CS137
Electronic Design Automation
Day 9: February 25, 2004
Placement II
André DeHon, Michael Wrighton
Today


Continue with Placement
Improving Quality


Avoiding local minima
Techniques:


Simulated Annealing
Exhaustive (Branch-and-bound)
DeHon/Wrighton
CALTECH CS137 Winter2004
Review: Cost Functions



Total Wirelength
Wirelength2
Critical Path Delay


More on this later
Close Things Placed Close
DeHon/Wrighton
CALTECH CS137 Winter2004
First Cut




Randomly place netlist
Consider moves of nodes
Perform those moves which improve some
global cost metric
When no more moves possible -- terminate
DeHon/Wrighton
CALTECH CS137 Winter2004
Force-Directed



Model netlist as being
connected with springs
F=kx
Perform swaps when F
is reduced
DeHon/Wrighton
CALTECH CS137 Winter2004
Force Directed Swaps
DeHon/Wrighton
CALTECH CS137 Winter2004
Problem: Greedy Approach

Any greedy approach will tend to get wedged in
some local minima

FM attacked problem by swapping all nodes even if
negative gain
DeHon/Wrighton
CALTECH CS137 Winter2004
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
DeHon/Wrighton
CALTECH CS137 Winter2004
Key Benefit

Avoid Local Minima
Allowed to take locally not improving moves in
order to avoid being stuck
Cost of Solution

Local Minimum
Better Solution
Best Solution
Algorithm Progression
DeHon/Wrighton
CALTECH CS137 Winter2004
Simulated Annealing

At high temperature can move around




not trapped to only make “improving” moves
free energy from temperature allows exploration of nonminimum states
avoid being trapped in local minima
As temperature lowers


less energy to take big, non-minimizing moves
more local / greedy moves
DeHon/Wrighton
CALTECH CS137 Winter2004
Design Optimization
Components:
 “Energy” (Cost) function to minimize


Moves


represent entire state, drives system forward
local rearrangement/transformation of solution
Cooling schedule



initial temperature
temperature steps (sequence)
time at each temperature
DeHon/Wrighton
CALTECH CS137 Winter2004
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 Dcost
if less than zero, accept
else if RND<e-Dcost/T, accept
update T
DeHon/Wrighton
CALTECH CS137 Winter2004
Details

Initial Temperature


T0=Davg/ln(Paccept)
Cooling schedule

fixed ratio: T=lT



(e.g. l=0.85)
temperature dependent
Time at each temperature



fixed number of moves?
Fixed number of rejected moves?
Fixed fraction of rejected moves?
DeHon/Wrighton
CALTECH CS137 Winter2004
Cost Function


Can (Should) be very general
Drive entire solution in right direction


reward each good move
Should be cheap to compute delta costs


e.g. FM
O(1) ideally
DeHon/Wrighton
CALTECH CS137 Winter2004
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
DeHon/Wrighton
CALTECH CS137 Winter2004
Cost Functions


Total Wire Length
Bounding Box Semiperimeter


Exact estimate for up to 3 node nets
VPR Bounding Box (improves apprx):
N nets

Cost   q(i)  bbx (i)  bby (i)

i 1

q(i) varies from 1 -> 2.79 by the number of
nodes on the net
DeHon/Wrighton
CALTECH CS137 Winter2004
Timing Cost Function




TimingCost(i,j) = Dly(i,j) ∙Critclty(i,j)CritcltyExp
Maintain all delay’s (i -> j) in a lookup table
Recompute Net Criticality @ Every
Temperature
Balance w/ Wiring Cost Function
Marquardt, Betz, & Rose, 2000
DeHon/Wrighton
CALTECH CS137 Winter2004
Multiple Cost Functions
800
60
50
600
40
400
30
20
200
10
0
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Lambda
DTimingCost
DWiringCost
DCost  l
 (1  l )
OldTimingC ost
OldWiringC ost
DeHon/Wrighton
CALTECH CS137 Winter2004
1
Timing Cost

“Auto-Normalize”
Function is not
weighted on ‘absolute’
values
Wiring Cost

Handling ‘Max’ Style Cost Functions


Max(a1, a2, a3, a4…) is difficult to ‘drive’
annealing
Instead: Minimize
ea1+ea2+ea3+ea4+…
DeHon/Wrighton
CALTECH CS137 Winter2004
Initial Solution



