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87
Fast Estimation of Diameter and Shortest Paths (without Matrix Multiplication)
, 1996
"... this paper is organized as follows. We begin by presenting some definitions and useful observations in Section 2. In Section 3, we describe the algorithms for distinguishing between graphs of diameter 2 and 4, and the extension to obtaining a ratio 2=3 approximation to the diameter. Then, in Section ..."
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Cited by 79 (2 self)
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this paper is organized as follows. We begin by presenting some definitions and useful observations in Section 2. In Section 3, we describe the algorithms for distinguishing between graphs of diameter 2 and 4, and the extension to obtaining a ratio 2=3 approximation to the diameter. Then, in Section 4, we apply the ideas developed in estimating the diameter to obtain the promised algorithm for an additive approximation for APSP. Finally, in Section 5 we present an empirical study of the performance of our algorithm for allpairs shortest paths.
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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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
Finding the Hidden Path: Time Bounds for AllPairs Shortest Paths
, 1993
"... We investigate the allpairs shortest paths problem in weighted graphs. We present an algorithmthe Hidden Paths Algorithmthat finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra&ap ..."
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Cited by 75 (0 self)
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We investigate the allpairs shortest paths problem in weighted graphs. We present an algorithmthe Hidden Paths Algorithmthat finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra's algorithm. We argue that m* is likely to be small in practice, since m* = O(n log n) with high probability for many probability distributions on edge weights. We also prove an Ω(mn) lower bound on the running time of any pathcomparison based algorithm for the allpairs shortest paths problem. Pathcomparison based algorithms form a natural class containing the Hidden Paths Algorithm, as well as the algorithms of Dijkstra and Floyd. Lastly, we consider generalized forms of the shortest paths problem, and show that many of the standard shortest paths algorithms are effective in this more general setting.
Allpairs shortest paths with real weights in O(n³ / log n) time
 PROC. OF THE 9TH WADS, LECTURE NOTES IN COMPUTER SCIENCE 3608
, 2005
"... We describe an O(n³ / log n) ..."
Faster subtree isomorphism
 Journal of Algorithms
, 1999
"... We study the subtree isomorphism problem: Given trees H and G, find a subtree of G which is isomorphic to H or decide that there is no such subtree. We give an O((k 1.5 / log k)n)time algorithm for this problem, where k and n are the number of vertices in H and G, respectively. This improves over t ..."
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Cited by 41 (2 self)
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We study the subtree isomorphism problem: Given trees H and G, find a subtree of G which is isomorphic to H or decide that there is no such subtree. We give an O((k 1.5 / log k)n)time algorithm for this problem, where k and n are the number of vertices in H and G, respectively. This improves over the O(k 1.5 n) algorithms of Chung and Matula. We also give a randomized (Las Vegas) O(k 1.376 n)time algorithm for the decision problem. 1
Sampling community structure
 In WWW
, 2010
"... We propose a novel method, based on concepts from expander graphs, to sample communities in networks. We show that our sampling method, unlike previous techniques, produces subgraphs representative of community structure in the original network. These generated subgraphs may be viewed as stratified ..."
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Cited by 34 (2 self)
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We propose a novel method, based on concepts from expander graphs, to sample communities in networks. We show that our sampling method, unlike previous techniques, produces subgraphs representative of community structure in the original network. These generated subgraphs may be viewed as stratified samples in that they consist of members from most or all communities in the network. Using samples produced by our method, we show that the problem of community detection may be recast into a case of statistical relational learning. We empirically evaluate our approach against several realworld datasets and demonstrate that our sampling method can effectively be used to infer and approximate community affiliation in the larger network.
Hyperclique pattern discovery
 DATA MINING AND KNOWLEDGE DISCOVERY JOURNAL
, 2006
"... Existing algorithms for mining association patterns often rely on the supportbased pruning strategy to prune a combinatorial search space. However, this strategy is not effective for discovering potentially interesting patterns at low levels of support. Also, it tends to generate too many spuriou ..."
