Chan, P., Schlag, D., and Zien, J., "Spectral K-Way Ratio Cut Partitioning and Clustering," Proceedings of the 30th ACM/IEEE Design Automation Conference, July 1993, pp. 749-754.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ....

....twice; 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 ....

[Article contains additional citation context not shown here]

P. K. Chan, M. D. F. Schlag and J. Zien, "Spectral K-Way Ratio Cut Partitioning and Clustering", IEEE Trans. on CAD,

....of L corresponding to L s (K 1) smallest positive eigenvalues in nondecreasing order (so S N Y = Z) then the lower bound is attained for X = 1 K u N u t K N K Z R t K . Solutions that relax all but the third constraint in (P2) e.g. due to Barnes [4] Bolla [7] and Chan et al. [12], result in the geometric representation X = u N N : Z] instead. The most efficient algorithm for computing the eigenvalues and eigenvectors of a large, sparse, and symmetric N N matrix is the Lanczos algorithm with O(N 1.4 ) runtime [14, 45] Thus, O(N 1.4 ) is the running time for ....

....running time of this algorithm is O(N log K) A variation of this technique is the K means classification method [40] in which each new vertex is added to the partition with the nearest mean. The mean of a partition is the mean of its vertex geometric representatives. KP Technique. Chan et al. [12] suggested the KP algorithm which is based on information from the N N partition matrix P = X X t . p ij is one if v i and v j are in the same partition, and zero otherwise. A set of partition centers is obtained as in the KC algorithm; however, placement of a new vertex v i in a partition with ....

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, Vol. 13, No. 9, pp. 1088--1096, 1994.

.... algorithms include the recursive bipartitioning by Kernighan and Lin [22] a generalization of the FM algorithm with lookahead by Sanchis [27] the primal dual algorithm [30] and a generalization of the graph spectral based partitioning method to multi way ratio cut by Chan, Schlag, and Zien [5]. To reduce the computational complexity for partitioning very large circuits, cluster based partitioning methods have been introduced. In this approach clusters are identified and collapsed, and the resulting clustered network is partitioned using existing partitioning methods. Clustering methods ....

P. Chan, M. Schlag, and J. Zien, "Spectral K-Way Ratio-Cut Partitioning and Clustering," Proc. 30th ACM/IEEE Design Automation Conf., June 1993.

....according to the Theorem, we associate the overlapped part to the larger candidate such that the smaller one can become even smaller. In procedure Expand, each candidate is given a cost. The cost is a ratio which is equal to the 10 number of crossing nets divided by the size of that candidate [4]. We then run the FM algorithm on the candidates according to the order of ascending cost, trying to expand each candidate as much as possible without violating the I O and the capacity constraints. Overlapped clusters are allowed but their gain values are penalized linearly during the ....

P. K. Chan, M. D. Schlag, and J. Y. Zien, "Spectral K-Way Ratio-Cut Partitioning and Clustering," Proc. 30 th ACM/IEEE Design Automation Conf., Jun. 1993, pp. 749 - 754.

....our claim in Chapter 2 about the superior efficiency compared to bit parallel circuits. 102 I O D r a[26] r a[25] a[2] in a[1] a[3] a[4] 2 3 1 5 6 8 D r 4 r r r r D D r D 7 D D r a[28] a[8] a[6] a[9] a[7] a[10] a[11] a[27] a[5] a[12] a[13] a[14] a[30] a[15] a[20] a[21] a[20] a[19] a[18] a[16] a[17] a[32] a[33] a[23] out 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 1 2 24 critical loop period = 26 Figure 5.9: 5th order wave elliptic filter. This filter architecture is derived from the analogue filter design ....

....Chapter 2 about the superior efficiency compared to bit parallel circuits. 102 I O D r a[26] r a[25] a[2] in a[1] a[3] a[4] 2 3 1 5 6 8 D r 4 r r r r D D r D 7 D D r a[28] a[8] a[6] a[9] a[7] a[10] a[11] a[27] a[5] a[12] a[13] a[14] a[30] a[15] a[20] a[21] a[20] a[19] a[18] a[16] a[17] a[32] a[33] a[23] out 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 1 2 24 critical loop period = 26 Figure 5.9: 5th order wave elliptic filter. This filter architecture is derived from the analogue filter design methodology. The ....

