Modern Floor-planning Based on B∗-Tree

and Fast Simulated Annealing

Paper by Chen T. C. and Cheng Y. W (2006)

Presented by Gal Itzhak

22.7.15

Agenda

Introduction

What is a Floor-Planning?

Motivation for global floor-planning optimization

B* tree blocks representation

Simulated annealing

Classic simulated annealing for floor-planning optimization

Fast and adaptive simulated annealing for floor-planning optimization

An addition and comparison

A generalization and comparison: Simulated Tempering

What is floor-planning?

What is floor-planning?

What is floor-planning?

A schematic representation of the major blocks of a hardware design

Applies for both ASIC and FPGA designs

What is floor-planning?

A sliceable floor-planning

A non- sliceable floor-planning

Motivation

Modern chips and hardware designs complexity is growing dramatically:

Smaller transistor widths

Faster clocks

Much more logics and memory

⇒ Less margin for floor-planning ⇒ global optimization is required

I’ll use the floor-planning optimization problem to present some important stochastic global optimization algorithms

All make use of the simulated annealing framework

B* tree representation

A generalization of the well known binary tree

Does not limit the number of children of a node to 2

provides a unique and convenient representation of both sliceable and non-sliceable floor-planning

For a sliceable floor-planning: reduced to a binary tree

B* tree representation

Basic rules (similar to DFS in some manner):

1. set the lower left block as the root.

recursively build:

2. the adjacent right hand side set of blocks as the left child.

3. the lowest block above as the right child.

B* tree representation

Example:

B* tree representation

Adjacent B* trees may differ by:

A single block rotation

A single block (node) move

As single swap of blocks (nodes)

Resize a single soft block

MCMC M-H

MCMC M-H

Simulated Annealing

The most common stochastic global optimization method

Uses MCMC techniques (Metropolis-Hastings sampler) to optimize and evaluate some cost function C(x)

Sampling is taken from the Boltzman distribution over the state space:

The proposal distribution is usually uniform:

Thus, for a fixed temperature we get an acceptance probability of:

}=min{1, }

Simulated Annealing -

Simulated Annealing for floor-planning

We define the following cost function:

Where A and W are the total planning area and wire length respectively

is the required aspect ratio of the fixed outline

Aspect ratio

Denote by the (user defined) maximal percentage of dead space. Then:

=1= 2= 3= 4

Simulated Annealing for floor-planning

Sampling Algorithm:

Set

Do

Pick a neighbor state y

Accept a move to state y, w.p.

Check the new solution cost and update the best so far

Update temperature

Until converged or cooled down

Return the best solution

Obviously, the cooling schedule of T has a dramatic impact on the algorithm’s performance:

Cooling too fast would fail to optimize globally

Cooling too slow would result in very long running time

Common schedules are (0<<1) or decay rate 1/k

[GG84] suggests a logarithmic schedule, though in practice it’s too moderate

Simulated Annealing for floor-planning

Fast Simulated Annealing for floor-planning

Fast Simulated Annealing for floor-planning

1. High temperature random search- accepts inferior solutions and avoid getting trapped in a local minimum

2. Low temperature local search- converge to some local minimum with high probability

3. Up-hill climbing search- instantly increase temperature to allow acceptance of uphill solutions. Then slowly cool to converge.

As the two first stages usually require only a few iterations, one may allow to devote the spare time to the third stage

Fast and adaptive Simulated Annealing for floor-planning

Suppose now, for simplicity, that

Choosing the right α beforehand is impractical

Trying all possible α’s is inefficient

𝐶 (𝑥 )=𝛼 ∙ 𝐴+(1−𝛼 ) ∙ (𝑅−𝑅∗ )2

Fast and adaptive Simulated Annealing for floor-planning

Sampling Algorithm:

Set

Do

Pick a neighbor state y (perturb the current b* tree)

Accept a move to state y, w.p.

Check the new solution cost and update the best one so far

Adapt the weights in the cost function C(x)

Update temperature: according to the fast SA schedule

Until converged or cooled down

Return the best solution

SA algorithms for floor-planning comparison

SA algorithms for floor-planning comparison

The importance of α 𝐶 (𝑥 )=𝛼 ∙ 𝐴+(1−𝛼 ) ∙ (𝑅−𝑅∗ )2

A generalized variation: Simulated Tempering

Looking to accelerate optimization convergence time, applications dictated a non-logarithmic cooling schedule

Therefore, convergence to the global minimum is no longer guaranteed with high probability [GG84]

Simulated Tempering: let T be a RV chosen from a finite set of temperatures: where

for each

Simulated Tempering

In ST the markov chain state space is augmented to (x,i), where i indicates the random temperature.

An adjacent temperature transition or is accepted according to the Metropolis-Hastings method:}=}

The proposal distribution is

Observe that unlike SA, the temperature may also increase

Simulated Tempering for floor-planning

Sampling Algorithm:

Set

While stopping criteria not met

Run S iterations of M-H or Gibbs. At each, accept transition to state y (which is a perturbation of the current B* tree) w.p. that was previously mentioned. Keep the best solution.

Propose a move to an adjacent temperature by

Accept the proposal by

Return the best solution

Simulated Tempering for floor-planning Classical

SA ST

Timber Wolf SA It seems that ST provides a better solution than

SA in terms of wirelength, for roughly similar running time

Timber wolf fails to outperform the classical SA 

To conclude

The notion of floor-planning, its importance, and stochastic methods for its construction optimization have been introduced (all based on the SA framework)

On the on hand, the method of Fast and Adaptive SA was shown, resulting in classical SA W* and A* results; however obtained much quicker

On the other, the Simulated Tempering algorithm induced a better solution in terms of W*, for similar running time

References

Chen T. C. and Cheng Y. W. “Modern Floorplanning Based on B∗-Tree and Fast Simulated Annealing.” IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 25, NO. 4, APRIL 2006, pp 637-650.

Cong J. et al. “Relaxed simulated tempering for VLSI floorplan designs.” Design Automation Conference, 1999. Proceedings of the ASP-DAC ’99. Asia and South Pacific.

Geman, Stuart, and Donald Geman. “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images." IEEE Transactions on Pattern Analysis and Machine Intelligence (1984) pp 721-741.