teCEivED
BY
TIC
OEC
37
1975!
This
is
a
preprint
of
a
paper intended
for
publication
m
± journal
or
proceedings. Since changes may
be
made
before publication, this preprint
is
made available with
the understanding that
it
will
not
be
cited
or
reproduced
without
the
permission
at
the
author.
UCRL
-
77548
PREPRINT
Go
MV-ftfiicfi-
m
LAWRENCE UVERMORE LABORATORY
University olCalitotnm/Livermore.Calitorma
AN ANALYTIC TECHNIQUE
FOR
ROUTER COMPARISON
David C. Wilson and Robert J. Smith, II
November 26, 1975
This paper was prepared for submission to the
13th Design Automation Conference, Palo Alto, CA
June 27-29, 1976
OISTRiELi;.
AN ANALYTIC TECHMIQuC njH POUTER COMPARISON
David C. Wilson
2602 D Custer Faraway
Richardson, Tends 75080
and
Robert J. S-iith, II
Electronics Engineering Department
Laurence Liver':orr> Laboratory
Livermore, California 94550
INTRODUCTION
Design automation of electronic systems is
generally separated into a number of distinct
areas of effort. Orcucr [!] has divided design
automation into the ureas of logic synthesis, gate
simulation,
partitioning, placemen, rooting, and
fault detection and diagnosis, fcnile tins separa-
tion may not be complete or entirely accurate,
these functions generally must be performed.
At sone point in the design process the com-
ponents have been chosen, the logical interconnec-
tions specified, and the composert<, placed on a
printed circuit board; the next siep is the
physical interconnection of electrically co-mon
elements. Nu-nerous techniques hd*e been proposed
to solve the resuH'ny interconnection or routing
problem,
kith almost endless minor variations on
these techniques possible [3].
Lee's algoritnn [?] if one of the few true
algorithm ipp!:;;t!r ts '.tic rc^t:^^ srab!t~; it
guarantees tuat an interconnect"" tct-ecn two
points will be found if a satisfactory path exists.
It is an exhaustive alyarithm. as nany true
algorithms are, exylorin.j all possible patis.
Typically it is qute slow in execution bocduse
all possible paths ore explored in parallel. A
number of modifications to this basic technique
have been proposed; most change the algorithm to
a heuristic {it can no longer guarantee a solution
will be found) in orde.' to realize a gain in
execution speed.
There Are also many heuristics of very differ-
ent kinds of solving the routing proDlem; nearly
any possible technique for tracing lines is
feasible, but results cannot be guaranteed.
Honever, many heuristics run very rapidly in
comparison to exhaustive algorithm and appear to
do a "reasonable" interconnection job.
One general classification scheme applicable
to most routers is whp'her they are depth- or
breadth-first routers. That is, do they explore a
••ath to a greaL depth ct-oocing to e\plore another
when failure is encountered, or do they explore
multiple
path:,
in parallel, stopping i»hcn the
endpoint is reached by any
path.
Lee's algorithm
is a breadth-first algor'lhm. Host heuristics use
This work was perfoi'^ed mder the auspices of the
U.S. Energy p2sejrch end Developr.er.i Adninistration
under contract rubber W-?591^EhG-:3-
7W
depth-first approaches which attcmnt to reduce
execution time.
There has been seme experlmem tfon with
polyrouters. I.e., two or morfe rout-—s operating 1n
tandum.
Often a fast heuristic dep -first router
is used to make as many connections possible,
followed by a slower, more expensive nd exhaustive
router attempting only those connect ns tnat could
not be made by the first state cf the router.
Host of the routers discussed in current
literature are experirentally evaluated to give
some indication of their success, with performance
generally expressed as a percentage completion
figure for the number of wires routed on a board,
compared to the nunber of wires attempted. There
are few analytical tools, however, to indicate
or predict router performance.
This paper introduc'S a model for printed
circuit boards
»ti'.
t
\
can 5c u;;d is predict the
probability that a router will Successfully rake
a connection. The model reflects certain
characteristics of the circuit board that is
being routed and the model incrementally changes
as the board is routed. Routing procedures
typically have certa'n parameters which influence
selection of the set of paths that are explored,
and determine the order of exoloration. These
pa rasters obviously influence the probability
that a particular connection can be made. These
parameters nay also be used to formulate a model
of the behavior of a particular routing procedure.
The purpose of the board and router models
proposed here is to al?o.( (comparative} predictions
of router performance. More specifically, it 1s
assumed that the features of a router which select
the set of paths explore! should impact the
probability that a router will be successful, as
well as router execution time. Relationships
between execution tine and performance can be
analyzed using the concepts developed below.
