Lars Hagen and Andrew Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proceedings of IEEE International Conference on Computer Aided Design, pages 10--13, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is to use graphs. The document to document similarity graph G s [2]is obtained by treating the pairwise similarity matrix of the dataset as the adjacency matrix of G s . Viewing the documents in this fashion, a number of edge cut based criterion functions can be used to cluster document datasets [7, 17, 34, 13, 43, 10]. The # 1 (Equations 6) function [13] views the clustering process as that of partitioning the documents into groups that minimize the edge cut of each partition. However, because this edge cut based criterion function may have trivial solutions the edge cut of each cluster is scaled by the sum of ....

Lars Hagen and Andrew Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proceedings of IEEE International Conference on Computer Aided Design, pages 10--13, 1991.

....one pair of objects (r i ; s j ) in the join index such that r i 2 v 1 and s j 2 v 2 . Figure 3 shows a page connectivity graph for the join index from Figure 1(b) Nodes (a; b) represent the pages of relation R, and nodes (A; B; C) represent the pages of relation S. A min cut node partition [17, 27] of graph BG = V 1 ; V 2 ; E) partitions the nodes in V into disjoint subsets while minimizing the number of edges whose incident nodes are in two different partitions. The cut set of a min cut partition is the set of edges whose incident nodes are in two different partitions. Fast and efficient ....

L. Hagen and A. Kahng. Fast Spectral Methods for Ratio Cut Partitioning and Clustering. In Proceedings of IEEE International Conference on Computer Aided Design, 1991.

....can be categorized into three classes. The first class strictly abides by the modeling graph, with no attempt to change the graph. High quality results have been reported by several algorithms which include iterative improvement based [1, 3] clustering based [4] and spectrum (eigenvector) based [5, 6]. The second class of algorithms may modify the graph through node replications [7, 8] Improvement is achieved by sacrificing some area due to node replications. These two classes both perform the partitioning task on the graph without considering the logic function of the circuit. The third ....

L. Hagen and A. B. Kahng, "Fast spectral methods for ratio cut partitioning and clustering," Proc. of the Int. Conf. on Computer-Aided Design, pp. 10--13, 1991.

....The graph partitioning problem is NP complete. However, many algorithms have been developed that find a reasonably good partition. Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity [4, 5, 12, 13, 15, 7, 31, 20, 19]. In these schemes, the original graph is successively coarsened down until it has only a small number of vertices, a partition of this coarsened graph is computed, and then this initial partition is successively refined by using a Kernighan Lin type heuristic as it is being projected back to the ....

Lars Hagen and Andrew Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proceedings of IEEE International Conference on Computer Aided Design, pages 10--13, 1991. 19

....of spatially related data items in large geographical information systems on disk to minimize disk I O requests, and mapping of task graphs to parallel processors. The graph partitioning problem is NP complete. However, many algorithms have been developed that find reasonably good partitionings [23, 22, 9, 24, 19, 18, 20, 2, 3, 7, 8, 12, 5, 21, 16, 13]. The k way partitioning problem is most frequently solved by recursive bisection. That is, we first obtain a 2 way partitioning of V, and then we recursively obtain a 2 way partitioning of each resulting partition. After log k phases, graph G is partitioned into k partitions. Thus, the problem ....

Lars Hagen and Andrew Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proceedings of 1EEE International Conference on Computer Aided Design, pages 1(13, 1991.

....E) is defined as a set of vertices V and a set of edges E, where each edge is a subset of the vertex set V. The graph partition problem is NP complete. Recently, a number of researchers have investigated a class of algorithms that can give a reasonably good solution for the bi partition problem [7, 8, 12, 13]. Most of these algorithms are more or less based on the FM algorithm, which was first proposed by Fiduccia and Mattheyses in 1982 [7] FM algorithm is a very effective heuristic for the bipartition problem. However, algorithm designers are more and more interested in the general k way partition ....

L. Hagen and A. B. Kahng, "Fast Spectral methods for Ratio Cut Partitioning and Clustering", in IEEE/ACM International Conference on Computer-Aided Design, 1991.

....the modules with a minimal length of the connections between them is calculated. Second, each partition which is larger than the target partition size is initially partitioned by a rectangular cut, as demonstrated by the dashed lines in Figure 4. The initial cut uses an adapted ratio cut objective [HK91] yielding well balanced partitions in terms of module area. Third, the number of cut nets of the current partitioning is reduced by moving modules between all partitions applying a multi way ratio cut method with a problemspecific objective function. These three steps, calculation of the ....

L. Hagen and A. B. Kahng. Fast spectral methods for ratio cut partitioning and clustering. In IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pages 10--13, 1991.