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P. Chan, M. Schlag, and J. Zien, "Spectral K-Way Ratio-Cut Partitioning and Clustering," Proc. 3th ACM/IEEE Design Automation Conference, 1993.

....we set W to be the average cluster size. The DP RP method of [2] can then be applied to optimally split the vertex ordering into a k way clustering. Our WINDOW DP RP derived clusterings are superior to a number of clustering methods in the literature in terms of three distinct objectives [5] [21] 7] The WINDOW orderings, by themselves, are also superior to previous graph ordering constructions [12] 16] 18] 2] Results for both clustering and multi way partitioning are quite encouraging. 1 Introduction: Clusterings and Vertex Orderings We seek to construct vertex orderings that ....

....can be repeated if desired. Hagen and Kahng [11] suggested using a random walk (RW ST) to find dense areas of the netlist, with cycles in the random walk forming the clusters. The suggested implementation requires O(n 3 ) time. In another direction, Alpert and Kahng [3] and Chan et al. [5] apply geometric or hybrid geometric topological clustering algorithms to multi dimensional spectral embeddings of the netlist. Generating the eigenvectors that yield the embedding can be expensive (see also [2] and such approaches can be complicated. Cong and Smith [6] present a bottom up ....

[Article contains additional citation context not shown here]

P. K. Chan, M. D. F. Schlag and J. Zien, "Spectral K-Way Ratio Cut Partitioning and Clustering", Proc. Symp. on Integrated Systems, Seattle, March 1993.

....as increase result quality. Many known clustering algorithms try to construct the natural hierarchy of a circuit by finding (strongly) connected components in a design. SC: min f CS n k e u v e u C v C C i i i i k ( 1 1 1 b g m r (2) The Scaled Cost (SC) value [ChSc93] is defined as the weighted sum of cluster outdegree over all clusters divided by cluster size and represents a generalized ratio cut objective. AB: max f CS e C e i e E e C i k i ( 1 1 1m r (3) Previous work can been grouped into approaches using local or global ....

Chan P.K.; Schlag M.D.F.; Zien J.Y.; "Spectral K-Way Ratio-Cut Partitioning and Clustering", Proc. Design Automation Conf., pp. 749-745, 1993

....to L s (K 1) smallest positive eigenvalues in nondecreasing order (so S N Y = Z) then the lower bound is attained for X = 1 K u N u t K N K Z R t K . Solutions that relax all but the third constraint in (P2) e.g. due to Barnes [4] Bolla [7] and Chan et al. [12], result in the geometric representation X = u N N : Z] instead. The most efficient algorithm for computing the eigenvalues and eigenvectors of a large, sparse, and symmetric N N matrix is the Lanczos algorithm with O(N 1.4 ) runtime [14, 45] Thus, O(N 1.4 ) is the running time for ....

....K means classification method [40] in which each new vertex is added to the partition with the nearest mean. The mean of a partition is the mean of its vertex geometric representatives. A HYPERGRAPH FRAMEWORK FOR OPTIMAL MODEL BASED DECOMPOSITION OF DESIGN PROBLEMS 27 KP Technique. Chan et al. [12] suggested the KP algorithm which is based on information from the N N partition matrix P = X X t . p ij is one if v i and v j are in the same partition, and zero otherwise. A set of partition centers is obtained as in the KC algorithm; however, placement of a new vertex v i in a partition ....

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, Vol. 13, No. 9, pp. 1088--1096, 1994.

....Objective for Circuit Decomposition J. H. Dennis Huang and Andrew B. Kahng UCLA Computer Science Department, Los Angeles, CA 90024 1596 USA jenhsin cs.ucla.edu, abk cs.ucla.edu Abstract Recent research on multi way partitioning has focused on the minimum cut [20, 26, 27] or generalized ratio cut [28, 29, 5] cost metrics. At the same time, clustering research has focused on such objectives as k l connectivity [12] DS metric [6] or clique finding [8] In this paper, we make the basic observation that cut objectives in partitioning, and density objectives in clustering, are fundamentally ....

