L. Hagen and A. Kahng, "New Spectral Methods for Ratio Cut Partitioning and Clustering," IEEE Trans. On CAD 11(9), pp. 1074-1085, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Ratio cut has many practical applications, most important being VLSI design, clustering and partitioning [20, 12, 1 ] Since ratio cut is a NP hard problem [13] we must seek for approximation algorithms to solve it in practically reasonable time. Many purely heuristic algorithms were developed [20, 22, 18, 6] most of them relying on simulated annealing, spectral methods or iterative movement of nodes from one side of the partition to the other. A common idea exploited by several authors [22, 2, 7, 18, 19] to improve their quality is using multi scale graph representation usually obtained by edge ....

L. Hagen and A. B. Kahng, New spectral methods for ratio cut partitioning and clustering, IEEE Trans. Computer-Aided Design, 11 (1992), pp. 1074-1085.

....a constant factor approximation algorithm for Cluster Editing when p = 2. 1 Introduction Problem Definition and Motivation: Clustering is a central optimization problem with applications in numerous fields including computational biology (cf. 15] image processing (cf. 16] VLSI design (cf. [7]) and many more. The input to the problem is typically a set of elements and pairwise similarity values between elements. The goal is to partition the elements into subsets, which are called clusters, so that two meta criteria are satisfied: Homogeneity elements inside a cluster are highly ....

C. Hagen and A.B. Kahng. New spectral methods for ratio cut partitioning and clustering. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 11(9):1074--1085, 1992.

....Ratio cut has many practical applications, most important being VLSI design, clustering and partitioning [23, 14, 1] Since ratio cut is a NP hard problem [15] we must seek for approximation algorithms to solve it in practically reasonable time. Many purely heuristic algorithms were developed [23, 25, 21, 8] most of them relying on simulated annealing, spectral methods or iterative movement of nodes from one side of the partition to the other. A common idea exploited by several authors [25, 2, 9, 21, 22] to improve their quality is using multi scale graph representation usually obtained by edge ....

L. Hagen and A. B. Kahng, New spectral methods for ratio cut partitioning and clustering, IEEE Trans. Computer-Aided Design, 11 (1992), pp. 1074 1085.

....an extension of spectral graph partitioning method in the case of nearly disconnected components. The second lowest eigenvalue is called the Fiedler value and the eigenvector is called the Fiedler vector. Our perturbative solution to Fielder value, Eq. 9) is identical to the ratio cut criteria [6, 16], where they use the objective function J = s=jAjjBj which can be written as J = s=jAj s=jBj) jV j and therefore is the same as Jrcut , since jV j = jAj jBj is xed] This new results further strengths the connection between ratio cut and Fiedler vector of (D W )y = y. We could equally ....

....are found to be quite informative: each component represents a clear topic area. We also discuss the relation to spectral graph partition and provided a new perspective on the objective function of the graph partitioning. In particular, from the perturbative analysis, we derived the ratio cut [6, 16], normalized cut [27] and min max cut [9] objective functions from the (normalized) Laplacian matrix of a graph. The perturbation analysis framework, starting from disconnected subgraphs as diagonal blocks in the Laplacian matrix and adding sparse edge connections (bridges) between subgraphs as ....

L. Hagen and A.B. Kahng. New spectral methods for ratio cut partitioning and clustering. IEEE. Trans. on Computed Aided Desgin, 11:1074-1085, 1992.

....for partitioning. In the simplest MINcut algorithm, a connected graph is partitioned into two subgraphs with the cutsize (cut set) minimized. However, MINcut often results in a skewed cut, i.e. a very small subgraph is cut away [4] Various constraints are introduced, such as the ratio cut [4, 14], the normalized cut [22] etc. to circumvent the problem. However, skewed cuts still occur when the overlaps between clusters are large. In this paper, we propose a new graph partition method based on the min max clustering principle: the similarity or association between two subgraphs (cut set) ....

....time. All these requirements can be simultaneously satis ed by the following objective function, Mcut = cut(A; B) W (A) cut(A; B) W (B) 3) We call this new objective function the min max cut function or Mcut for short. Mcut is inspired by previous works on spectral graph partition [21, 14, 22] (see section 3) It turns out that the continuous relaxation of Eq. 3) must be solved in a way that is di erent from existing graph partition relaxations [21, 14, 22] To reveal the solution, we reorder the rows and columns of W conformally with subgraphs A and B such that W = WA W A;B WB;A ....