.... or near critical nets are kept as short as possible [12, 27, 21] in this case, the optimization metrics proposed for partitioning are (weighted) mincut [27, 21] and weighted min max [12] Recursive 2 way partitioning is an efficient and popular approach to obtaining k way partitions for k 2 [9, 20, 30, 36, 35]. We will thus be concerned here with the 2 way min cut partitioning problem. Since 2 way min cut partitioning is NP complete [19] a number of approximate schemes have been proposed. These include iterative improvement methods [16, 17, 23, 24, 29, 30] simulated annealing [31, 32] and ....

....concerned here with the 2 way min cut partitioning problem. Since 2 way min cut partitioning is NP complete [19] a number of approximate schemes have been proposed. These include iterative improvement methods [16, 17, 23, 24, 29, 30] simulated annealing [31, 32] and clustering based techniques [7, 5, 18, 20, 28, 29, 36, 35]. An excellent survey on partitioning techniques appears in [6] In iterative improvement, we start with a random 2 way partition of the circuit, and iteratively improve it by either swapping pairs of nodes between the subsets, or moving one node at a time between them so that the cutset size is ....

[Article contains additional citation context not shown here]

L. Hagen and A. Kahng, "Fast spectral methods for ratio cut partitioning and clustering", Proc. Int'l. Conf. Computer-Aided Design, 1991, pp. 10-13.

....graph for a join index where nodes a; b represent the pages of relation R, and nodes A; B; C represent the pages of relation S. An edge represents a page join between a pair of pages. A page join represents the corresponding join between the tuples in the two pages. A min cut node partition [15, 21] of graph G = V; E) partitions the nodes in V into disjoint subsets while minimizing the number of edges whose incident nodes are in two different partitions. The cut set of a min cut partition is the set of edges whose incident nodes are in two different partitions. Fast and efficient heuristic ....

....encompass this node and its connecting nodes in the other relation. To construct a hyper edge graph, in addition to these pure hyperedges, we add hyperedges for each edge in the original bipartite graph. Figure 11 illustrates this construction with the help of an example. A min cut node partition [15, 21] of a hyper graph G partitions the nodes in V 1 and V 2 into disjoint subsets while minimizing the number of hyper edges whose incident nodes are in different partitions. The h cut set of a min cut partition is the set of hyper edges whose incident nodes are in two different partitions. Fast and ....

L. Hagen and A. Kahng. Fast Spectral Methods for Ratio Cut Partitioning and Clustering. In Proceedings of IEEE International Conference on Computer Aided Design, 1991.

....paradigm to the K MFFC FM algorithm, a K FM algorithm based on maximum fanout free cone (MFFC) clustering reported in [10] and show that the resulting algorithm, K DualMFFC FM reduces the net cutsize by 15 to 26 compared with K MFFC FM. Furthermore, we compare the K DualFM algorithm with EIG1[18] and Paraboli [26] two recently proposed spectral based bipartitioning algorithms. We showed that K DualFM reduces the net cutsize by 56 on average when compared with EIG1 and produces comparable results with Paraboli. 1. Introduction The K way partitioning problem is one of partitioning the ....

.... with look ahead scheme [24] and the simulated annealing approach [23, 17] The analytical methods include both the use of a linear placement formulation with the quadratic objective function, which is solved by computing the second smallest eigenvector of the Laplacian matrix of the given network [14, 2, 4, 18], and the use of the linear placement formulation with a linear objective function, which is solved by an iterative method in [26] The min cut based method uses the maximum flow algorithm to compute a series of minimum cuts in the given circuit in 2order to obtain an area balanced cut with ....

[Article contains additional citation context not shown here]

L. Hagen and A. B. Kahng, "Fast Spectral Methods for Ratio Cut Partitioning and Clustering," Proc. IEEE Int'l Conf. on Computer-Aided Design, pp. 10--13, 1991.

....of objects (r i ; s j ) in the join index such that r i 2 v i 1 and s j 2 v j 2 . Figure 2 shows a page connectivity graph for the join index from Figure 1(b) Nodes (a; b) represent the pages of relation R, and nodes (A; B; C) represent the pages of relation S. A min cut node partition [13, 22] of graph BG = V 1 ; V 2 ; E) partitions the nodes in V into disjoint subsets while minimizing the number of edges whose incident nodes are in two different partitions. The cut set of a min cut partition is the set of edges whose incident nodes are in two different partitions. Fast and efficient ....

L. Hagen and A. Kahng. Fast Spectral Methods for Ratio Cut Partitioning and Clustering. In Proceedings of IEEE International Conference on Computer Aided Design, 1991.

....it could be regarded as part of the same algorithmic approach to graph bisection as seed growth, since they both consider moving vertices based on edge crossings. We also implemented another standard technique that operates on a very different principle. This method, known as spectral bisection [1, 17, 15], uses an eigenvector computation to embed a graph along a line such that the sum of the squared distances between embedded connected vertices is minimized. Then we join the ends of the line to form a circle, and 22 consider each of the jV j=2 bisections resulting from breaking the ring into two ....

