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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 206 (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
Finding and counting given length cycles
 Algorithmica
, 1997
"... We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs. Most of the bounds obtained depend solely on the number of edges in the graph in question, and not on the number of vertices. The bounds obtained improve upon various previ ..."
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Cited by 82 (11 self)
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We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs. Most of the bounds obtained depend solely on the number of edges in the graph in question, and not on the number of vertices. The bounds obtained improve upon various previously known results. 1
PseudoRandom Graphs
 IN: MORE SETS, GRAPHS AND NUMBERS, BOLYAI SOCIETY MATHEMATICAL STUDIES 15
"... ..."
A new series of dense graphs of high girth
, 1995
"... Abstract. Let k ≥ 1 be an odd integer, t = ⌊ k+2 ⌋ , and q be a prime 4 power. We construct a bipartite, qregular, edgetransitive graph CD(k, q) of order v ≤ 2qk−t+1 and girth g ≥ k + 5. If e is the the number of edges 1+ 1 of CD(k, q) , then e = Ω(v k−t+1). These graphs provide the best known as ..."
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Cited by 40 (8 self)
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Abstract. Let k ≥ 1 be an odd integer, t = ⌊ k+2 ⌋ , and q be a prime 4 power. We construct a bipartite, qregular, edgetransitive graph CD(k, q) of order v ≤ 2qk−t+1 and girth g ≥ k + 5. If e is the the number of edges 1+ 1 of CD(k, q) , then e = Ω(v k−t+1). These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g, g ≥ 5, g ̸ = 11, 12. For g ≥ 24, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for 5 ≤ g ≤ 23, g ̸ = 11, 12, it improves on or ties existing bounds. 1.
A simple linear time algorithm for computing a (2k − 1)spanner of O(n 1+1/k ) size in weighted graphs
 In Proceedings of the 30th International Colloquium on Automata, Languages and Programming
, 2003
"... ) edges are required in the worst case for any (2k \Gamma 1)spanner, which has been proved for k = 1; 2; 3; 5. There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn 1=k) expected running ti ..."
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Cited by 34 (5 self)
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) edges are required in the worst case for any (2k \Gamma 1)spanner, which has been proved for k = 1; 2; 3; 5. There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn 1=k) expected running time. In this paper, we present an extremely simple linear time randomized algorithm that computes a (2k \Gamma 1)spanner of size matching the conjectured lower bound. An important feature of our algorithm is its local approach. Unlike all the previous algorithms which require computation of shortest paths, the new algorithm merely explores the edges in the neighborhood of a vertex or a group of vertices. This feature leads to designing simple externalmemory and parallel algorithms for computing sparse spanners, whose running times are optimal up to logarithmic factors.
Compactness results in extremal graph theory
 Combinatorica
, 1982
"... Dedicated to Tibor Gallai on his seventieth birthday ..."
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Cited by 33 (1 self)
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Dedicated to Tibor Gallai on his seventieth birthday
An Extremal Problem for Random Graphs and the Number of Graphs With Large EvenGirth
, 1995
"... We study the maximal number of edges a C2k free subgraph of a random graph Gn;p may have, obtaining best possible results for a range of p = p(n). Our estimates strengthen previous bounds of Furedi [12] and Haxell, Kohayakawa, and / Luczak [13]. Two main tools are used here: the first one is an ..."
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Cited by 22 (11 self)
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We study the maximal number of edges a C2k free subgraph of a random graph Gn;p may have, obtaining best possible results for a range of p = p(n). Our estimates strengthen previous bounds of Furedi [12] and Haxell, Kohayakawa, and / Luczak [13]. Two main tools are used here: the first one is an upper bound for the number of graphs with large evengirth, i.e., graphs without short even cycles, with a given number of vertices and edges, and satisfying a certain additional pseudorandom condition; the second tool is the powerful result of Ajtai, Koml'os, Pintz, Spencer, and Szemer'edi [1] on uncrowded hypergraphs as given by Duke, Lefmann, and Rodl [7].
Finding even cycles even faster
 In Proceedings of the 21st International Colloquium on Automata, Languages and Programming
, 1994
"... Abstract. We describe efficient algorithms for finding even cycles in undirected graphs. Our main results are the following: (i) For every k ≥ 2, there is an O(V 2) time algorithm that decides whether an undirected graph G =(V,E) contains a simple cycle of length 2k, and finds one if it does. (ii) T ..."
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Cited by 19 (4 self)
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Abstract. We describe efficient algorithms for finding even cycles in undirected graphs. Our main results are the following: (i) For every k ≥ 2, there is an O(V 2) time algorithm that decides whether an undirected graph G =(V,E) contains a simple cycle of length 2k, and finds one if it does. (ii) There is an O(V 2) time algorithm that finds a shortest even cycle in an undirected graph G =(V,E).
On Graphs which Contain All Small Trees
, 1978
"... We investigate those graphs G, with the property that any tree on N vertices occurs as subgraph of G,. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by $z log II and nl+l/log log %, respecti ..."
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Cited by 17 (4 self)
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We investigate those graphs G, with the property that any tree on N vertices occurs as subgraph of G,. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by $z log II and nl+l/log log %, respectively.