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UT EE 382V - Lecture 5. Circuit Partitioning (III)

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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 learnedKL AlgorithmFM algorithm Variation and ExtensionMultilevel partition (hMetis)Simulated Annealing301/14/19Partitioning AlgorithmsTwo elegant partition algorithms  although not the fastestLearn 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 dcbaAdcba10000 000300050004 dcbaDdcbaD: 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).nnnnnnnnnxaxaxaxaxaxaxxaaaLMLML221112121111nnnnaaaL2111211xA x A ...601/14/19A Basic Property jiijjinniininiiinnnnnnaxxxxaxaxxxaaaaxx,11,1,1111111,,MLMLMLLxTAx701/14/19Basic Idea of Laplacian Spectrum Based Graph PartitioningGiven 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 minimumLinear placement interpretation:minimize total squared wirelength' 11 .s.t ),,,(21XiXi xxxxxinLxxEjiji)(),(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 x0S )( 4 ) According to Courant-Fischer minimax principle: the 2nd smallest eigenvalue satisfies:Property of Laplacian Matrix (Cont’d)0jiijTxxQQxx2||minxQxxTx in S1001/14/19Results on Spectral Based Graph PartitioningMin bisection cost c(x, x’)  n/4Min ratio-cut cost c(x,x’)/|x||x’|  /n The second smallest eigenvalue gives the best linear placementCompute the best bisection or ratio-cutbased on the second smallest eigenvector1101/14/19Computing the EigenvectorOnly need one eigenvector(the second smallest one)Q is symmetric and sparseUse block Lanczos Algorithm1201/14/19Interpreting the EigenvectorLinear Ordering of modulesTry 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 GraphsGBUI(2n, d, b) [Bui-Chaudhurl-Leighton 1987]2n nodesd-regularmin-bisectionbn nGBUI ResultsValue of XiCluster 2Cluster 1Module i1401/14/19Some Applications of Laplacian SpectrumPlacement 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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UT EE 382V - Lecture 5. Circuit Partitioning (III)

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