L. Hagen and A. B. Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proceedings of the IEEE International Conference on Computer-Aided Design, pages 10--13, 1991.

....higher level lookahead gains and improved the results for small circuits. All these algorithms improve an initial partition through a sequence of node moves (based on node gains ) and thus fall under the class of iterative improvement partitioners (IIPs) Recently, a number of non IIP algorithms [3, 4, 12, 20, 21] have been proposed and excellent results have been obtained. FM and LA are the most commonly used two way partitioning algorithms largely due to their excellent run times, simple implementations and flexibility. However, this class of IIP algorithms have a common weakness, viz. they only find ....

L. Hagen and A. B. Kahng, "Fast Spectral Methods for Ratio Cut Partitioning and Clustering", Proc. IEEE Intl. Conf. Computer-Aided Design, 1991, pp. 10-13.

....The graph partitioning problem is NP complete. However, many algorithms have been developed that find a reasonably good partition. Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity [4, 5, 9, 10, 12, 7, 21, 15, 14]. In these schemes, the original graph is successively coarsened down until it has only a small number of vertices, a partition of this coarsened graph is computed, and then this initial partition is successively refined by using a Kernighan Lin type heuristic as it is being projected back to the ....

Lars Hagen and Andrew Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proceedings of IEEE International Conference on Computer Aided Design, pages 10--13, 1991.

....Pseudo peripheral vertices [22] are widely used for the starting vertices of the distance. The neighbor split method [67] removes an edge from the graph, and split the graph according to the difference of the distance form the two nodes incident to the removed edge. The spectral bisection method [55, 26] is a high performance algorithm. It constructs the Laplacian matrix of the graph and computes the Fiedler vector (the eigenvector of the second smallest eigenvalue) Recently a lot of propositions on modifications of the spectral bisection method are presented [3] Some methods are intended to ....

L. Hagen and A. Kahang, "Fast Spectral Methods for Ratio Cut Partitioning and Clustering," Proc. ICCAD '91, pp. 10--13.

....FM by adding higher level lookahead gains and improved the results for small circuits, while Hagen et al. 15] investigated implementation and tie breaking techniques for improving the performance of FM type algorithms. A number of clustering based, i.e. bottom up, partitioning algorithms [9, 10, 12, 13, 14, 18] have also been proposed and good results have been obtained. FM and LA are the most commonly used two way partitioning algorithms largely due to their excellent run times, simple implementations and flexibility. However, this class of iterative improvement algorithms have a common weakness, ....

L. Hagen and A. B. Kahng, "Fast Spectral Methods for Ratio Cut Partitioning and Clustering", Proc. IEEE Intl. Conf. Computer-Aided Design, 1991, pp. 10-13.

....than a non critical one to ensure that the length of critical or near critical nets are kept as short as possible [8] The min cut problem is to obtain a P k so that its cost is minimized. Recursive 2 way partitioning is an efficient and popular approach to obtaining k way partitions for k 2 [3, 7, 13, 14]. We will thus be concerned here with the 2 way min cut partitioning problem. Since the problem is NP complete, a number of approximate schemes have been proposed. These include iterative improvement methods [3, 6, 9, 10] simulated annealing [12] and clustering based techniques [1, 2, 7, 11, 13, ....

.... 2 [3, 7, 13, 14] We will thus be concerned here with the 2 way min cut partitioning problem. Since the problem is NP complete, a number of approximate schemes have been proposed. These include iterative improvement methods [3, 6, 9, 10] simulated annealing [12] and clustering based techniques [1, 2, 7, 11, 13, 14]. In iterative improvement methods, we start with a random 2 way partition of the circuit, and iteratively improve it by either swapping pairs of nodes between the subsets, or moving one node at a time between them so that the cutset size is reduced. Clustering based methods try to find natural ....

[Article contains additional citation context not shown here]

L. Hagen and A. Kahng, "Fast spectral methods for ratio cut partitioning and clustering", Proc. Int'l. Conf. Computer-Aided Design, 1991, pp. 1013.

....The graph partitioning problem is NP complete. However, many algorithms have been developed that find a reasonably good partition. Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity [4, 5, 11, 12, 14, 7, 28, 17, 16]. In these schemes, the original graph is successively coarsened down until it has only a small number of vertices, a partition of this coarsened graph is computed, and then this initial partition is successively This work was supported by NSF CCR 9423082, by Army Research Office contract ....

Lars Hagen and Andrew Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proceedings of IEEE International Conference on Computer Aided Design, pages 10--13, 1991.

