P. Chan, D. Schlag, and J. Zien. Spectral K-way ratio-cut partitioning and clustering. IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, 13(9):1088-- 1096, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....nodes of the graph into two disjoint subsets A 1 and A 2 with corresponding index sets I 1 ; I 2 (that is, A 1 = B i and A 2 = B i ) This bisection may be described by a vector x 2 f 1g , with x(i) 1 if node i is in I 1 and x(i) 1 if node i is in I 2 . A standard approach [17, 5, 2] to the minimal cut bisection problem is to consider the form: min x(i)2f 1g : 3.7) The condition that i x(i) 0 forces both I 1 and I 2 to contain the same number of elements; later we will relax this condition. Note that we will get a contribution to (3.7) only if two nodes i and j ....

....expect based on the relaxation of 0 s and 1 s argument . Remark 7.3: The use of multiple eigenvectors x 1 ; x considered as points in R was rst suggested in Hall [17] in the context of placement of points in R so as to minimise a connection cost function. Chan et al. [5] provide a very readable introduction to the approach of nding a q way minimal graph cut using several eigenvectors of L, and introduce a di erent cluster identi cation heuristic. One could consider searching for clusters in V = f( 1 x 1 ; x )g where the k s are ....

Pak K. Chan, Martine D.F. Schlag, and Jason Y. Zien. Spectral k-way ratio-cut partitioning and clustering. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 13(9):1088-1096, 1994.

.... [9, 6] cuts a graph in two by thresholding the second eigenvector of the graph s normalized Laplacian matrix, and numerous clustering algorithms use selected eigenvectors of dot product or kernel matrices to re represent the data for clustering by simpler heuristics such as thresholding or Kmeans [25, 2, 5, 7, 27, 21, 29, 13, 3, 18, 19, 4, 20, 22]. While the statistical basis and optimality of PCA is well understood, virtually all other spectral inethods are motivated by imperfect analogies between data derived graphs and physical problems (e.g. harmonic analysis 2 and random walks3) or as approximations to other problems (e.g. vector ....

....in Ng et al. 20] the normalized row vectors of the matrix formed by the first k weighted eigenvectors are used as the input to a k means clusterer, and a perturbational analysis was used to show that the results should be stable if the data is already nearly clustered . In Chan et al. [5], the directional angle between the row vectors of the first k eigenvectors of the Laplacian matrix was used as a new distance measure for partitioning. Alperr Yao [2] equated partitioning with the problem of clustering these row vectors, and found that the more eigenvectors used, the better. ....

P. Chan, D. Schlag, and J. Zien. Spectral K-way ratio-cut partitioning and clustering. IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, 13(9):1088 1096, 1994.

....two disjoint subsets A n 1 and A n 2 with corresponding index sets I 1 ; I 2 (that is, A n 1 = S i2I1 B i and A n 2 = S i2I2 B i ) This bisection may be described by a vector x 2 f 1g n , with x(i) 1 if node i is in I 1 and x(i) 1 if node i is in I 2 . A standard approach [17, 5, 2] to the minimal cut bisection problem is to consider the form: min x(i)2f 1g P i x(i) 0 n X i;j=1 P ij (x(i) x(j) 2 : 3.7) The condition that P i x(i) 0 forces both I 1 and I 2 to contain the same number of elements; later we will relax this condition. Note that we will get a ....

....expect based on the relaxation of 0 s and 1 s argument 10 . Remark 7.3: The use of multiple eigenvectors x 1 ; x considered as points in R was rst suggested in Hall [17] in the context of placement of points in R so as to minimise a connection cost function. Chan et al. [5] provides a very readable introduction to the approach of nding a q way minimal graph cut using several eigenvectors of L, and introduce a di erent cluster identi cation heuristic. Alpert et al. 2] introduce the MELO (Multiple Eigenvector Linear Ordering) algorithm to approximate minimal graph ....

Pak K. Chan, Martine D.F. Schlag, and Jason Y. Zien. Spectral k-way ratio-cut partitioning and clustering. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 13(9):1088-1096, 1994.

....and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [Sim91, Wil90] solving sparse linear systems [PSW92] and partitioning for domain decomposition [CR87, CS93] It is also used in VLSI circuit design and simulation [CSZ93, HK92, AK95]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [BS92, HL92, HL93, PSL90, Sim91, Wil90] However, the quality of the partition that these methods should produce has so far eluded precise ....

P. K. Chan, M. Schlag, and J. Zien. Spectral k-way ratio cut partitioning and clustering. In Symp. on Integrated Systems, 1993.

....algorithms determine a partitioning from the graph describing the circuit or system, whereas iterative methods aim at improving the quality of an existing partitioning solution. Constructive partitioning approaches are mainly based on clustering[AV96, DD96] spectral or eigenvector methods[CSZ94], placement based partitioning, mathematical programming or network flow computations. 1.1.1 Interchange Methods To date, iterative improvement techniques that make local changes to an initial partition are still the most successful partitioning algorithms in practice. The advantage of these ....

