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16
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
All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication
 Journal of the ACM
, 2000
"... We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves... ..."
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Cited by 60 (6 self)
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We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves...
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.
Undirected Single Source Shortest Paths in Linear Time
 J. Assoc. Comput. Mach
, 1997
"... The single source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a weighted graph G with a source vertex s, find the shortest path from s to all other vertices in the graph. Since 1959 all theoretical developments in SSSP have been based on Dijkstra' ..."
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Cited by 49 (3 self)
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The single source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a weighted graph G with a source vertex s, find the shortest path from s to all other vertices in the graph. Since 1959 all theoretical developments in SSSP have been based on Dijkstra's algorithm, visiting the vertices in order of increasing distance from s. Thus, any implementation of Dijkstra 's algorithm sorts the vertices according to their distances from s. However, we do not know how to sort in linear time. Here, a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with integer weights. The algorithm avoids the sorting bottleneck by building a hierechical bucketing structure, identifying vertex pairs that may be visited in any order. 1 Introduction Let G = (V; E), jV j = n, jEj = m, be an undirected connected graph with an integer edge weight function ` : E ! N and a distinguished source vertex...
A faster algorithm for minimum cycle basis of graphs
 IN 31ST INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING
, 2004
"... 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(m ω n) algorithm, where ω is the best exponent of matrix multiplication. It is prese ..."
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Cited by 21 (10 self)
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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(m ω n) algorithm, where ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. We obtain an O(m 2 n + mn 2 log n) algorithm for this problem. Our algorithm also uses fast matrix multiplication. When the edge weights are integers, we have an O(m 2 n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ω) time. For any ɛ> 0, we also design a 1 + ɛ approximation algorithm to compute a cycle basis which is at most 1 + ɛ times the weight of a minimum cycle basis. The running time of this algorithm is O ( mω ɛ log(W/ɛ)) for reasonably dense graphs, where W is the largest edge weight.
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
I/Oefficient undirected shortest paths
 In Proc. 11th Annual European Symposium on Algorithms, volume 2832 of LNCS
, 2003
"... Abstract. We show how to compute singlesource shortest paths in undirected graphs with nonnegative edge lengths in O ( p nm/B log n + MST (n, m)) I/Os, where n is the number of vertices, m is the number of edges, B is the disk block size, and MST (n, m) is the I/Ocost of computing a minimum spann ..."
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Cited by 11 (4 self)
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Abstract. We show how to compute singlesource shortest paths in undirected graphs with nonnegative edge lengths in O ( p nm/B log n + MST (n, m)) I/Os, where n is the number of vertices, m is the number of edges, B is the disk block size, and MST (n, m) is the I/Ocost of computing a minimum spanning tree. For sparse graphs, the new algorithm performs O((n / √ B) log n) I/Os. This result removes our previous algorithm’s dependence on the edge lengths in the graph. 1
AverageCase Complexity of ShortestPaths Problems
"... We study both upper and lower bounds on the averagecase complexity of shortestpaths algorithms. It is proved that the allpairs shortestpaths problem on nvertex networks can be solved in time O(n 2 log n) with high probability with respect to various probability distributions on the set of inpu ..."
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Cited by 2 (0 self)
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We study both upper and lower bounds on the averagecase complexity of shortestpaths algorithms. It is proved that the allpairs shortestpaths problem on nvertex networks can be solved in time O(n 2 log n) with high probability with respect to various probability distributions on the set of inputs. Our results include the first theoretical analysis of the average behavior of shortestpaths algorithms with respect to the vertexpotential model, a family of probability distributions on complete networks with arbitrary real arc costs but without negative cycles. We also generalize earlier work with respect to the common uniform model, and we correct the analysis of an algorithm with respect to the endpointindependent model. For the algorithm that solves the allpairs shortestpaths problem on networks generated according to the vertexpotential model, a key ingredient is an algorithm that solves the singlesource shortestpaths problem on such networks in time O(n 2 ) with high probability. All algorithms mentioned exploit that with high probability, the singlesource shortestpaths problem can be solved correctly by considering only a rather sparse subset of the arc set. We prove a lower bound indicating the limitations of this approach. In a fairly general probabilistic model, any algorithm solving the singlesource shortestpaths problem has to inspect# n log n) arcs with high probability. Kurzzusammenfassung. In dieser Arbeit werden sowohl obere als auch untere Schranken f ur die averagecaseKomplexit at von K urzesteWegeAlgorithmen untersucht. Wir beweisen f ur verschiedene Wahrscheinlichkeitsverteilungen auf Netzwerken mit n Knoten, dass das allpairs shortestpaths problem mit hoher Wahrscheinlichkeit in Zeit O(n 2 log n) gel ost werden kann. Insbeso...
Cacheoblivious planar shortest paths
 In Proc. 32nd International Colloquium on Automata, Languages, and Programming. LNCS
, 2005
"... Abstract. We present an efficient cacheoblivious implementation of the shortestpath algorithm for planar graphs by Klein et al., and prove that it incurs no more than O ` N B1/2−ɛ + N B log N ´ block transfers on a graph with N vertices. This is the first cacheoblivious algorithm for this problem ..."
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Cited by 2 (1 self)
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Abstract. We present an efficient cacheoblivious implementation of the shortestpath algorithm for planar graphs by Klein et al., and prove that it incurs no more than O ` N B1/2−ɛ + N B log N ´ block transfers on a graph with N vertices. This is the first cacheoblivious algorithm for this problem that incurs o(N) block transfers. 1