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44
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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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
Approximate distance oracles for unweighted graphs . . .
"... ������������ � Let be an undirected graph � on vertices, and ���������� � let denote the distance � in between two � vertices � and. Thorup and Zwick showed that for any +ve � integer, the � graph can be preprocessed to build a datastructure that can efficiently � reportapproximate distance betwee ..."
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Cited by 57 (9 self)
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������������ � Let be an undirected graph � on vertices, and ���������� � let denote the distance � in between two � vertices � and. Thorup and Zwick showed that for any +ve � integer, the � graph can be preprocessed to build a datastructure that can efficiently � reportapproximate distance between any pair of vertices. That is, for �������� � any, the distance � ���������� � reported satisfies The remarkable feature of this datastructure is that, ���� � for, it occupies subquadratic space, i.e., it does not store allpairs distances information explicitly, and still it can answer � anyapproximate distance query in constant time. They named the datastructure “oracle ” because of this feature. Furthermore the tradeoff between � stretch and the size of the datastructure is essentially optimal. In this paper we show that we can actually construct approximate distance oracles in ��������� expected time if the graph is unweighted. One of the new ideas used in the improved algorithm also leads to the first linear time algorithm for computing an optimal �������� � sizespanner of an unweighted graph.
On dynamic shortest paths problems
 In ESA: Annual European Symposium on Algorithms
, 2004
"... Abstract. We obtain the following results related to dynamic versions of the shortestpaths problem: (i) Reductions that show that the incremental and decremental singlesource shortestpaths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static ..."
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Cited by 39 (2 self)
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Abstract. We obtain the following results related to dynamic versions of the shortestpaths problem: (i) Reductions that show that the incremental and decremental singlesource shortestpaths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static allpairs shortestpaths problem. We also obtain slightly weaker results for the corresponding unweighted problems. (ii) A randomized fullydynamic algorithm for the allpairs shortestpaths problem in directed unweighted graphs with an amortized update time of Õ(m n) and a worst case query time is O(n3/4). (iii) A deterministic O(n2 log n) time algorithm for constructing a (log n)spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining singlesource shortestpaths tree up to a given distance. 1
Faster algorithms for approximate distance oracles and allpairs small stretch paths
 In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2006
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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 19 (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:
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 18 (5 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
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 12 (5 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.
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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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.