Analytic Placement Algorithms

Chung-Kuan ChengCSE Department, UC San Diego, CA 92130

Contact: ckcheng@ucsd.edu

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Outline

• Introduction• Nesterov’s Method for Convex Space• Density Distribution• Remarks

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Introduction

• Analytic Placement• Obj: length + density distr + timing + routing

congestion• Nonlinear Programming Algorithms– Convex Space

• Density Distribution– Mass transportation

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Convex Optimization: min f(X)• Newton’s Method: Second ordered method– Find F(X)= df(X)/dX= 0– Xk

= Xk-1 - dF(X)/dX|X=Xk-1-1 F(Xk-1)

• Krylov Space Method: First ordered method– Gradient Descent– Conjugate Gradient– Nesterov’s Method

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IntroductionGlobal Rate of Convergence: Let k be the number of iterations.•Newton method: : O(L/k2)-O(L/k3)•Gradient method: O(L/k)•Quasi-Newton or conjugate gradient: Not better or even worse. (Y.L. Yu, Alberta)•Nesterov’s method: O(L/k2), the order is the optimum for first order approaches.

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Introduction• Nesterov: Three gradient projection methods

published in 1983, 1988, 2005.• Beck & Teboulle: FISTA, a proximal gradient

version in 2008.• Nesterov: basic book in 2004.• Tseng: overview and unified analysis in 2008.

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Nesterov’s MethodMinimize f(X) under certain constraints, where f(X) and constraints are convex functions satisfying Lipshitz condition.•Convex function– f(X)>= f(Y)+ grad f(Y)(X-Y)

•Lipshitz condition: there exists a constant a – |grad f(X) - grad f(Y)| <= a|X-Y|

•Definition– L(X,Y)= f(Y)+ grad f(Y)(X-Y) + 0.5a |X-Y|2

– P(Y)= min X { L(X,Y), X is feasible} 7

Nesterov’s Method: definitions• Set QL(Y)= Y-1/a grad f(Y)

• L(QL(Y),Y)=f(Y)-0.5a |QL(Y)-Y|2

=f(Y)-0.5/a|grad f(Y)|2

• Lemma: f(QL(Y))-f(Z) >= 0.5a {|Z-Y|2-|Z-QL(Y)|2}

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Nesterov’s Method: AlgorithmInitial: Y1=X0, t1= 1

Step (k>0)•Xk=P(Yk)

•tk+1= ½{1+(1+4tk2)½}

•Yk+1=Xk+(tk-1)/tk+1 (Xk –Xk-1)

Lemma: tk>= 0.5 (k+1)

Theorem: f(Xk)-f(X*)<= 2a |X0-X*|2/(k+1)2

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Density DistributionMass transport formulation: Given a map and its mass density, transport the mass evenly to the whole map1.Min sum_i |xi-yi|b

2.Constraint: new mass density is a constantxi location of mass i

yi new location of mass i

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Density Distribution: Algorithm• Linear assignment: High complexity• Min cost flow: Linear cost• Algorithm:– Input: mass density with mass locations xi: D(X)

– Derive 2D Fourier transform, D(w), of the mass– Do inverse transform on -jwD(w) which is the

force to move to the new locations. The solution is: f(X)= grad -D(X).

• Property: curl f(X)= 0.11

Summary• Nesterov’s method has been successfully applied to

different fields, e.g. compressed sensing. No report on the placement yet.

• Mass transport is heavily studied in image processing. The gradient can be derived from Fourier transform.

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Thank You!

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