# UT EE 382V - Lecture 5. Circuit Partitioning (III) (14 pages)

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## Lecture 5. Circuit Partitioning (III)

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## Lecture 5. Circuit Partitioning (III)

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Lecture Notes

Pages:
14
School:
University of Texas at Austin
Course:
Ee 382v - New Topics in Computer Engineering
##### New Topics in Computer Engineering Documents

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VLSI Physical Design Automation Lecture 5 Circuit Partitioning III Spectral and Flow Prof David Pan dpan ece utexas edu Office ACES 5 434 01 14 19 1 Recap of what you have learned KL Algorithm FM algorithm Variation and Extension Multilevel partition hMetis Simulated Annealing 01 14 19 2 Partitioning 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 algorithm 01 14 19 3 Spectral Based Partitioning Algorithms a 1 b c 3 4 3 d a b A c d a 0 1 0 3 b 1 0 0 4 c 0 0 0 3 d 3 4 3 0 a b D c d a 4 0 0 0 b 0 5 0 0 c 0 0 3 0 d 0 0 0 10 D degree matrix A adjacency matrix D A Laplacian matrix Eigenvectors of D A form the Laplacian spectrum of Q 01 14 19 4 Eigenvalues and Eigenvectors x A a11 a12 an1 an 2 If Ax x1 a11 x1 a12 x2 L a1n xn L a1n M M L ann xn an1 x1 an 2 x2 L ann xn Ax x then 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 01 14 19 5 A Basic Property xTAx a11 x1 L xn a n1 n xi ai1 i 1 xi x j aij L L a1n x1 M M L ann xn x1 n xi ain M i 1 xn i j 01 14 19 6 Basic Idea of Laplacian Spectrum Based Graph Partitioning Given a bisection X X define a partitioning vector 1 x x1 x2 L xn s t xi 1 i X i X clearly x 1 x 0 1 1 1 1 0 0 0 0 For a partition vector x x i j E i xj 2 4 C X 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 xi xj 2 i j E Linear placement interpretation minimize total squared wirelength 01 14 19 x1 x2 x3 xn 7 Property of Laplacian Matrix 1 x Qx Qij xi x j T xT Dx xT Ax di xi2 di xi2 x i j E i A x x 2 a x x ij i i j E j ij i xj 2 j x1 x2 x3 xn squared wirelength 4 C X X If x is a partition vector T Therefore we want to minimize x Qx 01 14 19 8 Property of Laplacian Matrix Cont d 2 Q is symmetric and semi definite i e i x Qx T Q x x ij i j 0 ii all eigenvalues of Q are

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