P.K. Chan, D.F. Schlag, and J.Y. Zien. Spectral K-way Ratio-Cut Partitioning and Clustering. IEEE Transactions on Computer Aided Design, 13(9):1088--1096, 1994.

....used method for p way partitioning, when p is a power of 2, is recursive bisection. It first divides a graph into two equal sized pieces, by a good bisection algorithm, and then recursively divides the two pieces. When p is not a power of 2, simple variants of recursive bisection are used [8]. Ideally, we would like to use an optimal bisection algorithm in recursive bisection. However, because the optimal bisection problem, that divides a graph into two equal sized subgraphs to minimize the number of edges cut, is NP complete, practical RB algorithms use more efficient heuristics in ....

P. K. Chan, M. Schlag, and J. Zien. Spectral k-Way Ratio Cut Partitioning and Clustering. Proceedings of Symposium on Integrated Systems Seattle, WA, 1993.

....problem. The most commonly used method for p way partitioning, when p is a power of two, is RB. It first divides a graph into two equal sized pieces, by a good bisection algorithm, and then recursively divides the two pieces. When p is not a power of two, simple variants of RB are used [8]. Ideally, we would like to use an optimal bisection algorithm in RB. However, because the optimal bisection problem that divides a graph into two equal sized subgraphs to minimize the number of edges cut is NP complete, practical RB algorithms use more e#cient heuristics in place of an optimal ....

P. K. CHAN, M. SCHLAG, AND J. ZIEN, Spectral k-way ratio cut partitioning and clustering, in Proc. Symposium on Integrated Systems, Seattle, WA, 1993.

....researchers have investigated a number of k way partitioning algorithms that try to compute a k way partitioning directly, rather than via recursive bisection. The most notable of them are the generalization of the FM algorithm for k way partitioning [4, 7] the spectral multi way ratio cut [6], the primal dual algorithm of [5] the geometric embedding [9] the dual net method [12] and the K PM LR algorithm [20] A key problem faced by some of these algorithms is that the k way FM refinement algorithm easily gets trapped in local minima. The recently developed K PM LR algorithm by Cong ....

P. Chan, M. Schlag, and J. Zien. Spectral k-way ratio-cut partitioning and clustering. In Proc. of the Design Automation Conference, pages 749--754, 1993.

....cost measure, but in different ways. Shin Kim [26] suggested the use of the gradual enforcement of balance criterion in FM algorithm during partitioning. There are many other approaches to circuit partitioning that are not based on KL algorithm such as Simulated Evolution [21] Spectral Methods [5, 9], Mean Field Annealing [3, 4] and Simulated Annealing [10] Simulated Annealing algorithm [14] SA algorithm) is one of the most successful approaches to graph and circuit partitioning. Johnson et al. 10] performed an extensive experimental evaluation of SA algorithm on graph partitioning in ....

P. K. Chan, D. F. Schlag, and J. Y. Zien. Spectral K-way ratio-cut partitioning and clustering. In Proceedings of the 30th Design Automation Conference, pages 749--754. ACM/IEEE, 1993.

....and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [Sim91, Wil90] solving sparse linear systems [PSW92] and partitioning for domain decomposition [CR87, CS93] It is also used in VLSI circuit design and simulation [CSZ93, HK92, AK95]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [BS92, HL92, HL93, PSL90, Sim91, Wil90] However, the quality of the partition that Computer Science Division, U.C. Berkeley, CA 94720, ....

P. K. Chan, M. Schlag, and J. Zien. Spectral k-way ratio cut partitioning and clustering. In Symp. on Integrated Systems, 1993.

....cost measure, but in different ways. Shin Kim [26] suggested the use of the gradual enforcement of balance criterion in FM algorithm during partitioning. There are many other approaches to circuit partitioning that are not based on KL algorithm such as Simulated Evolution [21] Spectral Methods [5], 9] Mean Field Annealing [3] 4] and Simulated Annealing [10] Simulated Annealing algorithm [14] SA algorithm) is one of the most successful approaches to graph and circuit partitioning. Johnson et al. 10] performed an extensive experimental evaluation of SA algorithm on graph ....

Chan, P. K., Schlag, D. F., and Zien, J. Y. Spectral K-way ratio-cut partitioning and clustering. In Proceedings of the 30th Design Automation Conference (1993), ACM/IEEE, pp. 749--754.

....is to minimize the number of nets among all partitions while satisfying various constraints such as lower and upper bounds on the area and pin count of each partition. Some of the previous works include recursive KL [9] generalization of FM [10, 11] primal dual [12] spectral multiway ratio cut [3], geometric embedding [2] multilevelbased [8] and dual net based [4] method. There are two primary approaches for generating multiway partitioning solution; recursive or f lat. The recursive approach applies bipartitioning recursively until the desired number of partitions is obtained, whereas ....