Spectral Placement
Random
Constructive Placement
DeHon/Wrighton
CALTECH CS137 Winter2004
Moves


Swap two cells
swap regions




…rows, columns, subtrees
rotate cell (when feasible)
flip (mirror) cell
permute cell inputs (equivalent inputs)
DeHon/Wrighton
CALTECH CS137 Winter2004
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
DeHon/Wrighton
CALTECH CS137 Winter2004
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)


DeHon/Wrighton
CALTECH CS137 Winter2004
Theory

If stay long enough at each cooling stage


If cool long enough


will achieve tight error bound
will find optimum
Long Enough

If P ≠ NP – will be very long
DeHon/Wrighton
CALTECH CS137 Winter2004
Practice

Good results



ultimately, what most commercial tools use...
Slow convergence
Tricky to pick schedules to accelerate
convergence
DeHon/Wrighton
CALTECH CS137 Winter2004
Black Magic

VPR Details

Rlimitnew = Rlimitold (1-0.44+α)




Maintain moves accepted at
44%
Lam and Delosme
cN4/3
Moves @ Temp =
Temperature Update
Schedule

Tnew = Told ∙γ
α
γ
α > 0.96
0.5
0.8 < α ≤ 0.96
0.9
0.15 < α ≤ 0.8
0.95
α ≤ 0.15
0.8
Empirical
DeHon/Wrighton
CALTECH CS137 Winter2004
Big Hammer


Costly, but general
Works for most all problems


Can have hybrid/mixed cost functions


as long as weight to single potential
With care, can attack multiple levels


(part, placement, route, retime, schedule…)
place and route
Ignores structure of problem

resignation to finding/understanding structure
DeHon/Wrighton
CALTECH CS137 Winter2004
Optimal/Exhaustive
More on Long Enough…

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….
DeHon/Wrighton
CALTECH CS137 Winter2004
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
DeHon/Wrighton
CALTECH CS137 Winter2004
Optimum Placement

Simplest case:




Gate/array (FPGA) w/ fixed cell locations
N locations
M cells
Try all permutations of N choose M
N!/(N-M)! cases
DeHon/Wrighton
CALTECH CS137 Winter2004
Improving

Prune off symmetry cases



Rotate 90, 180, 270
Mirror X, Y, XY
Reject provably bad starts

(return to in a minute)
DeHon/Wrighton
CALTECH CS137 Winter2004
Exhaustive Placement

More general:



Modules have variable size
Modules can be rotated/flipped…
To explore all cases:

For each module

For each orientation
DeHon/Wrighton
CALTECH CS137 Winter2004
Branch-and-Bound


Recall from Logic Optimization
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
DeHon/Wrighton
CALTECH CS137 Winter2004
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
DeHon/Wrighton
CALTECH CS137 Winter2004
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
DeHon/Wrighton
CALTECH CS137 Winter2004
Caldwell et. al. Results
DeHon/Wrighton
CALTECH CS137 Winter2004
[TRCAD v19n11p1304-13]
Runtime
DeHon/Wrighton
CALTECH CS137 Winter2004
[TRCAD v19n11p1304-13]
Faster?

Accounting and gain complexity



Make it linear time
But do make each update somewhat complex
Exhaustive case less bookkeeping

Not an atypical result…
DeHon/Wrighton
CALTECH CS137 Winter2004
Admin



Will Meet on Monday
Should have feedback from Dehon
Reminder: No Support for Non-Standard
Tools


Jython, Optik, M3, C, Perl, Tcl/Tk, etc.
Reading

Scheduling
DeHon/Wrighton
CALTECH CS137 Winter2004
Summary

Simulated Annealing




General purpose solution


use randomness to explore space
accept “bad” moves to avoid local minima
decrease tolerance over time
costly in runtime
Small (sub)problems


May solve exhaustively
Can prune to accelerate…
DeHon/Wrighton
CALTECH CS137 Winter2004
Big Ideas:

Use randomness to explore large (nonconvex) space


Simulated Annealing
Use dominance to quickly skip over
obviously bad solutions

Branch and Bound
DeHon/Wrighton
CALTECH CS137 Winter2004