This study also attempts to address several
other questions: what factors cause a router to
be good and how could a routers' performance be
increased keeping the sane gene"
1
heuristic;
which router in a polyrouter
S'IOUU
be used for a
particular connection or when should the change-
over between re
(
ers be madt; and finally ir, what
order should thi connections be made to give the
greatest expected nusfier of completed connections*
Admittedly n-jne of thec questions are answered
fully in this paper. However, th>- gineral jiijivsis
trthnii|ue I'uy hold the t-utential fur trtMim-j Such
issuer. It She jl(J jiso MI- "• ;.iij-.ij-fd tlu' buth
bend envtmnn'ent <ii'd rj-ii^y !>oHjvior ".nui.-is are
Stati.ticrtlly iJsed. jml thpreFurc do niH uurantee
that a particular connection «.an lie
!•.)«,•
Cet-T-en
any Hit) points; the j(i^i'o.t%ii liH"i •.til c to
produce, for each m:ui«"_lii.n. j probability that a
satisfactory path will ;-c Uxjid. VJ rojU'-i
funclioits are perfurra-d Jurui-j me c<j-;>u;.itiun of
this probjb>lit.v.
THE
B^SIC
MODELS
Tht fliode) of a printrd circuit board is based
itrictly on the 'density' t.F
<J
board. Several
authors [4,6] have noted that as a boare" becomes
more crowded (adduienul wires conpletedl U
becomes harder to route uires, *•* would be expected.
At a certain density it bc-coi'ies nearly impossible
to route an additional wire. This idea forms the
basis of the problem board riude'.
The model assumes that each layer of a printed
circuit board can be represented
dS d
rectangular
grid of small squares, each of which is either
empty (can be used) or occupied [f-jrtner use is
illegal].
The density of occupied sr.jares (or
Just density) is defined by equation (I] as the
ratio of the number of o upit;d squares to t^e
total nitfiier of squires all layers of a boara. «s
density is just the proha'blity of a square being
used,
equation (2) defines tne •jrofcjbi lity F of
a square being fre^.
(I) Density • C . ; /-^PJIiiSi!™?-
1
' * * occupied sqs. • = erpty sqs.
(?) Free = F = 1 - D
The initial modelinu efforts reported here
have been focused on three general classes of
boards:
HDDEL 1. A board is filled '."tt: randoiily distrib-
uted occupied squares, but no lines or
groups of connected grid squares are
included.
This rodel .-night resemble
a board containing randomly placed
component pads.
MODEL ?. The board has ra:tdi/-ly distributed
con-
tinuous segnents of occupied squares,
with all wires ruving both expected
width and expected length equal to
constant values. This model so-rewhat
resembles a pj-'tiaUy routed bon-d, but
is more reguljr i" its layout.
MODEL 3. The board hdS rjna. ly distributed
con-
tiguous seginT.ls of ci.cjp'cd scs.ijrei,
with each scg ."•: r..i', ir.g a c;ns:>«t
exoecteu wi-it'i
*TVJ
a r-andon le»'ii.h
chosen Iror--
s.-~i:
yi»cn ilistrU-.j'.ion.
This rrodel attests to ^ore accurately
represent a partially routed printed
circuit board.
Dur modeling effur
argument. If tne den«.ii
lated by equation (I), i
a square is occupied is
. !)•.'»•?J on the following
;'. thy board is
CJ
Icu-
• tie probability that
mi tne piosatility
Lriat a square i*. free i; F by equation I."), for a
»-th of length n, the probability that J line can
)- ioUPd exactly tti.it oislance is th'. probability
that n free sqyiies co«- be found fnliuwed by n
wtcupied square, anrf 15 given by equation (3).
tip nrubabitity th.it a line can be routed between
10 point*-, however, i
%
r.ot the probability that
I can be routed exactly tne distance between
.hobe points, but rdther the probability it can
>e routed at least ttut distance, and i~ given by
'qujtion (4), winch is simplified to equation (b).