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L. Hagen and A.B. Kahng. New spectral methods for ratio cut partitioning and clustering. IEEE. Trans. on Computed Aided Desgin, 11:1074-1085, 1992.

....weight(V 1 ) cut(V 1 ; V 2 ) 2 : 3.3 Ratio cut and Normalized cut objectives Thus far we have not speci ed the particular choice of vertex weights in (3.7) A simple choice is to have weight(i) 1 for all vertices i. This leads to the ratio cut objective which has been considered in [5, 15] (for circuit partitioning) Ratio cut(V 1 ; V 2 ) An interesting choice is to make the weight of each vertex equal to the sum of the weights of edges that are incident on it, i.e. E ik : This leads to the normalized cut criterion that was used in [34] for image segmentation. Note ....

L. Hagen and A. B. Kahng. New spectral methods for ratio cut partitioning and clustering. IEEE Transactions on Computer-Aided Design, 11:1074-1085, 1992.

....of G on the diagonal, and entry b ij = 1 if G has the edge (v i , v j ) and 0 otherwise. The eigenvector u 2 corresponding to # 2 (the second smallest eigenvalue of B) is computed, and the vertices of the graph are partitioned according to the values of their corresponding entries in u 2 [23, 18]. The goal is to compute a small separator; that is, as few edges or vertices as possible should be deleted from the graph to achieve the partition. Additionally, the sizes of the resulting components should be roughly comparable. Although spectral methods are popular, there has been little ....

....#(1) bisectors; the spectral bisection algorithm produces a #(n) bisection, which is as far from the optimum as possible (to within a constant) The spectral bisection algorithm can be modified to generate a better separator for the bisection counterexample. Some modifications are presented in [18]; they still use a partition based on u 2 . We consider a simple spectral separator algorithm, the best threshold cut algorithm, based on the most general of these suggested modifications. In such an algorithm, best is measured in terms of the cut quotient, the ratio between the number of ....

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L. Hagen and A. B. Kahng, New spectral methods for ratio cut partitioning and clustering, IEEE Transactions on Computer-Aided Design, 11 (1992), pp. 1074--1085.

....exactly reduces to vector partitioning. Motivated by this theoretical result, we use low dimensional vector partitioning instances to construct linear orderings of the graph vertices. These linear orderings yield high quality multi way partitionings that significantly outperform the EIG1 [11], KP [7] and SFC [2] algorithms, and also produce balanced 2 way partitionings. Our experimental results suggest that solving vector partitioning is an effective approach to graph partitioning; we believe that this approach potentially opens the door to a new class of effective heuristics. 1 ....

....we do this to avoid 1 2 terms throughout this work. Min cut graph partitioning is known to be NP complete, so heuristic methods must be invoked. Previous approaches have included seeded epitaxial growth, iterative improvement [16] genetic algorithms [6] etc. Spectral methods [1] 2] 4] 7] 8] [11] [13] 15] have been successful in recent years and are of particular interest for our present work. These works share a common trait of using eigenvectors to construct some type of geometric representation of the graph. We note four such representations: ffl Linear ordering or 1 dimensional ....

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L. Hagen and A. B. Kahng, "New Spectral Methods for Ratio Cut Partitioning and Clustering", IEEE Trans. on CAD 11(9), Sept. 1992, pp. 1074-1085.

....connected if and only if 0 is an eigenvalue of the Laplacian of G with multiplicity 1 (see, e.g. 10] The following characterization of # 2 holds (see, e.g. 10] # 2 = min x T 1=0 x T Lx x T x . 3. 1) For any vector x and Laplacian matrix L of the graph G, we have (see, e.g. [19]) the following: x T Lx = # (v i ,v j )#E(G) x i x j ) 2 . 3.2) An edge weighted graph is a graph for which a real, nonzero weight w ij is associated with each edge (v i , v j ) Fiedler extended the notion of the Laplacian to graphs with positive edge weights [11] he referred to ....

