J. Frankle, R.M. Karp, Circuit placements and cost bounds by eigenvector decomposition, Proc. IEEE Int. Conf. Comput. Aided Des. (1986) 414--417.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the relatively recent subfield of graph theory dealing with graph spectra. Early theoretical work connecting graph spectra and partitioning is due to Barnes, Donath and Hoffman [1] 6] 7] More recent eigenvector and eigenvalue methods have dealt with both module placement (Frankle and Karp [11] and Tsay and Kuh [41] and graph min cut bisection (Boppana [3] and Blanks [2] In general, these previous works formulate the partitioning problem as the assignment or placement of nodes The reader will note that our discussion omits several popular approaches, including simulated annealing ....

J. Frankle and R. M. Karp. Circuit placement and cost bounds by eigenvector decomposition. In Proc. ICCAD-82, pages 414--417, 1982.

....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 ....

....as a point in space. Their KP algorithm uses the directional cosine between two vectors as a similarity measure between vertices, and finds clusters accordingly. ffl Solution vectors: Any two way partitioning solution can be represented as an n dimensional 0 1 solution vector. Frankle and Karp [8] proposed an algorithm that sends multiple probes into n dimensional space, and uses eigenvectors to find the best solution vector with similar direction to a given probe. 2 We propose a new geometric representation that utilizes a transformation from graph partitioning into a vector ....

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J. Frankle and R. M. Karp, "Circuit Placements and Cost Bounds by Eigenvector Decomposition," IEEE Conf. Computer Aided Design,

....routing layers. To exhibit the profit derived from the failure prediction algorithm, each circuit is placed on the gate array such that over 95 of the available area is covered by modules, ensuring congested routing areas. Two placement algorithms are used, one based on eigenvector decomposition [10] (table 2) the other on simulated annealing [11] table 3) The routing algorithm is run with and without failure prediction. For each circuit the number of nets that failed to route, the number of nets for which search wave expansion did not start and the cpu time used are given in columns 3 ....

J. Frankle and R.M. Karp, "Circuit placement and cost bounds by eigenvector decomposition," Proceedings International Conference on Computer Aided Design, pp. 414--417, Santa Clara, 1986.

.... of a graph and its spectrum (the eigenvalues eigenvectors of its associated matrices) has been an area of active research for several years [13, 8, 16, 20] The spectra of the adjacency matrix A or the Laplacian Q of a graph are the basis for both partitioning [3, 9, 12] and placement techniques [11]. III B Spectral ratio cut partitioning using the Laplacian Q Spectral partitioning forms clusters of vertices based on the embedding implied by the eigenvectors V of a graph matrix, which can be the Laplacian Q, or the adjacency matrix A of the graph. To minimize the ratio cut cost metric, ....

....are presented. We refer to our k way spectral based partitioning algorithm as KPF. We ran the graphs derived from the MCNC FPGA partitioning 1993 benchmarks for the Xilinx XC3000 series. The hypergraphs of the benchmarks were transformed into graphs by using Frankle s clique expansion net model [11], or the star graph net model. Unlike the clique net model, the star graph net model produces a very sparse graph matrix, which accelerates the eigensolver. On the other hand, this hypergraph model generates auxiliary vertices that have to be filtered out from the eigenvector embedding. We have ....

J. Frankle and R. M. Karp. Circuit placements and cost bounds by eigenvector decomposition. In IEEE International Conference on Computer-Aided Design ICCAD-86, pages 414--417, Santa Clara, California, November 1986.

....into two equal halves, keeping contiguous vertices together. To take advantage of additional computation time, we can incrementally compute additional eigenvectors and their associated embeddings. More sophisticated variants of spectral bisection can be constructed by combining multiple embeddings [10]. Table 10 compares KL with spectral bisection. Unfortunately, even though we used a popular implementation of the Lanczos algorithm for sparse matrices, 2 the eigenvector computation takes O(jV j 1:4 ) time in the worst case, and occasionally took longer than the time allotted for a ....

