Analytic Placement Algorithms

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Analytic Placement Algorithms

• Chung-Kuan Cheng
• CSE Department, UC San Diego, CA 92130
• Contact ckcheng_at_ucsd.edu

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Outline

• Introduction
• Nesterovs 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)

• Newtons Method Second ordered method
• Find F(X) df(X)/dX 0
• Xk Xk-1 - dF(X)/dXXXk-1-1 F(Xk-1)
• Krylov Space Method First ordered method
• Gradient Descent
• Conjugate Gradient
• Nesterovs 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)
• Nesterovs 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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Nesterovs Method

• Minimize f(X) under certain constraints, where
  f(X) and constraints are convex functions
  satisfying Lipshitz condition.
• Convex function
• f(X)gt f(Y) grad f(Y)(X-Y)
• Lipshitz condition there exists a constant a
• grad f(X) - grad f(Y) lt aX-Y
• Definition
• L(X,Y) f(Y) grad f(Y)(X-Y) 0.5a X-Y2
• P(Y) min X L(X,Y), X is feasible

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Nesterovs Method definitions

• Set QL(Y) Y-1/a grad f(Y)
• L(QL(Y),Y)f(Y)-0.5a QL(Y)-Y2
• f(Y)-0.5/agrad f(Y)2
• Lemma
• f(QL(Y))-f(Z) gt 0.5a Z-Y2-Z-QL(Y)2

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Nesterovs Method Algorithm

• Initial Y1X0, t1 1
• Step (kgt0)
• XkP(Yk)
• tk1 ½1(14tk2)½
• Yk1Xk(tk-1)/tk1 (Xk Xk-1)
• Lemma tkgt 0.5 (k1)
• Theorem f(Xk)-f(X)lt 2a X0-X2/(k1)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
• Min sum_i xi-yib
• 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

• Nesterovs 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!