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37
Approximate distance oracles
, 2004
"... 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 273 (9 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.
Compact Routing with Minimum Stretch
"... 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(n2j3 log413 n) and achieves paths that are most 3 times the length of the shortest path distances for all nodes in a ..."
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Cited by 118 (4 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(n2j3 log413 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 Q(n) local space at some vertex.
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 91 (7 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.
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 86 (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 68 (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.
All Pairs Shortest Paths in Undirected Graphs with Integer Weights
 In IEEE Symposium on Foundations of Computer Science
, 1999
"... We show that the All Pairs Shortest Paths (APSP) problem for undirected graphs with integer edge weights taken from the range f1; 2; : : : ; Mg can be solved using only a logarithmic number of distance products of matrices with elements in the range f1; 2; : : : ; Mg. As a result, we get an algorith ..."
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Cited by 56 (7 self)
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We show that the All Pairs Shortest Paths (APSP) problem for undirected graphs with integer edge weights taken from the range f1; 2; : : : ; Mg can be solved using only a logarithmic number of distance products of matrices with elements in the range f1; 2; : : : ; Mg. As a result, we get an algorithm for the APSP problem in such graphs that runs in ~ O(Mn ! ) time, where n is the number of vertices in the input graph, M is the largest edge weight in the graph, and ! ! 2:376 is the exponent of matrix multiplication. This improves, and also simplifies, an ~ O(M (!+1)=2 n ! ) time algorithm of Galil and Margalit. 1. Introduction The All Pairs Shortest Paths (APSP) problem is one of the most fundamental algorithmic graph problems. The APSP problem for directed or undirected graphs with real weights can be solved using classical methods, in O(mn + n 2 log n) time (Dijkstra [4], Johnson [10], Fredman and Tarjan [7]), or in O(n 3 ((log log n)= log n) 1=2 ) time (Fredman [6], ...
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 53 (10 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.
All Pairs Shortest Paths in weighted directed graphs  exact and almost exact algorithms
, 1998
"... 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 the APSP problem for weighted directed graphs in which the edge weights are integers of small abso ..."
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Cited by 39 (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 the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in ~ O(n 2+ ) time, where satisfies the equation !(1; ; 1) = 1 + 2 and !(1; ; 1) is the exponent of the multiplication of an n \Theta n matrix by an n \Theta n matrix. The currently best available bounds on !(1; ; 1), obtained by Coppersmith and Winograd, and by Huang and Pan, imply that ! 0:575. The running time of our algorithm is therefore O(n 2:575 ). Our algorithm improves on the ~ O(n (3+!)=2 ) time algorithm, where ! = !(1; 1; 1) ! 2:376 is the usual exponent of matrix multiplication, obtained by Alon, Galil and Margalit, whose running time is only known to be O(n 2:688 ). The second
Roundtrip Spanners and Roundtrip Routing in Directed Graphs
"... We introduce the notion of roundtripspanners of weighted directed graphs and describe ecient algorithms for their construction. For every integer k 1 and any > 0, we show that any directed graph on n vertices with edge weights in the range [1; W ] has a (2k + )roundtripspanner with O( ..."
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Cited by 32 (0 self)
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We introduce the notion of roundtripspanners of weighted directed graphs and describe ecient algorithms for their construction. For every integer k 1 and any > 0, we show that any directed graph on n vertices with edge weights in the range [1; W ] has a (2k + )roundtripspanner with O( edges. We then extend these constructions and obtain compact roundtrip routing schemes. For every integer k 1 and every > 0, we describe a roundtrip routing scheme that has stretch 4k + , and uses at each vertex a routing table of size ~ O( log(nW )). We also show that any weighted directed graph with arbitrary positive edge weights has a 3roundtripspanner with O(n ) edges. This result is optimal. Finally, we present a stretch 3 roundtrip routing scheme that uses local routing tables of size ~ O(n ). This routing scheme is essentially optimal. The roundtripspanner constructions and the roundtrip routing schemes for directed graphs that we describe are only slightly worse than the best available spanners and routing schemes for undirected graphs. Our roundtrip routing schemes substantially improve previous results of Cowen and Wagner. Our results are obtained by combining ideas of Cohen, Cowen and Wagner, Thorup and Zwick, with some new ideas.
Faster algorithms for approximate distance oracles and allpairs small stretch paths
 In 47th Annual IEEE Symp. on Foundations of Computer Science (FOCS
, 2006
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