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Cited by 33 (12 self)
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Existing algorithms for mining association patterns often rely on the supportbased pruning strategy to prune a combinatorial search space. However, this strategy is not effective for discovering potentially interesting patterns at low levels of support. Also, it tends to generate too many spurious patterns involving items which are from different support levels and are poorly correlated. In this paper, we present a framework for mining highlycorrelated association patterns called hyperclique patterns. In this framework, an objective measure called hconfidence is applied to discover hyperclique patterns. We prove that the objects in a hyperclique pattern have a guaranteed level of global pairwise similarity to one another as measured by the cosine similarity (uncentered Pearson’s correlation coefficient). Also, we show that the hconfidence measure satisfies a crosssupport property which can help efficiently eliminate spurious patterns involving items with substantially different support levels. Indeed, this crosssupport property is not limited to hconfidence and can be generalized to some other association measures. In addition, an algorithm called hyperclique miner is proposed to exploit both crosssupport and antimonotone properties of the hconfidence measure for the efficient discovery of hyperclique patterns. Finally, our experimental results show that hyperclique miner can efficiently identify hyperclique patterns, even at extremely low levels of support.
Augment or Push? A computational study of Bipartite Matching and Unit Capacity Flow Algorithms
 ACM J. EXP. ALGORITHMICS
, 1998
"... We conduct a computational study of unit capacity flow and bipartite matching algorithms. Our goal is to determine which variant of the pushrelabel method is most efficient in practice and to compare pushrelabel algorithms with augmenting path algorithms. We have implemented and compared three pus ..."
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Cited by 32 (1 self)
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We conduct a computational study of unit capacity flow and bipartite matching algorithms. Our goal is to determine which variant of the pushrelabel method is most efficient in practice and to compare pushrelabel algorithms with augmenting path algorithms. We have implemented and compared three pushrelabel algorithms, three augmenting path algorithms (one of which is new), and one augmentrelabel algorithm. The depthfirst search augmenting path algorithm was thought to be a good choice for the bipartite matching problem, but our study shows that it is not robust. For the problems we study, our implementations of the fifo and lowestlevel selection pushrelabel algorithms have the most robust asymptotic rate of growth and work best overall. Augmenting path algorithms, although not as robust, on some problem classes are faster by a moderate constant factor. Our study includes several new problem families and input graphs with as many as 5 \Theta 10 5 vertices.
On the Information Rate of Secret Sharing Schemes
 Theoretical Computer Science
, 1992
"... We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1=2 + ff ..."
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Cited by 30 (5 self)
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We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1=2 + ffl, where ffl is an arbitrary positive constant. We also consider the problem of testing if one of these access structures is a substructure of an arbitrary access structure and we show that this problem is NPcomplete. We provide several general lower bounds on information rate and average information rate of graphs. In particular, we show that any graph with n vertices admits a secret sharing scheme with information rate\Omega\Gammate/3 n)=n). 1 Introduction A secret sharing scheme is a method to distribute a secret s among a set of participants P in such a way that only qualified subsets of P can reconstruct the value of s whereas any other subset of P ; nonqualified to know s; cannot ...
A decomposition theorem for maximum weight bipartite matchings with applications to evolutionary trees
 in Proceedings of the 7th Annual European Symposium on Algorithms, Lecture Notes in Comput. Sci
, 1999
"... Abstract. Let G be a bipartite graph with positive integer weights on the edges and without isolated nodes. Let n, N, and W be the node count, the largest edge weight, and the total weight of G. Let k(x, y) be log x / log(x2 /y). We present a new decomposition theorem for maximum weight bipartite ma ..."
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Cited by 28 (2 self)
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Abstract. Let G be a bipartite graph with positive integer weights on the edges and without isolated nodes. Let n, N, and W be the node count, the largest edge weight, and the total weight of G. Let k(x, y) be log x / log(x2 /y). We present a new decomposition theorem for maximum weight bipartite matchings and use it to design an O ( √ nW/k(n, W/N))time algorithm for computing a maximum weight matching of G. This algorithm bridges a longstanding gap between the best known time complexity of computing a maximum weight matching and that of computing a maximum cardinality matching. Given G and a maximum weight matching of G, we can further compute the weight of a maximum weight matching of G −{u} for all nodes u in O(W) time. Key words. allcavity matchings, maximum weight matchings, minimum weight covers, graph algorithms, unfolded graphs