Analytic Placement Algorithms
Chung-Kuan Cheng
CSE Department, UC San Diego, CA 92130
Contact: [email protected]
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Outline
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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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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.
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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 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}
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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: 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
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Density Distribution
Mass transport formulation: Given a map and its mass
density, transport the mass evenly to the whole map
1. Min sum_i |xi-yi|b
2. Constraint: new mass density is a constant
xi 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.
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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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