Modern Floor-planning Based on B∗-Tree and Fast Simulated Annealing

1. 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

2. 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

3. What is floor-planning?

4. What is floor-planning?

5. What is floor-planning? • A schematic representation of the major blocks of a hardware design • Applies for both ASIC and FPGA designs

6. What is floor-planning? A sliceable floor-planning A non- sliceable floor-planning

7. 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

8. 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

9. 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.

10. B* tree representation Example:

11. 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

12. MCMC M-H

13. MCMC M-H

14. 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,}

15. Simulated Annealing -

16. 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

17. Aspect ratio Denote by the (user defined) maximal percentage of dead space. Then: =1 =2 =3 =4

18. 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

19. Simulated Annealing for floor-planning • 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

20. Fast Simulated Annealing for floor-planning

21. 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

22. 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

23. 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

24. SA algorithms for floor-planning comparison

25. SA algorithms for floor-planning comparison

26. The importance of α

27. 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

28. 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

29. 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

30. Simulated Tempering for floor-planning Classical SA ST • 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 Timber Wolf SA

31. 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

32. 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 onPattern Analysis and Machine Intelligence(1984) pp 721-741.‏