(3) Probability of success For length n =
? (length-n) = fn • D = (1-D)
n
* "
(4) P (length - n) = £ (1-D)
L
*D •
L=n L=0
P
S
U - n) = 1 - D • (l-D) • D - <1-D)
2
* D
- - (1-0)""' • 0
(1-0) P(l - n) «• (1-D) - (1-0) * 0 - (l-D)
5
P
$
U - n)(l-UD) = (1-D)
n
• D
(5) P
5
{i 2 n) - (i-0)
n
= r
n
In a similar rinr.er the expected length for a
routed wire can he calculated using the normal
forumla for expected value calculations, equation
(6):
(C) EC'tr.gthJ * 5Z Itnyth • F Hfcngth)
let^gthsO
(7) E(length) = "£, » " \1-0)
n
' D
E(length)= (l-D) • D + 2(1-D>
2
" D • 3(1-D)
3
(l-D) E(length) = (l-D)
11
* D * ?(1-D
J
) * D
- 3(1-D)
4
• 0 *
E(length) - (l-D) r{i
eng
th) = (1-0)
-
n
• (l-D)' • D * (l-D)
3
' D •
D ' E(length) = £ {l-D-)
n
" D
n=l
(8) E(length) - £ (1-D)
r
1
n=l
»
- 0-0)
The expected length value of equation (fl) is
of a very simple
font.,
just inversely proportional
to the density <-f the hoard. Using rodel 1 a
series of experiments were conducted to demonstrate
that this was in fdet an accurate «odel of the
environment. For this experiment (and all others
referenced here) the wire width was taken to be
One unit. Hires of gn.-dtrr width '.ouf0 re- handled
wjlhl'iiutUaUy in a very similar nrtnn'T. Jhij
falluwiiej jn.iJyiii
11
Ju'-t'ot'.*- de yoner,! trcr:-
hiquc, wliicti uses tin- unit wire width a'„^u--ution.
Fliurp I shows the results of routing experi-
ment'' , bjsed en bOJrd
i.«..lf]
I- A very M'jn degree
of agreei^nt betw?en
n,....Li(j»
fl and the i-Tt-ri-
mentally determined
VIIH-.-'J
was iound. 7 ,£ close-
ness of the agree
i.tnt
is i-ai Surprisiny, R'usuSr?
the rMlhc">at ical model developed >s i*M>lwuiy for
ttic situation that wjs mode-led; single, randomly
placed used squares.
H'jc-n
the used squares are not single points
(tf.q..
Ihrre arc conti-juous occupied s*»g flits of
squares such as wires), then the probability of
encountering an occupied square is q-eatly "wiifieo.
When iwdi-Mng a rovtr-r »Mch pl.tcfs wjrei in o
single direction on each
l.iyc-r
of a printed circuit
board,
the probability \ji mtinference »s related
to the number of distinct segments of sqjjres,
rather than the number of Occunteo saudrc-o-
7o develop equations for the tffectivt density
and expected length for boards with wire se^-enls,
the modeling technique shoivn in figure 2 t*. used,
figure ?a shows a line to be run, with one already-
routed line segment in the
patn.
The previously
routed linps are coHap'-rd '"to s>r>jie paints
(shown in Figure 2b). A correction ten" is then
applied to the expected length calculation to
account for the finite length of the previously
routed wires; this
tern;
is the average se^ent
length divided by 2. Equations 9 and 10 reflect
the results of these calculations.
<»> "effect™
=
°e
:
(10) £tU-n9th) ' EC1)
H^m^m^
Running parallel wires has reduced the
effective density of a board; intuitively this
approach has reduced the congestion. This suggests
the important of minimising the number of
distinct wire segments that must be used. Figures
3 and A sufircariie experiments for board c<orjeN I
and 3. Tim average segment length of equation
JO requrjef a priori l.nowledge of the segment
lengths developed during routing. Obviously this
value cannot be computed e,«<n.tly before rating.
but it can be approximated before interconnection,
or could be accumulated as routing proceeds if it
is needed.
Figures 3 and 4 show agreenent between the
second model and the corresponding experi:
i-nta
1
values, althojgh with increased error over the
values, in Figure 2. This increased error is the
result of the variance in the average hiirc lengths,
and could have been anticipated, liquations 9 and
10 could be applied to borird nodcl 1 as -.ft 1.
since the number of used segments >s
the
number of
©ctupird squares in this case.
Uiing the expected density values, the
probability of success, on a path (equations 3 and
4) can tie calculated, these probability values
arc mxjified because of the finite segment length
for the wires that are present, and are given
3<]em as equations (1*1 and (4a):
(3d)
P
s
(l=nJ
= (1-D}
n
* (average segment Igth/ZJ'D
(4a)
P
i
(l=nJ
- (i-D)
n
'' (average segment lgth/2)
Note that using equations 9 and 10 the
expected number of SjtcssfuJ ca'.v, on a complete
route ma/ ue calcul.itrd, as the wibcr of patns
tri^d ti<*jes the uff-
i'l't/
o' su'.-.ess for each
pjtn.
Ine ve-ionce u* 'quation (ID) can also be
calculated to give J confidence factor for routing.
It is also of ir.u-v?sl. though, to obtain equations
for the probability of success P for several
routers ami alienor cc-parisoi'. using this type Of
analysis.
