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73
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 67 (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.
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 61 (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.
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 52 (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
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 34 (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
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) ..."
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 30 (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 24 (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 ...
Averagecase Analysis of Algorithms for Matchings and Related Problems
 Journal of the ACM
, 1994
"... We analyze the behavior of augmenting paths in random graphs. Our results show that in almost every graph, any nonmaximum 01 flow admits a short augmenting path. This enables us to prove that augmenting path algorithms, which are fast in the worst case, also perform exceedingly well on the average ..."
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Cited by 23 (0 self)
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We analyze the behavior of augmenting paths in random graphs. Our results show that in almost every graph, any nonmaximum 01 flow admits a short augmenting path. This enables us to prove that augmenting path algorithms, which are fast in the worst case, also perform exceedingly well on the average. In particular, we show that the O(&radic;(V) E) algorithms for bipartite and general matchings run in almost linear time with high probability. It is also shown that the expected running time of the matching algorithms is O(E) on input graphs chosen uniformly at random from the set of all graphs. We establish that the permanent of almost every bipartite graph can be approximated in polynomial time. We extend our results to the analysis of the running time of Dinic's algorithm for finding factors of graphs.
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 23 (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
Algorithms for dense graphs and networks on the random access computer
 ALGORITHMICA
, 1996
"... We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size L = f2 (log n) a maximal matching in an nvertex bipartite graph in time O (n 2 + n2"5/~.) = O (n2"5/log n), how to com ..."
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Cited by 18 (4 self)
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We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size L = f2 (log n) a maximal matching in an nvertex bipartite graph in time O (n 2 + n2"5/~.) = O (n2"5/log n), how to compute the transitive closure of a digraph with n vertices and m edges in time O(n 2 + nm/,k), how to solve the uncapacitated transportation problem with integer costs in the range [0..C] and integer demands in the range [U..U] in time O ((n 3 (log log / log n) 1/2 + n 2 log U) log nC), and how to solve the assignment problem with integer costs in the range [0..C] in time O(n 2"5 log nC/(logn/loglog n)l/4). Assuming a suitably compressed input, we also show how to do depthfirst and breadthfirst search and how to compute strongly connected components and biconnected components in time O(n~. + n2/L), and how to solve the single source shortestpath problem with integer costs in the range [0..C] in time O(n²(log C)/log n). For the transitive closure algorithm we also report on the experiences with an implementation.