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19
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
An NC Algorithm for Minimum Cuts
 IN PROCEEDINGS OF THE 25TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
"... We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from ..."
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Cited by 46 (3 self)
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We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from the minimum cut problem to the problem of obtaining a (2 + ffl)approximation to the minimum cut. This reduction involves a natural combinatorial SetIsolation Problem that can be solved easily in RNC. The third result is a derandomization of this RNC solution that requires a combination of two widely used tools: pairwise independence and random walks on expanders. We believe that the setisolation approach will prove useful in other derandomization problems. The techniques extend to two related problems: we describe NC algorithms finding minimum kway cuts for any constant k and finding all cuts of value within any constant factor of the minimum. Another application of these techni...
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 New GraphTheoretic Approach to Clustering, with Applications to Computer Vision
, 2004
"... This work applies cluster analysis as a unified approach for a wide range of vision applications, thereby combining the research domain of computer vision and that of machine learning. Cluster analysis is the formal study of algorithms and methods for recovering the inherent structure within a given ..."
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Cited by 44 (4 self)
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This work applies cluster analysis as a unified approach for a wide range of vision applications, thereby combining the research domain of computer vision and that of machine learning. Cluster analysis is the formal study of algorithms and methods for recovering the inherent structure within a given dataset. Many problems of computer vision have precisely this goal, namely to find which visual entities belong to an inherent structure, e.g. in an image or in a database of images. For example, a meaningful structure in the context of image segmentation is a set of pixels which correspond to the same object in a scene. Clustering algorithms can be used to partition the pixels of an image into meaningful parts, which may correspond to different objects. In this work we focus on the problems of image segmentation and image database organization. The visual entities to consider are pixels and images, respectively. Our first contribution in this work is a novel partitional (flat) clustering algorithm. The algorithm uses pairwise representation, where the visual objects (pixels,
The capacitated vehicle routing problem
"... We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Travelin ..."
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Cited by 34 (5 self)
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We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two wellstudied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decompositionbased separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY.
Finding Maximum Flows in Undirected Graphs Seems Easier than Bipartite Matching
 In Proc. of the 30th Annual ACM Symposium on Theory of Computing
, 1997
"... Consider an nvertex, medge, undirected graph with maximum flow value v. We give a method to find augmenting paths in such a graph in amortized sublinear (O(n p v)) time per path. This lets us improve the time bound of the classic augmenting path algorithm to O(m + nv 3=2 ) on simple graphs. T ..."
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Cited by 14 (0 self)
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Consider an nvertex, medge, undirected graph with maximum flow value v. We give a method to find augmenting paths in such a graph in amortized sublinear (O(n p v)) time per path. This lets us improve the time bound of the classic augmenting path algorithm to O(m + nv 3=2 ) on simple graphs. The addition of a blocking flow subroutine gives a simple, deterministic O(nm 2=3 v 1=6 )time algorithm. We also use our technique to improve known randomized algorithms, giving O(m+nv 5=4 )time and O(m+n 11=9 v)time algorithms for capacitated undirected graphs. For simple graphs, in which v n, the last bound is O(n 2:2 ), improving on the best previous bound of O(n 2:5 ), which is also the best known time bound for bipartite matching. 1 Introduction In this paper we consider the problem of finding maximum flows in undirected graphs with small flow values. Traditionally, only a special case of this problem has been considered: unitcapacity graphs with no parallel edges (...
Greedy splitting algorithms for approximating multiway partition problems
 Math. Programming
, 2005
"... Abstract. Given a system (V, T, f, k), where V is a finite set, T ⊆ V, f: 2 V → R is a submodular function and k ≥ 2 is an integer, the general multiway partition problem (MPP) asks to find a kpartition P = {V1, V2,..., Vk} of V that satisfies Vi ∩T � = ∅ for all i and minimizes f(V1)+f(V2)+ · · ..."
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Cited by 6 (0 self)
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Abstract. Given a system (V, T, f, k), where V is a finite set, T ⊆ V, f: 2 V → R is a submodular function and k ≥ 2 is an integer, the general multiway partition problem (MPP) asks to find a kpartition P = {V1, V2,..., Vk} of V that satisfies Vi ∩T � = ∅ for all i and minimizes f(V1)+f(V2)+ · · ·+f(Vk), where P is a kpartition of V if (i) Vi � = ∅, (ii) Vi ∩ Vj = ∅, i � = j, and (iii) V1 ∪ V2 ∪ · · · ∪ Vk = V hold. MPP formulation captures a generalization in submodular systems of many NPhard problems such as kway cut, multiterminal cut, target split and their generalizations in hypergraphs. This paper presents a simple and unified framework for developing and analyzing approximation algorithms for various MPPs. Key words. approximation algorithm – hypergraph partition – kway cut – multiterminal cut – multiway partition problem – submodular function 1.
Fast Randomized Algorithms for Computing Minimum {3,4,5,6}Way Cuts
"... A minimum kway cut of an nvertex, medge, weighted, undirected graph is a partition of the vertices into k sets that minimizes the total weight of edges with endpoints in dierent sets. We give new randomized algorithms to nd minimum 3way and 4way cuts, which lead to time bounds of O(mn k 2 log ..."
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Cited by 4 (0 self)
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A minimum kway cut of an nvertex, medge, weighted, undirected graph is a partition of the vertices into k sets that minimizes the total weight of edges with endpoints in dierent sets. We give new randomized algorithms to nd minimum 3way and 4way cuts, which lead to time bounds of O(mn k 2 log 3 n) time for k 6. This improves on the best previous time bounds by a factor of ~ n 2 ). 1 Introduction A minimum kway cut of an nvertex, medge, weighted, undirected graph is a partition of the vertices into k sets that minimizes the total weight of edges with endpoints in dierent sets. Asking for a minimum kway cut is equivalent to asking for the edge set of minimum total weight whose removal would break the graph into at least k connected components. Our main motivations for studying minimum kway cuts are that they are a natural property of graphs, and that they have received considerable attention in the past. Nagamochi and Ibaraki [11] also point to a number of appl...