Multilevel Optimization for LargeScale Circuit Placement

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Multilevel Optimization for Large-Scale Circuit
Placement

• Tony F. Chan, Jason Cong, Tianming Kong,
  Joseph R. Shinnerl

Mathematics Department Computer Science
Department, University of California, Los
Angeles, CA 90095 Research Supported by Intel,
SRC, NSF.
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The Circuit Placement Problem

• Given
• N circuits, a.k.a blocks, modules, or cells
• A rectangle (the chip) in which the circuits
  must be placed without overlapping
• Connectivity specs (netlist)
• Constraints, e.g., timing, heat dissipation,
  routability
• Problem Find an arrangement of the circuits on
  the chip that minimizes total wirelength subject
  to all constraints above.
• Difficulty Modeling all O(N2) constraints when
  104 ? N ? 107.

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Overview

• Challenges for Circuit Placement
• Large Design Sizes (over one million moveable
  objects)
• Complex Design Constraints (delay, noise,
  manufacturability, etc.)
• Existing methodologies are inadequate.
• Simulated Annealing-based methods can handle
  complex design constraints, but their runtime and
  quality scale poorly.
• Quadratic Programming-based methods are very
  efficient, but they cannot handle complex
  constraints well.
• We have implemented a fast placement engine
  capable of handling complex constraints and
  producing good placements.

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Key Components of Our Method

• Nonlinear Programming Formulation
• Multilevel Decomposition
• recursive clustering and declustering
• continuous and discrete refinement
• interior-point solution at coarsest level
• truncated preconditioned conjugate gradients on
  the Newton Equations
• Fast Multipole Method for nonoverlap constraints
• Legalization and Postprocessing (Domino)

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Hypergraph modelA NET is a subset of
(interconnected) CELLS.A hyperedge is a subset
of vertices (its like a clique).
Cell 1
Cell 4
Cell 3
Cell 5
Cell 2
Problem arrange the cells to minimize total
wirelength.
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Clique modelReplace hyperedges by cliques
collapse edgeswith common endpoints.
Cell 1
1/3
Cell 4
Cell 3
1
1/3
1/3
Cell 5
Cell 2
Problem arrange the cells to minimize total
wirelength.
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Clique Model

• Replace each hyperedge i by a clique. Weight each
  edge in the clique by bi 2/(ni(ni-1)), where
  ni is the clique size.
• Collapse edges with the same endpoints into
  single edges, summing weights bi .
• The length of each edge is then weighted by this
  sum.

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Nonoverlap Constraints
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Nonlinear Programming Formulation

• min f(x)
• subject to c(x) ? 0 (NP)
• where x?Rn.
• f Rn?R objective function
  (n?2N or 3N)
• c Rn?Rm constraint functions (m
  ?N(N-1)/2 N)
• F?x?Rn c(x) ?0 feasible region
• x local solution to
  NP (KKT conditions)
• Assumption f and c are smooth
• Difficulty active set A?ici(x)0 is
  unknown

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Multilevel Framework
Final Fine-Grain Problem. Slot Assignment,
Discrete Refinement, DOMINO
Initial Fine-Grain Problem
Initial Fine-Grain Problem
MESC Clustering
Decluster
Intermediate Level Discrete Refinement Only
Intermediate Level
etc.
Decluster
MESC Clustering
etc.
Decluster
MESC Clustering
Intermediate Level Continuous and
Discrete Refinement
Intermediate Level
Decluster
MESC Clustering
Coarse-Grain Problem Solve Nonlinear Programming
Problem with Interior-Point Method.
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MESC--Multilevel Edge Separability Clustering

• Edge separability q of edge (s,t) approximates
  the s-t min-cut of G.
• Using CAPFOREST (SIDMA 1992), we can approximate
  q for all edges in N log N time.
• MESC ranks edge e (s,t) as r(e)
  q(e)/min(area(s),area(t))in order to balance
  cluster sizes.
• Cell locations are not used.

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Interpolation (P) and Restriction (PT)

• Suppose cells x1h and x2h are clustered together
  to form cluster x1H. Declusteringis trivial
  x1h x2h x1H .
• Interpolation is simply xh PxH, wherePij 1
  iff xih belongs to cluster xjH and 0
  otherwise.
• f(xh) (1/2)xhTQxh bTx
  (1/2)xHTPTQPxH bTPxH fH(xH)

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Declustering and Slot Assignment

• Linear interpolation places cluster components
  concentrically at the cluster center.
• Linear assignment is used to distribute these
  concentric components to nearby locations.

