1. CALTECH CS137 Fall2005 -- DeHon 1 CS137: Electronic Design Automation Day 17: November 11, 2005 Placement (Simulated Annealing…)

2. CALTECH CS137 Fall2005 -- DeHon 2 Today Placement Improving Quality –Avoiding local minima Techniques: –Simulated Annealing –Exhaustive (Branch-and-bound)

3. CALTECH CS137 Fall2005 -- DeHon 3 Simulated Annealing Physically motivated approach Physical world has similar problems –objects/atoms seeking minimum cost arrangement –at high temperature (energy) can move around –at low temperature, no free energy to move –cool quickly  freeze in defects (weak structure) –cool slowly  allow to find minimum cost

4. CALTECH CS137 Fall2005 -- DeHon 4 Key Benefit Avoid Local Minima –Allowed to take locally non-improving moves in order to avoid being stuck

5. CALTECH CS137 Fall2005 -- DeHon 5 Simulated Annealing At high temperature can move around –not trapped to only make “improving” moves –free energy from temperature allows exploration of non-minimum states –avoid being trapped in local minima As temperature lowers –less energy to take big, non-minimizing moves –more local / greedy moves

6. CALTECH CS137 Fall2005 -- DeHon 6 Design Optimization Components: 1.“Energy” (Cost) function to minimize –represent entire state, drives system forward 2.Moves –local rearrangement/transformation of solution 3.Cooling schedule –initial temperature –temperature steps (sequence) –time at each temperature

7. CALTECH CS137 Fall2005 -- DeHon 7 Basic Algorithm Sketch Pick an initial solution Set temperature (T) to initial value while (T>T min ) –for time at T pick a move at random compute  cost if less than zero, accept else if (RND<e -  cost/T ), accept –update T

8. CALTECH CS137 Fall2005 -- DeHon 8 Details Initial Temperature –T 0 =  avg/ln(P accept ) Cooling schedule –fixed ratio: T= T (e.g. =0.85) –temperature dependent –function of both temperature and acceptance rate example to come Time at each temperature –fixed number of moves? –fixed number of rejected moves? –fixed fraction of rejected moves?

9. CALTECH CS137 Fall2005 -- DeHon 9 Cost Function Can be very general –Combine area, timing, energy, routability… Should drive entire solution in right direction –reward each good move Should be cheap to compute delta costs –e.g. FM –Ideally O(1)

10. CALTECH CS137 Fall2005 -- DeHon 10 Example Cost Functions Total Wire Length –Linear, quadratic… Bounding Box (semi-perimeter) –Surrogate for routed net length Channel widths –probably wants to be more than just width Cut width bb y bb x

11. CALTECH CS137 Fall2005 -- DeHon 11 Bad Cost Functions Update cost –rerun maze route on every move –rerun timing analysis –recalculate critical path delay Drive toward solution: –size < threshold ? –Critical path delay

12. CALTECH CS137 Fall2005 -- DeHon 12 VPR Wire Costs VPR Bounding Box Swartz, Betz, & Rose FPGA 1998 Original table: Cheng ICCAD 1994

13. CALTECH CS137 Fall2005 -- DeHon 13 VPR Timing Costs Criticality(e)=1-Slack(e)/Dmax TCost(e)=Delay(e)*Criticality(e) CriticalityExp Keep all edge delays in a table Recompute Net Criticality at each Temperature Marquardt, Betz, & Rose FPGA2000

14. CALTECH CS137 Fall2005 -- DeHon 14 VPR Balance Wire and Time Cost Marquardt, Betz, & Rose FPGA2000

15. CALTECH CS137 Fall2005 -- DeHon 15 Initial Solution Spectral Placement Random Constructive Placement –Fast placers start at lower temperature; assume constructive got global right.

16. CALTECH CS137 Fall2005 -- DeHon 16 Moves Swap two cells –Within some distance limit? (ex. to come) swap regions –…rows, columns, subtrees rotate cell (when feasible) flip (mirror) cell permute cell inputs (equivalent inputs)

17. CALTECH CS137 Fall2005 -- DeHon 17 Variant Allow non-legal solutions –capture badness in cost function –E.g. -- allow cells to overlap Just make sure cost function makes very expensive as cool – settle out to legal solutions

18. CALTECH CS137 Fall2005 -- DeHon 18 Variant: “Rejectionless” Order moves by cost –compare FM Pick random number first Use random to define range of move costs will currently accept Pick randomly within this range Idea: never pick a costly move which will be rejected

19. CALTECH CS137 Fall2005 -- DeHon 19 Theory If stay long enough at each cooling stage –will achieve tight error bound If cool long enough –will find optimum

