CS137: Electronic Design Automation Day 10: February 6, 2002 - - PDF document

▶
cs137 electronic design automation
SMART_READER_LITE
LIVE PREVIEW

CS137: Electronic Design Automation Day 10: February 6, 2002 - - PDF document

Aug 31, 2023 275 likes •445 views

CS137: Electronic Design Automation Day 10: February 6, 2002 Placement (Simulated Annealing) CALTECH CS137 Winter2002 -- DeHon Today Placement Improving Quality Avoiding local minima Techniques: Simulated Annealing

slide-1
SLIDE 1

1

CALTECH CS137 Winter2002 -- DeHon

CS137: Electronic Design Automation

Day 10: February 6, 2002 Placement (Simulated Annealing…)

CALTECH CS137 Winter2002 -- DeHon

Today

  • Placement
  • Improving Quality

– Avoiding local minima

  • Techniques:

– Simulated Annealing – Exhaustive (Branch-and-bound)

slide-2
SLIDE 2

2

CALTECH CS137 Winter2002 -- DeHon

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

CALTECH CS137 Winter2002 -- DeHon

Key Benefit

  • Avoid Local Minima

– Allowed to take locally not improving moves in order to avoid being stuck

slide-3
SLIDE 3

3

CALTECH CS137 Winter2002 -- DeHon

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

CALTECH CS137 Winter2002 -- DeHon

Design Optimization

Components:

  • “Energy” (Cost) function to minimize

– represent entire state, drives system forward

  • Moves

– local rearrangement/transformation of solution

  • Cooling schedule

– initial temperature – temperature steps (sequence) – time at each temperature

slide-4
SLIDE 4

4

CALTECH CS137 Winter2002 -- DeHon

Basic Algorithm Sketch

  • Pick an initial solution
  • Set temperature (T) to initial value
  • while (T> Tmin)

– 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

CALTECH CS137 Winter2002 -- DeHon

Details

  • Initial Temperature

– T0=∆avg/ln(Paccept)

  • Cooling schedule

– fixed ratio: T=λT

  • (e.g. λ=0.85)

– temperature dependent

  • Time at each temperature

– fixed number of moves? – Fixed number of rejected moves? – Fixed fraction of rejected moves?

slide-5
SLIDE 5

5

CALTECH CS137 Winter2002 -- DeHon

Cost Function

  • Can be very general
  • Should drive entire solution in right

direction

– reward each good move

  • Should be cheap to compute delta costs

– e.g. FM

CALTECH CS137 Winter2002 -- DeHon

Cost Functions

  • Total Wire Length
  • Channel widths

– probably wants to be more than just width

  • Cut width
slide-6
SLIDE 6

6

CALTECH CS137 Winter2002 -- DeHon

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

CALTECH CS137 Winter2002 -- DeHon

Initial Solution

  • Spectral Placement
  • Random
  • Constructive Placement
slide-7
SLIDE 7

7

CALTECH CS137 Winter2002 -- DeHon

Moves

  • Swap two cells
  • swap regions

– …rows, columns, subtrees

  • rotate cell (when feasible)
  • flip (mirror) cell
  • permute cell inputs (equivalent inputs)

CALTECH CS137 Winter2002 -- DeHon

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

slide-8
SLIDE 8

8

CALTECH CS137 Winter2002 -- DeHon

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
  • (never pick a costly move which will reject)

CALTECH CS137 Winter2002 -- DeHon

Theory

  • If stay long enough at each cooling

stage

– will achieve tight error bound

  • If cool long enough

– will find optimum

slide-9
SLIDE 9

9

CALTECH CS137 Winter2002 -- DeHon

Practice

  • Good results

– ultimately, what most commercial tools use...

  • Slow convergence
  • Tricky to pick schedules to accelerate

convergence

CALTECH CS137 Winter2002 -- DeHon

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

  • With care, can attack multiple levels

– place and route

  • Ignores structure of problem

– resignation to finding/understanding structure

slide-10
SLIDE 10

10

CALTECH CS137 Winter2002 -- DeHon

Optimal/Exhaustive

CALTECH CS137 Winter2002 -- DeHon

  • 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….

slide-11
SLIDE 11

11

CALTECH CS137 Winter2002 -- DeHon

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

CALTECH CS137 Winter2002 -- DeHon

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

slide-12
SLIDE 12

12

CALTECH CS137 Winter2002 -- DeHon

Improving

  • Prune off symmetry cases

– Rotate 90, 180, 270 – Mirror X, Y, XY

  • Reject provably bad starts

– (return to in a minute)

CALTECH CS137 Winter2002 -- DeHon

Exhaustive Placement

  • More general:

– Modules have variable size – Modules can be rotated/flipped…

  • To explore all cases:

– For each module

  • For each orientation
slide-13
SLIDE 13

13

CALTECH CS137 Winter2002 -- DeHon

Branch-and-Bound

  • Flavor now

– More next week

  • 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

CALTECH CS137 Winter2002 -- DeHon

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
slide-14
SLIDE 14

14

CALTECH CS137 Winter2002 -- DeHon

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

CALTECH CS137 Winter2002 -- DeHon

Caldwell et. al. Results

[TRCAD v19n11p1304-13]

slide-15
SLIDE 15

15

CALTECH CS137 Winter2002 -- DeHon

Runtime

[TRCAD v19n11p1304-13]

CALTECH CS137 Winter2002 -- DeHon

Faster?

  • Accounting and gain complexity

– Make it linear time – But do make each update somewhat complex – Exhaustive case less bookkeeping

  • Not an atypical result…
slide-16
SLIDE 16

16

CALTECH CS137 Winter2002 -- DeHon

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…

CALTECH CS137 Winter2002 -- DeHon

Big Ideas:

  • Use randomness to explore large (non-

convex) space

– Simulated Annealing

  • Use dominance to quickly skip over
  • bviously bad solutions

– Branch and Bound