....of signal nets crossing between the two partitions U and V . This metric is natural in that it addresses both the minimum cut objective and the balance requirement for partition sizes. Hagen and Kahng [17] used spectral methods to obtain good ratio cut partitioning solutions. Later, Chan et al. [5] introduced the scaled cost metric, which seeks a clustering P 1 ; P 2 ; P k that minimizes 1 n(k Gamma1) P k i=1 C i jP i j ; here, C i is the number of signal nets crossing the boundary of the partition P i . This is shown in [5] to be a generalization of the two way ratio cut ....

[Article contains additional citation context not shown here]

P. K. Chan, M. D. F. Schlag, and J. Y. Zien, "Spectral K-Way Ratio-Cut Partitioning and Clustering ", Proc. 30th ACM/IEEE Design Automation Conference, 1993, pp. 749-754.

....ordering of the modules according to a specific attraction function and then uses dynamic programming to split the linear ordering optimally based on the clustering objective function. The orderings used correspond to the scaled cost (generalized ratio cut) metric proposed by Chan et al. [4]. gains by 1, for k 2. We avoid changing the first level gain since it should always reflect the actual gain resulting from a move of this module. However, we add 1 to all the other gain levels so that the increased priority is guaranteed to affect the tie breaking. a b c d e a b c d e a b c ....

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

.... include the recursive bipartitioning by Kernighan and Lin [KeLi70] a generalization of the FM algorithm with lookahead by Sanchis [Sa89] the primal dual algorithm [YeCL91] and a generalization of the graph spectral based partitioning method to multi way ratio cut by Chan, Schlag, and Zien [ChSZ93]. To reduce the computational complexity for partitioning very large circuits, cluster based partitioning methods have been introduced based on various clustering techniques, such as random walk clustering [CoHK91, HaKa92] multicommodity flow based clustering [YeCL92] clique based clustering ....

Chan, P., M. Schlag, and J. Zien, "Spectral K-Way Ratio-Cut Partitioning and Clustering," Proc. 30th ACM/IEEE Design Automation Conf., June 1993.

....clusters: maximize f(P k ) k X i=1 w(C i ) where w(C i ) X fe2E j e C i 6= g je C i j Gamma 1 jej Gamma 1 i.e. net e incident to cluster C i adds absorption (p Gamma 1) Delta 1 jej Gamma1 to the cluster, where p is the number of pins of e in the cluster. ffl Scaled Cost [5] is a k way generalization of the ratio cut objective: minimize f(P k ) 1 n(k Gamma 1) k X i=1 w(C i ) where w(C i ) jfe j 9u; v 2 e; u 2 C i ; v = 2 C i gj jC i j i.e. w(C i ) is the outdegree of a cluster, divided by the cluster size. 1.2 Vertex Orderings Given vertices V = ....

P. K. Chan, M. D. F. Schlag and J. Zien, "Spectral K-Way Ratio Cut Partitioning and Clustering", Proc. Symp. on Integrated Systems, Seattle, March 1993.

....Spectral geometric embeddings of a circuit netlist can lead to fast, high quality multi way partitioning solutions. Furthermore, it has been shown that d dimensional spectral embeddings (d 1) are a more powerful tool than single eigenvector embeddings (d = 1) for multi way partitioning [2] [4]. However, previous methods cannot fully utilize information from the spectral embedding while optimizing netlist dependent objectives. This work introduces a new multi way circuit partitioning algorithm called DP RP. We begin with a d dimensional spectral embedding from which a 1 dimensional ....

....curve. The 1dimensional ordering retains useful information from the multi dimensional embedding while allowing application of efficient algorithms. We show that for a new Restricted Partitioning formulation, dynamic programming efficiently finds optimal solutions in terms of Scaled Cost [4] and can transparently handle userspecified cluster size constraints. For 2 way ratio cut partitioning, DP RP yields an average of 45 improvement over KP [4] and EIG1 [6] and 48 improvement over KC [2] 1 Introduction Systems with several million transistors entail problem complexities that ....

[Article contains additional citation context not shown here]

P. K. Chan, M. D. F. Schlag and J. Zien, "Spectral K-Way Ratio Cut Partitioning and Clustering", Proc. Symp. on Integrated Systems, Seattle, March 1993. (also see J. Zien, "Spectral K-Way Ratio Cut Graph Partitioning", M.S. Thesis, Computer Engineering Dept., UC Santa Cruz, March 1993, for experimental results).