20. CALTECH CS137 Fall2005 -- DeHon 20 Practice Good results –ultimately, what most commercial tools use...what vpr uses… Slow convergence Tricky to pick schedules to accelerate convergence

21. CALTECH CS137 Fall2005 -- DeHon 21 Range Limit Want to tune so accepting 44% of the moves – Lam and Delosme DAC 1988 VPR –Define Rlimit – defines maximum  x and  y accepted –Tune Rlimit to maintain acceptance rate –Rlimit new =Rlimit old ×(1-0.44+  )  is measured acceptance rate

22. CALTECH CS137 Fall2005 -- DeHon 22 VPR Cooling Schedule Moves at Temperature = cN 4/3 Temperature Update –Tnew=Told×  –Idea: advance slowly in good  range Betz, Rose, & Marquardt Kluwer 1999

23. CALTECH CS137 Fall2005 -- DeHon 23 Range Limiting? Eguro alternate [DAC 2005] – define P=D -M –Tune M to control 

24. CALTECH CS137 Fall2005 -- DeHon 24 Range Limiting Eurgo, Hauck, & Sharma DAC 2005

25. CALTECH CS137 Fall2005 -- DeHon 25 Big Hammer Costly, but general Works for most all problems –(part, placement, route, retime, schedule…) Can have hybrid/mixed cost functions –as long as weight to single potential –(e.g. wire/time from VPR) With care, can attack multiple levels –place and route Ignores structure of problem –resignation to finding/understanding structure

26. CALTECH CS137 Fall2005 -- DeHon 26 Optimal/Exhaustive

27. CALTECH CS137 Fall2005 -- DeHon 27 If you run simulated annealing long enough…. –It should converge to optimum If you have enough monkeys typing at keyboards keyboards long enough –They’ll eventually produce the works of Shakespeare….

28. CALTECH CS137 Fall2005 -- DeHon 28 Brute Force? If you are really going to give up on structure and explore the entire space –…there are more efficient ways to do it …and maybe they’re not terrible –For small/modest problems –As our computers get faster

29. CALTECH CS137 Fall2005 -- DeHon 29 Optimum Placement Simplest case: –Gate/array (FPGA) w/ fixed cell locations N locations M cells Try all permutations of N choose M N!/M! cases

30. CALTECH CS137 Fall2005 -- DeHon 30 Improving Prune off symmetry cases –Rotate 90, 180, 270 –Mirror X, Y, XY Reject provably bad starts –(return to in a minute)

31. CALTECH CS137 Fall2005 -- DeHon 31 Exhaustive Placement More general: –Modules have variable size –Modules can be rotated/flipped… To explore all cases: –For each module For each orientation

32. CALTECH CS137 Fall2005 -- DeHon 32 Branch-and-Bound Consider dense 1D placement –Search space of all placements –Tree branch is choice of logical cell for physical cell position –Keep track of best solution so far as reach leaves –If partial solution worse than best solution, prune branch

33. CALTECH CS137 Fall2005 -- DeHon 33 Pruning: 1D example Reducing channel width –Have solution with width=10 –When find a partial solution with width>=10 Can abort that branch Reducing Delay –Have solution with delay=20 –When find a partial solution with delay>=20 Can abort branch

34. CALTECH CS137 Fall2005 -- DeHon 34 Viable Only on small problems –But “small” growing with machine speed Use for end-case in constructive –Flatten bottom of hierarchy –Maybe even in iteration/overlap relaxation

35. CALTECH CS137 Fall2005 -- DeHon 35 Caldwell et. al. Results [TRCAD v19n11p1304-13]

36. CALTECH CS137 Fall2005 -- DeHon 36 Runtime [TRCAD v19n11p1304-13]

37. CALTECH CS137 Fall2005 -- DeHon 37 Runtime [TRCAD v19n11p1304-13]

38. CALTECH CS137 Fall2005 -- DeHon 38 Faster? Accounting and gain complexity –Make it linear time –But does make each update somewhat complex –Exhaustive case less bookkeeping Not an atypical result…

39. CALTECH CS137 Fall2005 -- DeHon 39 Summary Simulated Annealing –use randomness to explore space –accept “bad” moves to avoid local minima –decrease tolerance over time General purpose solution –costly in runtime Small (sub)problems –May solve exhaustively –Can prune to accelerate…

40. CALTECH CS137 Fall2005 -- DeHon 40 Admin Class Monday …but not W, F (finish assignment)

41. CALTECH CS137 Fall2005 -- DeHon 41 Big Ideas: Use randomness to explore large (non- convex) space –Simulated Annealing Use dominance to quickly skip over obviously bad solutions –Branch and Bound