L. Hagen and A. B. Kahng, New spectral methods for ratio cut partitioning and clustering, IEEE Trans. Computer-Aided Design, 11 (1992), pp. 1074--1085.

....0(seee.g. 3] A graph G is connected if and only if 0 is a simple eigenvalue of the Laplacian of G (see e.g. 3] The following characterization of # 2 holds (see e.g. 10] # 2 =min x#1 x T Lx x T x . 3.1) For any vector x and Laplacian matrix L of the graph G,wehave(seee.g. [19]) x T Lx = # (v i ,v j )#E(G) x i x j ) 2 (3.2) An edge weighted graph is a graph for which a real, nonzero weight w ij is associated with each edge (v i ,v j ) Fiedler extended the notion of the Laplacian to graphs with positive edge weights [11] he referred to this representation ....

L. Hagen and A. B. Kahng, New spectral methods for ratio cut partitioning and clustering, IEEE Transactions on Computer-Aided Design, 11 (1992), pp. 1074--1085.

.... in computation time can indeed be achieved with little corresponding loss of accuracy in the Rent parameter value, simply by We note that in practice, a single execution of spectral ratio cut partitioning is significantly faster than 10 executions of the Fiduccia Mattheyses algorithm, e.g. [16] cites 83 sec of Sun Sparc 1 CPU time for the eigenvector computation in the top level bipartitioning of Primary1, while 10 FM executions required 204 sec of CPU time. However, the matrix computations used by spectral partitioning algorithms are potentially expensive, and we are therefore ....

L. Hagen and A. B. Kahng. New spectral methods for ratio cut partitioning and clustering. IEEE Trans. CAD, CAD-11(9):1074--1085, September 1992.

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L. Hagen and A. Kahng, "New Spectral Methods for Ratio Cut Partitioning and Clustering," IEEE Trans. On CAD 11(9), pp. 1074-1085, 1992.

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Lars Hagen and Andrew B. Kahng. New spectral methods for ratio cut partitioning and clustering. IEE Transactions on Computer-Aided Design, 11(9):1074--1085, September 1992.

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Hagen, L., & Kahng, A. (1992). New spectral methods for ratio cut partitioning and clustering. IEEE transaction on CAD, 11, 1074--1085.

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L. Hagen, A. B. Kahng, "New Spectral Methods for Ratio Cut Partitioning and Clustering", IEEE Transactions on Computer-Aided Design, Vol. 11, No. 9, pp. 1074-1085, September, 1992.

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L. Hagen and A. B. Kahng. New spectral methods for ratio cut partitioning and clustering. IEEE Transactions on Computer-Aided Design, 11(9):1074--1085, September 1992.

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L. Hagen, A.B. Kahng, New spectral methods for ratio cut partitioning and clustering, IEEE Trans. Comput. Aided Des. 11 (1992) 1074--1085.

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C. Hagen, A. Kahng, New spectral methods for ratio cut partitioning and clustering, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 11(9) (1992) 1074-1085.

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Lars Hagen and Andrew B. Kahng. New spectral methods for ratio cut partitioning and clustering. EEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 11(9):1074--1085, 1992. 18

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Hagen, L., Khang, A., "New Spectral Methods for Ratio Cut Partitioning and Clustering," IEEE Transactions on Computer-Aided Design, Vol. 11, No. 9, September 1992, pp. 1074-1085.

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C. Hagen and A.B. Kahng. New spectral methods for ratio cut partitioning and clustering. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 11(9):1074--1085, 1992. 12

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L. Hagen and A.B. Kahng. New spectral methods for ratio cut partitioning and clustering. IEEE. Trans. on Computed Aided Desgin, 11:1074-1085, 1992.

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L. Hagen and A. Kahng, "New Spectral Methods for Ratio Cut Partitioning and Clustering," IEEE Trans. On CAD 11(9), pp. 1074-1085, 1992.

No context found.

Lars Hagen and Andrew B. Kahng. New spectral methods for ratio cut partitioning and clustering. EEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 11(9):1074--1085, 1992. 18

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L. Hagen, A. B. Kahng, New spectral methods for ratio cut partitioning and clustering, IEEE Trans. Computer-Aided Design 11 (1992) 1074--1085.

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