....while of high quality, requires shortest path computations in the netlist graph and exhaustive enumeration of all partitions of disconnected components obtained through the shortest path deletion; its O(mn log n) time complexity also depends on two accuracy parameters b and 1 Delta . In [4], Chan et al. generalize the result of [10] from 2 way to k way ratio cut partitioning. Chan et al. use the first k eigenvectors of the netlist Laplacian to construct an orthogonal projector which maps an n dimensional space into a k dimensional space. Ideally, n elementary unit vectors in ....

....an orthogonal projector which maps an n dimensional space into a k dimensional space. Ideally, n elementary unit vectors in the n space (modules) will be mapped to exactly k distinct points in the k space (partitions) by this projector. Since this is not the case in practice, the authors of [4] use heuristic clustering methods in k space to obtain a k way partitioning. The approach requires additional matrix manipulations and a more complicated, netlist based clustering methodology than our methods below. The authors of [4] also generalize the ratio cut objective using a dimensionless ....

[Article contains additional citation context not shown here]

P. K. Chan, M. D. F. Schlag and J. Zien, "Spectral K-Way Ratio Cut Partitioning and Clustering", to appear in Proc. Symp. on Integrated Systems, Seattle, March 1993.

....nodes into a single large cluster, while other nodes remain singletons. The table also shows the total number of nets cut by the clustering, the sum of net degrees ( pins ) over all clusters, and the scaled cost value of the clustering (a multi way generalization of ratio cut, due to Chan et al. [7]) Although we have concentrated on the improved two way partitionings afforded by CAMS, it 9 The MBC method [5] relies on random matchings, it probabilistically identifies clusters; see also the recent randomized contraction strategy of Karger [20] for determining global minimum cuts in a ....

P.K. Chan, M.D.F. Schlag and J. Zien, "Spectral K-Way Ratio Cut Partitioning and Clustering, " in Proc. Symp. on Integrated Systems Seattle, March 1993.

....netcut being particularly amenable to such optimizations. 3 Evaluating Prospective Advances in Partitioning 3.1 Formulations and Metrics for VLSI Partitioning VLSI design presents many different flavors of hypergraph partitioning. Objective functions such as ratio cut [46] scaled cost [11], absorption cut [45] sum of degrees, number of vertices on the cut line [28] etc. have been applied for purposes ranging from routability driven clustering to multi level annealing placement. In top down coarse placement, partitioning involves fixed or propagated terminals [17, 44] tight ....

P. K. Chan and M. D. F. Schlag and J. Y. Zien, "Spectral K-Way Ratio-Cut Partitioning and Clustering", IEEE Transactions on Computer-Aided Design, vol. 13 (8), pp. 1088-1096.

....and abk cs.ucla.edu ABSTRACT Clustering has proven effective in improving the quality of VLSI netlist partitioning and placement algorithms. A wide variety of clustering schemes have been proposed, including random walks [13] iterative matching [7] and fairly complicated spectral techniques [1] [8]. Like [1] and [8] we use eigenvectors to compute a clustering, but do so in the simplest, most obvious manner. Our algorithm first computes a d digit code for each module v i according to the signs of the i th entries in a set of d eigenvectors. Then, modules with the same code are assigned to ....

....ABSTRACT Clustering has proven effective in improving the quality of VLSI netlist partitioning and placement algorithms. A wide variety of clustering schemes have been proposed, including random walks [13] iterative matching [7] and fairly complicated spectral techniques [1] 8] Like [1] and [8], we use eigenvectors to compute a clustering, but do so in the simplest, most obvious manner. Our algorithm first computes a d digit code for each module v i according to the signs of the i th entries in a set of d eigenvectors. Then, modules with the same code are assigned to the same cluster. ....

[Article contains additional citation context not shown here]

P. K. Chan, M. D. F. Schlag and J. Zien, "Spectral K-Way Ratio Cut Partitioning and Clustering", IEEE Trans. on CAD 13(9), 1994, pp. 1088-1096.

....: v n of a node set V = fv 1 ; v 2 ; vn g is defined by a bijection : f1 : ng f1 : ng. Node v i is the j th node in the ordering if (j) i. 2] presented various orderings, such as maxadjacency ordering, min perimeter ordering, max absorption [11] and min scaled cost [3], each of which may be achieved by defining some attraction from unordered nodes to the set of all previously ordered nodes. Let Nets(i) fe 2 Ejv i 2 eg be the set of nets incident on v i , and let S be the set of all previously ordered nodes, and for each unordered node v i let Attract(i) be ....