J. Frankle and R. M. Karp. Circuit placement and cost bounds by eigenvector decomposition. In Proceedings of the IEEE International COnference on Computer-Aided Design, pages 414--417, 1986.

....in the circuit network. The best partition solution obtained by the several runs of the clustered circuit network partitioning is used as the initial feasible partitioning input for the final partition run on the original circuit network. Another popular approach is the graph spectral method [39][12] 45] With this approach, one first needs to convert the hyper graph representation of the circuit network into a graph representation G(E; V ) and then 123 derive the eigenvectors of the matrix Q = D 0 A where D is the degree matrix and A is the adjacency matrix, each derived from graph G. ....

J. Frankle and R. M. Karp, "Circuit Placement and Cost Bounds by Eigenvector Decomposition," Proc. IEEE Conference on Computer-Aided Design, pp.414-417, 1986. 178

.... ) we obtain the n Theta n diagonal degree matrix D defined by D ii = d(v i ) Early theoretical work connecting graph spectra and partitioning is due to Barnes, Donath and Hoffman [1] 6] 7] Most recent eigenvector and eigenvalue methods have dealt with both module placement (Frankle and Karp [9], Kleinhans et al. 17] and Tsay and Kuh [21] and graph min cut bisection (Blanks [2] and Boppana [3] In general, these previous works formulate the partitioning problem as the assignment or placement of nodes into bounded size clusters, i.e. chip locations. The problem is then transformed ....

J. Frankle and R.M. Karp, "Circuit Placement and Cost Bounds by Eigenvector Decomposition", IEEE Intl. Conf. on Computer-Aided Design, 1986, pp. 414-417.

....routing space model (presented further on in this paper) which is stored in a common database [Boo90] On this database several placement and routing algorithms are defined. Currently two placement algorithms are implemented based on simulated annealing [Ott84] and eigenvalue decomposition [Fra86] To cope with the diversity of possible routing spaces (variable number of layers, different technologies and master slice structures) we are in need of a flexible router. The router must be capable to route through densely packed areas thereby ensuring that the wiring is distributed smoothly ....

....routing layers. To exhibit the profit derived from the failure prediction algorithm, each circuit is placed on the gate array such that over 95 of the available area is covered by modules, ensuring congested routing areas. Two placement algorithms are used, one based on eigenvector decomposition [Fra86] table 2) the other on simulated annealing [Ott89] table 3) The routing algorithm is run with and without failure prediction. For each circuit the number of nets that failed to route, the number of nets for which search wave expansion did not start and the cpu time used are given in columns ....

J. Frankle and R.M. Karp, "Circuit placement and cost bounds by eigenvector decomposition," Proceedings International Conference on Computer Aided Design, pp. 414--417, Santa Clara, 1986.

....with little quality degradation. The results of this thesis are limited to simulated annealing based placement algorithms. Recently, researchers have proposed alternate forms of placement algorithms such as simulated evolution [85, 86] force directed placement [87, 88] linear optimization based [89 91]. and min cut based placement [92 94] Very little work has been done in developing parallel algorithms for these alternate approaches [95 97] New and more efficient parallel algorithms for these approaches must be investigated. The algorithms in this thesis have been targeted for portable ....

J. Frankle and R. M. Karp, "Circuit placements and cost bounds by eigenvector decomposition, " in Digest of Papers, International Conference on Computer-Aided Design, Santa Clara, CA, Nov. 1986, pp. 414--417.

....edge; it is motivated by the linear placement into fixed locations at unit separation [11] ffl The partitioning specific model assigns cost 4 jej(jej Gamma1) Delta 2 jej Gamma2 2 jej to each clique edge so that the expected cost of each cut hyperedge is one. ffl The Frankle model [19] proposed assigning cost ( 2 jej ) 3=2 to each clique edge to address linear placement with a quadratic optimization objective. This model has been utilized in the partitioning heuristic of [10] ffl Costs of 2 jej [28] 2 jej ) 3 [41] and 4 jej 2 Gammajej [15] have also been ....

