Analytic Placement Algorithms

1. Analytic Placement Algorithms Chung-Kuan Cheng CSE Department, UC San Diego, CA 92130 Contact: ckcheng@ucsd.edu

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

3. Introduction • Analytic Placement • Obj: length + density distr + timing + routing congestion • Nonlinear Programming Algorithms • Convex Space • Density Distribution • Mass transportation

4. 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

5. Introduction Global 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.

6. 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.

7. Nesterov’s Method Minimize 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}

8. 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}

9. Nesterov’s Method: Algorithm Initial: 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

10. Density Distribution Mass transport formulation: Given a map and its mass density, transport the mass evenly to the whole map • Min sum_i |xi-yi|b • Constraint: new mass density is a constant xi location of mass i yi new location of mass i

11. 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.

12. 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.

13. Thank You!