Quadratic VLSI Placement

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Quadratic VLSI Placement

• Manolis Pantelias

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General

• Various types of VLSI placement
• Simulated-Annealing
• Quadratic or Force-Directed
• Min-Cut
• Nonlinear Programming
• Mix of the above
• Quadratic placers
• Try to minimize a quadratic wirelength objective
  function
• Indirect measure of the wirelength but can be
  minimized quite efficiently
• Results in a large amount of overlap among cells
• Additional techniques needed

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Proud

• R. S. Tsay, E. Kuh and C. P. Hsu, Proud A
  Sea-of-Gates Placement Algorithm

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Objective function

• Dont minimize the wirelength but the squared
  wirelength

• cij between module i and module j could be the
  number of nets connecting the two modules
• Connectivity matrix C cij
• B D C
• D is a diagonal matrix with
• Only the one-dimensional problem needs to be
  considered because of the symmetry between x and y

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Electric Network Analogy

• Objective function xTBx can be interpreted as the
  power dissipation of an n-node linear resistive
  network
• Vector x corresponds to the voltage vector
• x x1 x2T, x1 is of dimension m and is to be
  determined, x2 is due to the fixed I/O pads
• B contains the conductance of the nodes
• bij is the conductance between node i and node j

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Electric Network Analogy (Contd)

• Placement problem is equivalent to that of
  choosing voltage vector for which power is a
  minimum
• B11x1 B12x2 0
• B21x1 B22x2 i2
• Solve for Ax1 b
• Where A B11, b - B12x2
• Solved with iterating method (successive
  over-relaxation or SOR)

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Placement and Partitioning

• How to avoid module overlap?
• Solution Iterative partitioning in a
  hierarchical way
• Add module areas from left to right until roughly
  half of the total area, that defines
  partition-line
• Make modules to the right of partition-line
  fixed, modules to the left of partition-line
  movable
• Project fixed modules to center-line
• Perform global placement in the left-plane (of
  center-line)

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Placement and Partitioning (Contd)

• Make all modules in the left-plane fixed, project
  to the center line
• Make modules to the right of cut-line movables
• Perform global placement in the right-plane
• Proceed with horizontal cuts on each half
• Continue until each block contains one and only
  module

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BGS Iteration

• For a given hierarchy perform the previous
  partitioning more than once
• Assume a vertical cut
• y1 splits into y1a, y1b for each half

• y1a and y1b are perturbed solution from y1a and
  y1b because of the partitioning process
• 2-5 iterations give very good results at each
  hierarchy

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Problems

• A bad decision at a higher level affects the
  placement results of lower levels
• Difficult to overcome a bad partitioning of a
  higher level
• Backtracking?

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KraftWerk

• Hans Eisenmann and F. Johannes,
• Generic Global Placement and Floorplanning

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Force Directed Approach

• Transform the placement problem to the classical
  mechanics problem of a system of objects attached
  to springs.
• Analogies
• Module (Block/Cell/Gate) Object
• Net Spring
• Net weight Spring constant.
• Optimal placement Equilibrium configuration

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An Example
Resultant Force
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Force Calculation

• Hookes Law
• Force Spring Constant x Distance
• Can consider forces in x- and y-direction
  separately

(xj, yj)
F
Fx
(xi, yi)
Fy
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Problem Formulation

• Equilibrium Sj cij (xj - xi) 0 for all module
  i.
• However, trivial solution xj xi for all i, j.
  Everything placed on the same position!
• Need to have some way to avoid overlapping.
• Have connections to fixed I/O pins on the
  boundary of the placement region.
• Push cells away from dense region to sparse region

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Kraftwerk Approach

• Iteratively solve the quadratic formulation
• Spread cells by additional forces
• Density-based force proposed
• Push cells away from dense region to sparse
  region
• The force exerted on a cell by another cell is
  repelling and proportional to the inverse of
  their distance.
• The force exerted on a cell by a place is
  attracting and proportional to the inverse of
  their distance

// equivalent to spring force // equilibrium
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Some Details

• Let fi be the additional force applied to cell i
• The proportional constant k is chosen so that the
  maximum of all fi is the same as the force of a
  net with length K(WH)
• K is a user-defined parameter
• K0.2 for standard operation
• K1.0 for fast operation
• Can be extended to handle timing, mixed block
  placement and floorplanning, congestion,
  heat-driven placement, incremental changes, etc.

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Some Potential Problems of Kraftwerk

• Convergence is difficult to control
• Large K ? oscillation
• Small K ? slow convergence
• Example
• Layout of a multiplier
• Density-based force is expensive
• to compute

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FastPlace

• Natarajan Viswanathan and Chris Chu, FastPlace
  Efficient Analytical Placement using Cell
  Shifting, Iterative Local Refinement and a Hybrid
  Net Model

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FastPlace Approach

• Framework
• repeat
• Solve the convex quadratic program ?
• Reduce wirelength by iterative heuristic ?
• Spread the cells ?
• until the cells are evenly distributed ?
• Special features of FastPlace
• Cell Shifting
• Easy-to-compute technique ?
• Enable fast convergence ?
• Hybrid Net Model
• Speed up solving of convex QP ?
• Iterative Local Refinement
• Minimize wirelength based on linear objective ?

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Cell Shifting

1. Shifting of bin boundary

Uniform Bin Structure
Non-uniform Bin Structure

1. Shifting of cells linearly within each bin

• Apply to all rows and all columns independently

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Cell Shifting Animation
NBi
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Pseudo pin and Pseudo net
Pseudo pin

• Need to add forces to prevent cells from
  collapsing back
• Done by adding pseudo pins and pseudo nets
• Only diagonal and linear terms of the quadratic
  system need to be updated
• Takes a single pass of O(n) time to regenerate
  matrix Q (which is common for both x and y
  problems)

Pseudo net
Pseudo pin
Additional Force
Pseudo net
Target Position
Original Position
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Iterative Local Refinement

• Iteratively go through all the cells one by one
• For each cell, consider moving it in four
  directions by a certain distance
• Compute a score for each direction based on
• Half-perimeter wirelength (HPWL) reduction
• Cell density at the source and destination
  regions
• Move in the direction with highest positive score
  
• (Do not move if no positive score)
• Distance moved (H or V) is
• decreasing over iterations
• Detailed placement is handled
• by the same heuristic

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Clique, Star and Hybrid Net Models

• Runtime is proportional to of non-zero entries
  in Q
• Each non-zero entry in Q corresponds to one 2-pin
  net
• Traditionally, placers model each multi-pin net
  by a clique
• Hybrid Net Model is a mix of Clique and Star
  Models

pins Net Model
2 Clique
3 Clique
4 Star
5 Star
6 Star

Star Node
Hybrid Model
Clique Model
Star Model
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Equivalence of Clique and Star Models

• Lemma By setting the net weights appropriately,
• clique and star net models are
  equivalent.
• Proof When star node is at equilibrium position,
• total forces on each cell are the same for
• clique and star net models.

Star Node
Weight ?W
Weight ? kW for a k-pin net
Clique Model
Star Model
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Comparison

• FastPlace is fast compared to Kraftwerk (based on
  published data)
• 20-25x faster
• With cell shifting technique can converge in
  around 20 iterations.
• KraftWerk may need hundreds of iterations to
  converge.
• 10 better in wirelength
• hybrid net model