VLSI Physical Design AutomationRecap of what you have learnedPartitioning AlgorithmsSpectral Based Partitioning AlgorithmsEigenvalues and EigenvectorsA Basic PropertyBasic Idea of Laplacian Spectrum Based Graph PartitioningProperty of Laplacian MatrixProperty of Laplacian Matrix (Cont’d)Results on Spectral Based Graph PartitioningComputing the EigenvectorInterpreting the EigenvectorExperiments on Special GraphsSome Applications of Laplacian Spectrum01/14/191VLSI Physical Design AutomationProf. David [email protected]: ACES 5.434Lecture 5. Circuit Partitioning (III) Spectral and Flow201/14/19Recap of what you have learnedKL AlgorithmFM algorithm Variation and ExtensionMultilevel partition (hMetis)Simulated Annealing301/14/19Partitioning AlgorithmsTwo elegant partition algorithms although not the fastestLearn how to formulate the problem!=> Key to VLSI CAD(1) Spectral based partitioning algorithms(2) Max-flow based partition algorithm401/14/19Spectral Based Partitioning Algorithmsabcd13430343 343000001010 dcbaAdcba10000 000300050004 dcbaDdcbaD: degree matrix; A: adjacency matrix; D-A: Laplacian matrixEigenvectors of D-A form the Laplacian spectrum of Q501/14/19Eigenvalues and EigenvectorsIf Ax=xthen is an eigenvalue of A x is an eignevector of A w.r.t. (note that Kx is also a eigenvector, for any constant K).nnnnnnnnnxaxaxaxaxaxaxxaaaLMLML221112121111nnnnaaaL2111211xA x A ...601/14/19A Basic Property jiijjinniininiiinnnnnnaxxxxaxaxxxaaaaxx,11,1,1111111,,MLMLMLLxTAx701/14/19Basic Idea of Laplacian Spectrum Based Graph PartitioningGiven a bisection ( X, X’ ), define a partitioning vectorclearly, x 1, x 0 ( 1 = (1, 1, …, 1 ), 0 =(0, 0, …, 0))For a partition vector x: Let S = {x | x 1 and x 0}. Finding best partition vector x such that the total edge cut C(X,X’) is minimum is relaxed to finding x in S such that is minimumLinear placement interpretation:minimize total squared wirelength' 11 .s.t ),,,(21XiXi xxxxxinLxxEjiji)(),(2….x1 x2 x3 xn xxEjiji)(),(2= 4 * C(X, X’)801/14/19Property of Laplacian Matrix EjijiEjijiiijiijiiTTjiijTxxxaij xxdxxAxdAxxDxxxxQQxx),(2),(22)(2( 1 )squared wirelength= 4 * C(X, X’)Therefore, we want to minimize TQxxIf x is a partition vector….x1 x2 x3 xn901/14/19( 2 ) Q is symmetric and semi-definite, i.e.(i) (ii) all eigenvalues of Q are 0( 3 ) The smallest eigenvalue of Q is 0 corresponding eigenvector of Q is x0= (1, 1, …, 1 ) (not interesting, all modules overlap and x0S )( 4 ) According to Courant-Fischer minimax principle: the 2nd smallest eigenvalue satisfies:Property of Laplacian Matrix (Cont’d)0jiijTxxQQxx2||minxQxxTx in S1001/14/19Results on Spectral Based Graph PartitioningMin bisection cost c(x, x’) n/4Min ratio-cut cost c(x,x’)/|x||x’| /n The second smallest eigenvalue gives the best linear placementCompute the best bisection or ratio-cutbased on the second smallest eigenvector1101/14/19Computing the EigenvectorOnly need one eigenvector(the second smallest one)Q is symmetric and sparseUse block Lanczos Algorithm1201/14/19Interpreting the EigenvectorLinear Ordering of modulesTry all splitting points, choose the best oneusing bisection of ratio-cut metric23 10 22 34 6 21 823 10 22 34 6 21 81301/14/19Experiments on Special GraphsGBUI(2n, d, b) [Bui-Chaudhurl-Leighton 1987]2n nodesd-regularmin-bisectionbn nGBUI ResultsValue of XiCluster 2Cluster 1Module i1401/14/19Some Applications of Laplacian SpectrumPlacement and floorplan[Hall 1970][Otten 1982][Frankle-Karp 1986][Tsay-Kuh 1986]Bisection lower bound and computation[Donath-Hoffman 1973][Barnes 1982][Boppana 1987]Ratio-cut lower bound and computation[Hagen-Kahng 1991][Cong-Hagen-Kahng
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