....with respect to the number of pins in the circuit. This is done by moving one cell at a time and using an efficient bucket data structure. Krishnamurthy [4] enhanced FM by adding higher level lookahead gains and improved the results for small circuits. Recently, a number of clustering algorithms [9, 10, 11, 12, 15] have been proposed and excellent results have been obtained. FM and LA are the most commonly used two way partitioning algorithms largely due to their excellent run times, simple implementations and flexibility. However, this class of iterative improvement algorithms have a common weakness, viz. ....

L. Hagen and A. B. Kahng, "Fast Spectral Methods for Ratio Cut Partitioning and Clustering", Proc. IEEE Intl. Conf. Computer-Aided Design, 1991, pp. 10-13.

....about a design, except [ShKu92] and [MuBr91] who incorporate timing information. 4 CLUSTERING METRICS Recently, algorithms based on quadratic programming techniques, also referred to as spectral or eigenvalue based methods, have been presented showing good partitioning results [RiDo94] [HaKa91]. The general idea of spectral methods is to place the modules into an one dimensional (using one eigenvalue) or n dimensional (two or more eigenvalues) space and group neighbored elements into one part. In conjunction with clustering approaches many different clustering metrics exist, which will ....

....circuits presented in the Chapter 6 using three runs with different starting partitions for each design. We then compare these results (DDP) with already published results from PARABOLI [RiDo94] FM and FMC [CoSm93] FM and RW ST [HaKa92] MBC [Bui89] IG Match [CoHa92] EIG1 IG [HaKa92] EIG1 [HaKa91] and FM and RCut 1.0 [WeCh91] Note that we have transferred results published under primary1 2, which do not use module size information, to primSC1 2 and primGA1 2 respectively. 8 CONCLUSION AND FUTURE WORK As shown by the presented results design driven clustering produces comparable results ....

Hagen, L.; Kahng, A; "Fast Spectral Methods for Ratio Cut Partitioning and Clustering", Int. Conf. on Computer Aided Design, pp. 10-13, 1991

....Another class of graph partitioning algorithms reduces the size of the graph (i.e. coarsen the graph) by collapsing vertices and edges, partitions the smaller graph, and then uncoarsens it to construct a partition for the original graph. These are called multilevel graph partitioning schemes [4, 7, 19, 20, 26, 10, 43]. Some researchers investigated multilevel schemes primarily to decrease the partitioning time, at the cost of somewhat worse partition quality [43] Recently, a number of multilevel algorithms have been proposed [4, 26, 7, 20, 10] that further refine the partition during the uncoarsening phase. ....

....approaches have been proposed for obtaining coarser graphs. The first approach is based on finding a random matching and collapsing the matched vertices into a multinode [4, 26] while the second approach is based on creating multinodes that are made of groups of vertices that are highly connected [7, 19, 20, 10]. The later approach is suited for graphs arising in VLSI applications, since these graphs have highly connected components. However, for graphs arising in finite element applications, most vertices have similar connectivity patterns (i.e. the degree of each vertex is fairly close to the average ....

[Article contains additional citation context not shown here]

<F3.746e+05> L. Hagen and A.<F3.819e+05> Kahng,<F3.543e+05> Fast spectral methods for ratio cut partitioning and<F3.819e+05> clustering, in Proc. IEEE International Conference on Computer Aided Design, 1991, pp. 10--13.

....partitioning tree has Rent parameter essentially identical to the theoretical lower bound. This result has key implications in two areas. First, it affirms previous work of Wei and Cheng [43] who were the first to propose the ratio cut metric as a partitioning objective, and of Hagen and Kahng [15], who proposed using the spectral approach for ratio cut partitioning. Second, our work naturally leads to the development of better predictive layout models, enabling a more efficient search of the space of layout solutions. Indeed, to facilitate this latter application, we center on the ....

....partition: the numerator captures the minimum cut criterion while the denominator favors near bisection. It is well known that both minimum cut bisection and minimum ratio cut are NP complete [13] so heuristic methods must be used. Previous approaches fall into several classes, as surveyed in [6] [15] [29] The greedy iterative paradigm is popular either as a stand alone strategy or as a postprocessing refinement to other methods. Iterative methods are based on local perturbation of the current solution and typically entail variations of the Kernighan Lin method [20] 37] e.g. the ....

[Article contains additional citation context not shown here]

L. Hagen and A. B. Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proc. ICCAD-91, pages 10--13, 1991.

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Lars Hagen and Andrew Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proceedings of IEEE International Conference on Computer Aided Design, pages 10--13, 1991.

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L. Hagen and A. B. Kahng, "Fast Spectral Methods for Ratio Cut Partitioning and Clustering", Proc. IEEE Intl. Conf. Computer-Aided Design, 1991, pp. 10-13.

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Lars Hagen and Andrew Kahng. Fast spectral methods for ratio cut partitioning and clustering. In Proceedings of IEEE International Conference on Computer Aided Design, pages 10--13, 1991.

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