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1,248
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 75 (3 self)
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)), 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
The geometry of graphs and some of its algorithmic applications
 COMBINATORICA
, 1995
"... In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that res ..."
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Cited by 524 (19 self)
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that respect the metric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) Keep down the dimension of the host space and (ii) Guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the distances between
Loopy belief propagation for approximate inference: An empirical study. In:
 Proceedings of Uncertainty in AI,
, 1999
"... Abstract Recently, researchers have demonstrated that "loopy belief propagation" the use of Pearl's polytree algorithm in a Bayesian network with loops can perform well in the context of errorcorrecting codes. The most dramatic instance of this is the near Shannonlimit performanc ..."
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Cited by 676 (15 self)
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the real QMR network to converge if the priors were sampled randomly in the range [0, Small priors are not the only thing that causes oscil lation. Small weights can, too. The effect of both The exact marginals are represented by the circles; the ends of the "error bars" represent the loopy
The Average Distance in a Random Graph with Given Expected Degrees
"... Random graph theory is used to examine the “smallworld phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n / log ˜ d whe ..."
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Cited by 289 (13 self)
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Random graph theory is used to examine the “smallworld phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n / log ˜ d
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
Bipartite subgraphs of integer weighted graphs
 Discrete Math
, 1998
"... For every integer p> 0 let f(p) be the minimum possible value of the maximum weight of a cut in an integer weighted graph with total weight p. It is shown that for every large n and every m < n, f ( � � n n 2 + m) = ⌊ 2 n 4 ⌋ + min( ⌈ 2 ⌉, f(m)). This supplies the precise value of f(p) for ..."
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Cited by 8 (4 self)
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For every integer p> 0 let f(p) be the minimum possible value of the maximum weight of a cut in an integer weighted graph with total weight p. It is shown that for every large n and every m < n, f ( � � n n 2 + m) = ⌊ 2 n 4 ⌋ + min( ⌈ 2 ⌉, f(m)). This supplies the precise value of f
Turán problems for integerweighted graphs
 J. GRAPH THEORY
, 2002
"... A multigraph is (k, r)dense if every kset spans at most r edges. What is the maximum number of edges exN(n, k, r) in a (k, r)dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r> k3) by determining a constant m = m(k, r ..."
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Cited by 6 (1 self)
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A multigraph is (k, r)dense if every kset spans at most r edges. What is the maximum number of edges exN(n, k, r) in a (k, r)dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r> k3) by determining a constant m = m
Extremal Problems On IntegerWeighted Graphs
, 1999
"... . We consider complete graphs with integer edge weights, including the possibility of negative weights. Let ex(Kn ; Km ; r) denote the maximum weight of an integerweighted Kn such that no Km subgraph has weight at least r. In 1997, Bondy and Tuza [1] extended Tur'an's theorem for 0=1weig ..."
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. We consider complete graphs with integer edge weights, including the possibility of negative weights. Let ex(Kn ; Km ; r) denote the maximum weight of an integerweighted Kn such that no Km subgraph has weight at least r. In 1997, Bondy and Tuza [1] extended Tur'an's theorem for 0
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
Polynomial Time Approximation Schemes for Dense Instances of NPHard Problems
, 1995
"... We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiabi ..."
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Cited by 189 (35 self)
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: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints
Results 1  10
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1,248