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), Sept. 1994, pp. 1088-1096.

....a hyperedge e 2 E denotes a subset of V with size at least 2, and let E i = fe 2 E j 9u; v 2 e; u 2 C i ; v = 2 C i g be the set of hyperedges cut by cluster C i . The following objectives are monotone: ffl Min Cut: Minimize: f(P k ) X 1ik w(C i ) with w(C i ) jE i j ffl Scaled Cost (Chan , Schlag, and Zien 1994): Minimize: f(P k ) 1 n(k Gamma 1) X 1ik w(C i ) with w(C i ) jE i j jC i j ffl Absorption (Sun and Sechen 1993) Maximize: f(P k ) X 1ik w(C i ) with w(C i ) X fe2E j e C i 6= g je C i j Gamma 1 jej Gamma 1 Theorem 1: If f is monotone nondecreasing, DP RP returns an ....

CHAN, P. K., SCHLAG, M. D. F., and ZIEN, J. (1994), "Spectral K-Way Ratio Cut Partitioning and Clustering, " IEEE Transactions on Computer-Aided Design, 13(9), 1088-1096.

....MIP 9357582 and ASC 9157610. partitioning method [HaKa92b, CoHK92] The multi way partitioning algorithms include the recursive two way partitioning method [KeLi70] the generalization of the FMalgorithm with lookahead scheme [Sa89] and a generalization of the spectrum based partitioning method [ChSZ93]. A number of techniques have been introduced recently to further improve the quality of partitioning solutions, including cluster based partitioning methods [CoHK91, HaKa92, YeCL92, CoSm93] partitioning with module duplication [KrNe91, HwGa92] and communication complexity based partitioning ....

Chan, P., M. Schlag, and J. Zien, "Spectral K-Way Ratio-Cut Partitioning and Clustering," Proc. 30th ACM/IEEE Design Automation Conf., June 1993.

....various graph traversals and addressing particular clustering requirements. We then use dynamic programming [2] to optimally split the vertex ordering into a multi way clustering. Our approach outperforms several clustering methods in the literature in terms of three distinct clustering objectives [5] [8] 22] The ordering construction, by itself, also outperforms existing graph ordering constructions [2] 13] 17] 19] for this application. Tuning our approach to meta objectives , particularly clustering for two phase Fiduccia Mattheyses bipartitioning [9] remains an open area of research. ....

....based on their attraction to the previously ordered vertices. Various choices of attraction can capture DFS, BFS, max adjacency [17] min perimeter [18] and other well studied graph traversals. Other choices can capture the intuition behind various VLSI clustering objectives such as Scaled Cost [5] and Absorption [22] The window parameter allows the framework to adapt to possible cluster size constraints, by limiting attraction to only the most recently ordered vertices. While our focus is on circuit clustering, we believe that our construction directly applies to such problem classes as ....

[Article contains additional citation context not shown here]

P. K. Chan, M. D. F. Schlag and J. Zien, "Spectral K-Way Ratio Cut Partitioning and Clustering", IEEE Trans. on CAD 13(9), 1994, pp. 1088-1096.

No context found.

Chan, P., Schlag, D., and Zien, J., "Spectral K-Way Ratio Cut Partitioning and Clustering," Proceedings of the 30th ACM/IEEE Design Automation Conference, July 1993, pp. 749-754.

No context found.

P. K. Chan, M. Schlag, J. Y. Zien, "Spectral K-Way Ratio-Cut Partitioning and Clustering", Symposium on Integrated Systems, pp. 123-142, 1993.

No context found.

P. K. Chan, M. Schlag, J. Y. Zien, "Spectral K-Way Ratio-Cut Partitioning and Clustering", Symposium on Integrated Systems, pp. 123-142, 1993.

No context found.

P. K. Chan, M. D. F. Schlag, and J. Zien, "Spectral K-way Ratio-cut Partitioning and Clustering," IEEE Transactions on CAD, Vol 13, No. 9, 1994, pp. 1088-1096.

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