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Interior-Point Methods for Nonlinear
ProgrammingConstruct xk such that c(xk)gt0 for
all k and xk?x.
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The (Dynamically Weighted) Slack Variable
Formulation

• Initially, cells may overlap arbitrarily. As
  ? is reduced, however, the amount of overlap
  allowed is also reduced. Once ? decreases below
  zero, all constraints are satisfied.

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Example Test Run on Circuit sioo

• Initial Configuration (all cells along upper
  boundary)
• Next page subsequent major iteration
  configurations, from upper left to lower right.
  Initially, cells congregate in clusters. Then
  they rush'' to the boundary of the chip.
  Gradually, they move away from each other and
  from the boundary.

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Test Run on Circuit sioo (contd)
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Linesearch Algorithms

• Given initial iterate x x0
• repeat
• Calculate search direction p
• Calculate steplength ?
• Update x ? x ? p
• until done
• Notation
• g ? ?f Rn?Rn Gradient of f
• Hf Rn?Rn ? n Hessian of
  f
• J ? Dc Rn ?Rm ? n Jacobian of c

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Solving the Barrier subproblem

• Optimality conditions
• Newton-Barrier equations
• where
• Problem
• Solution

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The Fast Multipole Method (FMM)

• Designed to accelerate evaluation of decaying
  potential fields ?(r) ??i/ri used in
  computational particle simulations in astronomy,
  plasma physics, fluid dynamics, chemistry, etc.
• Recently applied successfully to large-scale
  computational problems in linear algebra and VLSI
• Can be used to evaluate the n2/2 non-overlap
  constraints in O(n) time

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Keys to FMM

• The contribution to the barrier function
  associated with the ith constraint is a sum
• To evaluate m sums of this form in O(pm)O(pn)
  time,FMM uses
• an efficient strategy for clustering particles
  according to adaptive, hierarchical partitions
  (the more distant the cluster, the larger it can
  be),
• exact formulas for merging clusters and
  recentering multipole expansions.

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Fast Evaluation of ?B?(x) and H?(x)v

• FMM is also applicable to potentials obtained
  from derivatives of the logarithm.
• Each component of ?B?(x) has the form
• Products H?(x)v are approximated by

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Discrete Refinement

• Combination with continuous optimization
  dramatically improves placement quality.
• Localized i.e., restricted to small subsets of
  cells.
• Current version is based on Goto 1981
  ?-neighborhoods and (randomized) ?-exchange
• The ?-neighborhood of location (i,j) is all
  locations (m,n) such that i-m j-n lt ? .
• ? denotes the total number of ?-neighborhoods
  we consider at each iteration.

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Discrete RefinementExample ? 1, ?3

• Find As optimal location assuming all cells
  except A are held fixed (relaxation). Suppose
  As optimal location is currently occupied by B.
• Pick an ?-neighbor of B at random, say D.
• Find Ds optimal location by relaxation,
  occupied by G, say.
• Pick an ?-neighbor of G at random, say K.
• Select the optimal permutation of A, D, and K.

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Test Results for mPL

• Sun UltraSparc-2/168MHz, 512MB memory
• 4 largest circuits from 1993 MCNC suite, all 18
  circuits from ISPD98 suite.
• Even though we minimize quadratic wirelength, we
  still compute the half-perimeter result for
  comparison with Gordian-LDomino.

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Gordian-L vs. mPL (ratios)
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Impact of Nonlinear Programmingon Circuit Biomed
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Increasing the Impact of Nonlinear Programming

• Adopt a floorplanning formulation at the coarsest
  levels correctly account for size differences of
  cell clusters at coarser levels
• allow rectangular cells of variable aspect ratio
• approximate rectangular cells by ellipsoids for
  near-field nonoverlap constraints
• hierarchical bin structure for far-field
  nonoverlap
• Smaller nonlinear programming problems can be
  solved to greater accuracy using off-the-shelf
  interior-point software.

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Comparison on Sun-Ultra60 360 MHz / 1024 MBwith
ITOOLS 4.0 (Timberwolf)
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ITOOLS Comparison RATIOS(Sun Ultra60
360MHz/1024MB, continued)
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Appendix 1 Basic Conjugate Gradients (CG)
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Appendix 2 GORDIAN (ca.1990)

• 1) Minimize quadratic wirelength subject to a
  single center-of-mass constraint.
• 2) Partition the result into two blocks.Add 2
  new center-of mass constraints, one for each
  block.
• 3) Minimize the same quadratic wirelength function
  subject to the new set of constraints.
• 4) Repeat step 2 on each block and continue,
  repeatedly adding new constraints at each step.