However, the traditional full stress model is not scalable due to its reliance on an initial allpairs shortest path calculation. A number of fast approximation algorithms have been proposed. While they work well for some graphs, the results are less satisfactory on graphs of intrinsically high dimension, because nodes overlap unnecessarily. We propose a solution, called the maxentstress model, which applies the principle of maximum entropy to cope with the extra degrees of freedom. We describe a forceaugmented stress majorization algorithm that solves the maxentstress model. Numerical results show that the algorithm scales well, and provides acceptable layouts for large, nonrigid graphs. This also has potential applications to scalable algorithms for statistical multidimensional scaling (MDS) with variable distances.
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