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13
A Clustering Algorithm based on Graph Connectivity
 Information Processing Letters
, 1999
"... We have developed a novel algorithm for cluster analysis that is based on graph theoretic techniques. ..."
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Cited by 99 (3 self)
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We have developed a novel algorithm for cluster analysis that is based on graph theoretic techniques.
A new approach to the minimum cut problem
 Journal of the ACM
, 1996
"... Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds th ..."
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Cited by 95 (8 self)
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Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n 2 log 3 n) time, a significant improvement over the previous Õ(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in �� � with n 2 processors; this gives the first proof that the minimum cut problem can be solved in ���. The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of � of the minimum cut’s in expected Õ(n 2 � ) time, or in �� � with n 2 � processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected Õ(n 2(r�1) ) time, or in �� � with n 2(r�1) processors. The “trace ” of the algorithm’s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the
Clique Partitions, Graph Compression and Speedingup Algorithms
 Journal of Computer and System Sciences
, 1991
"... We first consider the problem of partitioning the edges of a graph G into bipartite cliques such that the total order of the cliques is minimized, where the order of a clique is the number of vertices in it. It is shown that the problem is NPcomplete. We then prove the existence of a partition of s ..."
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Cited by 74 (3 self)
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We first consider the problem of partitioning the edges of a graph G into bipartite cliques such that the total order of the cliques is minimized, where the order of a clique is the number of vertices in it. It is shown that the problem is NPcomplete. We then prove the existence of a partition of small total order in a sufficiently dense graph and devise an efficient algorithm to compute such a partition. It turns out that our algorithm exhibits a tradeoff between the total order of the partition and the running time. Next, we define the notion of a compression of a graph G and use the result on graph partitioning to efficiently compute an optimal compression for graphs of a given size. An interesting application of the graph compression result arises from the fact that several graph algorithms can be adapted to work with the compressed representation of the input graph, thereby improving the bound on their running times, particularly on dense graphs. This makes use of the tradeoff ...
An Algorithm for Clustering cDNAs for Gene Expression Analysis
 In RECOMB99: Proceedings of the Third Annual International Conference on Computational Molecular Biology
, 1999
"... We have developed a novel algorithm for cluster analysis that is based on graph theoretic techniques. A similarity graph is defined and clusters in that graph correspond to highly connected subgraphs. A polynomial algorithm to compute them efficiently is presented. Our algorithm produces a clusterin ..."
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Cited by 45 (4 self)
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We have developed a novel algorithm for cluster analysis that is based on graph theoretic techniques. A similarity graph is defined and clusters in that graph correspond to highly connected subgraphs. A polynomial algorithm to compute them efficiently is presented. Our algorithm produces a clustering with some provably good properties. The application that motivated this study was gene expression analysis, where a collection of cDNAs must be clustered based on their oligonucleotide fingerprints. The algorithm has been tested intensively on simulated libraries and was shown to outperform extant methods. It demonstrated robustness to high noise levels. In a blind test on real cDNA fingerprint data the algorithm obtained very good results. Utilizing the results of the algorithm would have saved over 70% of the cDNA sequencing cost on that data set. 1 Introduction Cluster analysis seeks grouping of data elements into subsets, so that elements in the same subset are in some sense more cl...
A Faster Algorithm for Finding the Minimum Cut in a Directed Graph
 JOURNAL OF ALGORITHMS
, 1994
"... We consider the problem of finding the minimum capacity cut in a directed network G with n nodes. This problem has applications to network reliability and survivability and is useful in subroutines for other network optimization problems. One can use a maximum flow problem to find a minimum cut sepa ..."
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Cited by 31 (0 self)
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We consider the problem of finding the minimum capacity cut in a directed network G with n nodes. This problem has applications to network reliability and survivability and is useful in subroutines for other network optimization problems. One can use a maximum flow problem to find a minimum cut separating a designated source node s from a designated sink node t, and by varying the sink node one can find a minimum cut in G as a sequence of at most 2n 2 maximum flow problems. We then show how to reduce the running time of these 2n 2 maximum flow algorithms to the running time for solving a single maximum flow problem. The resulting running time is O(nm log(n 2 /m)) for finding the minimum cut in either a directed or an undirected network. © 1994 Academic Press, Inc. 1.
Improved algorithms for graph fourconnectivity
 J. Comp. System Sci
, 1991
"... We present a new algorithm based on open ear decomposition for testing vertex fourconnectivity and for finding all separating triplets in a triconnected graph. A sequential implementation of our algorithm runs in O(n 2) time and a parallel implementation runs in O(log 2 n) time using O(n 2) process ..."
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Cited by 22 (6 self)
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We present a new algorithm based on open ear decomposition for testing vertex fourconnectivity and for finding all separating triplets in a triconnected graph. A sequential implementation of our algorithm runs in O(n 2) time and a parallel implementation runs in O(log 2 n) time using O(n 2) processors on an ARBITRARY CRCW PRAM, where n is the number of vertices in the graph. This improves previous bounds for the problem for both the sequential and parallel cases. The sequential time bound is the best possible, to within a constant factor, if the input is specified in adjacency matrix form, or if the input graph is dense. 1.
Random Sampling and Greedy Sparsification for Matroid Optimization Problems.
 Mathematical Programming
, 1998
"... Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems ..."
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Cited by 9 (2 self)
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Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems on matroids: finding an optimum matroid basis and packing disjoint matroid bases. Applications of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts. An optimum matroid basis is typically found by a greedy algorithm that grows an independent set into an the optimum basis one element at a time. This continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm. We simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a given fixed basis is optimum, showing that the two problems can be solved in roughly the same ...
Representing and Enumerating Edge Connectivity Cuts in RNC
, 1991
"... An undirected edgeweighted graph can have at most \Gamma n 2 \Delta edge connectivity cuts. A succinct and algorithmically useful representation for this set of cuts was given by [4], and an efficient sequential algorithm for obtaining it was given by [12]. In this paper, we present a fast par ..."
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Cited by 7 (0 self)
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An undirected edgeweighted graph can have at most \Gamma n 2 \Delta edge connectivity cuts. A succinct and algorithmically useful representation for this set of cuts was given by [4], and an efficient sequential algorithm for obtaining it was given by [12]. In this paper, we present a fast parallel algorithm for obtaining this representation; our algorithm is an RNC algorithm in case the weights are given in unary. We also observe that for a unary weighted graph, the problems of counting and enumerating the connectivity cuts are in RNC.
How to find overfull subgraphs in graphs with large maximum degree, II
 Discrete Applied Math
, 2000
"... Let G be a simple graph with 3#(G) > V .TheOverfull Graph Conjecture states that the chromatic index of G is equal to #(G), if G does not contain an induced overfull subgraph H with #(H)=#(G), and otherwise it is equal to #(G) + 1. We present an algorithm that determines these subgraphs in O(n 5/3 ..."
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Cited by 6 (0 self)
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Let G be a simple graph with 3#(G) > V .TheOverfull Graph Conjecture states that the chromatic index of G is equal to #(G), if G does not contain an induced overfull subgraph H with #(H)=#(G), and otherwise it is equal to #(G) + 1. We present an algorithm that determines these subgraphs in O(n 5/3 m) time, in general, and in O(n 3 ) time, if G is regular. Moreover, it is shown that G can have at most three of these subgraphs. If 2#(G) #V ,thenG contains at most one of these subgraphs, and our former algorithm for this situation is improved to run in linear time. 1