P. Chan, M. Schlag, and J. Zien. Spectral k-way ratiocut partitioning and clustering. In Proc. Design Automation Conf., pages 749--754, 1993.

....eigenvectors. This is the correct approach since spectral embedding of the Laplacian appears to be closely related to the ratio cut cost metric. Let Pi be the set of all k way partitions of a graph G, and E h be the total weight of the edges in G having exactly one endpoint in partition P h . In [6] the authors show that, min X T X=I trace(X T QX) min P2 Pi k X h=1 E h jP h j (11) which is a lower bound on k way ratio cut cost metric. The first k eigenvectors V k of the Laplacian Q satisfy this inequality and the eigenvectors can be used to form the projector, V V T , as an ....

....example, use the Laplacian in place of the adjacency matrix. III D From Spectral Embeddings to Partitions Given the logic capacity and I O capacity constraints, we present a procedure to use the eigenvector embedding of a graph matrix to construct partitions that satisfy the constraints. As in [6], our approach to k way do not circulate Accepted for publication in Symposium FPGA 95, Monterey, California 7 KPF Partition(I O constraint, logic constraint) f Remove high fanout nets in the hypergraph and transform it to a graph Find k eigenvectors of this graph, V Associate each row of V ....

P. K. Chan, M. Schlag, and J. Zien. Spectral k-way ratio-cut partitioning and clustering. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 13(9):1088--1096, Sept. 1994.

.... [8] Fast bipartitioning methods were developed based on a linear ordering of the vertices using the eigenvector associated with the second smallest eigenvalue of the Laplacian of a graph in [11, 7] A k way spectral partitioning algorithm and new k way ratio cut cost function was presented in [4]. Recent methods for spectral k way ratio cut partitioning are presented in [1] Additional approaches to spectral partitioning are presented in [9, 3] A multi level partitioning algorithm using recursive applications of the ratio cut metric to form clusters is presented in [12] The first ....

....can be represented by an n Theta k assignment matrix Y where y ih is 1 when vertex i is in partition h and y ih is 0 otherwise. Given a partitioning, the n Theta k ratioed assignment matrix, R, has as entry r ih the value yih p jjPh jj . This definition differs slightly from the one in [4]. M is the n Theta n diagonal matrix whose m ii entry is the size of vertex i. Among the many variations of the k way partitioning problem, we focus on optimizing the k way ratiocut cost function [4] that is, finding a solution R such that P k h=1 Eh jjPh jj is minimized. Although it may ....

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P. K. Chan, M. D. F. Schlag, and J. Y. Zien. Spectral k-way ratio-cut partitioning and clustering. IEEE Trans. on CAD, 13(9):1088--1096, Sept. 1994.

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P. Chan, D. Schlag, and J. Zien. Spectral K-way ratio-cut partitioning and clustering. IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, 13(9):1088-- 1096, 1994.

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P. K. Chan, M. D. F. Schlag, and J. Y. Zien. Spectral K-way ratio-cut partitioning and clustering. IEEE Trans. CAD, 13(9):1088--1096, 1994.

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P. K. Chan, M. Schlag, and J. Zien. Spectral k-way ratio cut partitioning and clustering. In Symp. on Integrated Systems, 1993.

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P.K. Chan, M.D.F. Schlag, J. Zien, Spectral k-way ratio cut partitioning and clustering, IEEE Trans. CAD (1994) 1088--1096.

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P. K. Chan, M. D. F. Schlag, and J. Y. Zien. Spectral k-way ratio-cut partitioning and clustering. IEEE Transactions on Computer-aided Design of Integrated Circuits and Systems, 13(9):1088--96, 1994.

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P. Chan, D. Schlag, and J. Zien. Spectral K-way ratio-cut partitioning and clustering. IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, 13(9):1088-- 1096, 1994.

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P. K. Chan, M. D. F. Schlag, and J. Y. Zien. Spectral K-way ratio-cut partitioning and clustering. IEEE Trans. CAD, 13(9):1088--1096, 1994.

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P. K. Chan, M. D. F. Schlag, and J. Y. Zien. Spectral K-way ratio-cut partitioning and clustering. IEEE Trans. CAD, 13(9):1088--1096, 1994.

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P. K. Chan, M. Schlag, J. Zien, Spectral k-way ratio cut partitioning and clustering, Symp. on Integrated Systems, 1993.

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P. K. Chan, M. Schlag, J. Zien, Spectral k-way ratio cut partitioning and clustering, Symp. on Integrated Systems, 1993.

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P. K. Chan, M. D. Schlag, and J. Y. Zien. Spectral k-way ratio-cut partitioning and clustering. In 30th ACM/IEEE DAC Proceedings, 1993, pages 749--54, 1993.

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