....cost of each cut edge twice. Min cut graph partitioning is known to be NP complete, and many heuristic methods have been proposed (see [4] for a survey) Spectral methods are well established [7] 16] 18] 27] 36] and have also been the subject of extensive study in recent years [2] 5] 6] 10] [19] [23] 25] 29] 37] 40] These methods use eigenvectors of the Laplacian or adjacency matrix to construct various types of geometric representations of the graph. We loosely categorize previous methods according to five basic representations: ffl Linear orderings (spectral bipartitioning) The ....

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J. Frankle and R. M. Karp, Circuit placements and cost bounds by eigenvector decomposition, in: Proceedings of the IEEE International Conference on Computer-Aided Design (1986) 414-417.

....h=1 E h where E h = X v i 2Ch X v j = 2Ch a ij : 1) In other words, each E h is the edge cut of cluster C h . Min cut graph partitioning is NP complete, and many types of heuristic methods have been proposed (see the recent netlist partitioning survey [4] Spectral methods [1] 2] 6] 7] [8] [10] 12] 15] have been successful in recent years; these methods all use eigenvectors of the Laplacian or adjacency matrix to construct some type of geometric representation of the graph. We note examples of four such representations: ffl Linear orderings: Hall [12] showed that the second ....

....rather than as a point in space. Their KP algorithm constructs partitioning solutions using the directional cosine between two vectors as a similarity measure between vertices. ffl Indicator vectors: Any bipartitioning can be represented as an n dimensional 0 1 indicator vector. Frankle and Karp [8] proposed sending probes into the d dimensional space spanned by the best d eigenvectors; for a given probe, they find the indicator vector that maximally projects onto the probe in O(n log n) time. Our geometric representation is similar to that of the third approach, but we scale each ....

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J. Frankle and R. M. Karp, "Circuit Placements and Cost Bounds by Eigenvector Decomposition," IEEE Conf. Computer Aided Design, 1986, pp. 414-417

....this as an alternative to the standard 1 p Gamma1 net model [13] that has been used by e.g. 25] While experimental results in Section 3 and 6 support this choice, many other reasonable net model choices are available as well. For example, Hanan et al. 28] Huang [30] Frankle and Karp [20], and Tsay and Kuh [45] have proposed using the clique net model with weights 2 p , 6 p(p 1) 2 p ) 3 2 , and ( 2 p ) 3 respectively. Our choice of 4 p(p Gamma1) is motivated by 2 way partitioning, while most of the other net models are motivated by placement formulations. Hadley ....

....computed d dimensional spectral embeddings for 1 d 10 for both the standard net model and the partitioning specific net model, and then computed a heuristic SFC 3 Opt ordering for each embedding. To enable comparison with [12] we also computed orderings using the net model of Frankle and Karp [20], DP RP Algorithm for Linear Orderings Input: Circuit Netlist H(V; E) Linear Ordering fv 1 ; v 2 ; v n g L; U j Lower and upper cluster size bounds k j Number of clusters Output: P k j Optimal RP solution (without Condition 1(b) Vars: P k 0 [i;j] j Subsolutions k 0 j ....

J. Frankle and R. M. Karp. "Circuit Placement and Cost Bounds by Eigenvector Decomposition", Proc. IEEE Intl. Conf. on Computer-Aided Design, Nov. 1986, pp. 414-417.

No context found.

J. Frankle, R.M. Karp, Circuit placements and cost bounds by eigenvector decomposition, Proc. IEEE Int. Conf. Comput. Aided Des. (1986) 414--417.

No context found.

J. Frankle and R. M. Karp, "Circuit Placements and Cost Bounds by Eigenvector Decomposition, " IEEE Proc. ICCAD '86, pp. 414--417.

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J. Frankle and R. M. Karp, "Circuit Placement and Cost Bounds by Eigenvector Decomposition," Proceedings of the International Conference on CAD, 1986, pp. 414-417.

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