M0DIUN& *XftMPLES
A ratification of Lee's algorithm is the
'router-in-a-bo/.' version [7], As shown in Figure
5 the two points to ie interconnected form the
corners of a two-d''~i.Tisional rectangle, and Lee's
algorithm is applied mtnin that rectangle.
Intuitively this restriction appears justifiable,
since the refraining thrte quadrant* not used for
routing generally lead **-ay from the destination
point, and are therefore likely to be unprod.-.Uve.
figure 6 shows an enlargement of one area of
Hgure 5, wit. a squjre at location (x,y| in the
box, and the
TOUT
adjacent neighboring cells, the
routing procedure is assumed to Be up and to the
left; the proble-i is to find the probability of
routing from point i«,y) to point (1,1). This 1s
a function of the probabilities of success from
two neignbor points only (because of the algorithm
used), and is given by equation ill);
(11)
P
s
(x,y
to 1,1) = f *
P
s
U-l,y)
* F •
P
s
(x,y-1)
* (l-F) »F* f
s
(x-i,y}
• n-f) •
F
•
r
s
(x,y-n
• F •
P
s
(x-l,y)
• F •
(l-P
s
(x,y-t))
+ F *
P
s
(x,y-1)
* F *
<i-P
5
t*-i,y))
Equation (11) is an iterative equation, but
is easily tabulated ttecuase the boundary conditions
are known. Figure 7 shows the boundary conditions
for the two box sides, and the remainder of the
values r-j be tabulated by a straightforward
application of (11) alcna either the rows or the
columns.
For certain points in Figure S an intertStf/ig
phenomena occurs: Tne probability of Success
may be uniform in a regfun of the board. That is,
equation ill) yields
(12)
P
s
(x.y
to 1.1) =
x
=
F*x*F*x
• (1-F) • f * x * 2
+ F * x • F • O-x) * 2
which cun bi- simplified to
(IS)
P
(*.y to 1,1)
=
x
=
'
This equation is graphed as Figure 3.
This is an une*pected and Surprising result:
The probability of completing J nth bpfeen two
points in the uniform region is independent of the
path 'length between those
pjiin',
-md d.^t'iiv.-nt
on!/
upon l"<! bo*trd doiiiity. Tin', result «-S
Suggested as possible by LcoTijrd [b], tut is not
obvious. Clearly, however, it is 3 very tiesiruble
characteristic for any router! Figure B also shews
8 relatively sharp cut-off point 'cr routing at a
board density of about 35 , a figure which has been
observed [1j in other work.
Path length independence is r.ot valid over the
entire 'box' of Figure 5, The boundary conditions
cause certain region! to have different (and
generally much lower) P values; tnese regions are
shown as the shaded area of Figure 9. The extent
of this region is dependent only en the boara
density. The method used to calculate these
rtgions Is by tabulation of the P value for the
entire board, using equation 11.
The
p.ith
length independence formulas devel-
oped above do not hold for tht original Lee
algorithm. In this latter (.use the eo-jatioi
corresponding to equation (-3; is a cubic in P ,
implying that there are {potentially) three
distinct rcqions of stab'.e pronaaii »tv. RUiough
the original Lee algorithm] gives J higher prob-
ability of successful routing tns-i the heuristic
described t^ove, the analysis is .ilso mjch rorc
difficult. Eq<i*tmn (UJ minh* -e'l be used *s »
more directly obtainable lower b?'j".d o
n
performance
for the Lee algorithm.
Consider ne*t a quite different routing
heuristic used previously by the authors [61. !t
can be desenbea as a one-turn line prone
i-cjtfr
which attempts paths such as those shown in
figure 10a and 10b, witn o parameter k <Je5i;";3 tin?
how many lines are attempted m eac*i direction.
The equation for the P interconnection of two
points) is developed below, culc
equation (Iflf)
idting I
(14i) Probability of fai'ing to route at least
one line, i units long:
7T (1 - F'- " F*)
K=-k
(1461 Probability of success for previous
(l4aj:
1 - IT (1 - F-*" F'J
K=-k
|14cJ Probability of a Sjccessful 'interconnection
over one cross of Figure 9:
(1 - TT (1 - F
|r|
' f
1
)) " (1 - 7T
K=-k Rs-k
(i -
F^F"))
(14d) Probability of failure of (He):
* Iki ;
k
1 - 11 - TT Cl -
F^-F'))
" (1 - TT
K^-k K=-k
{1 -
F^'F*))
(Ifle) Probability of failure, on both crosses, of
a route:
(1 - (1 - 7T (1 -
F'MrM
• (1 - TT
K=-k K*-k
(\ - f'^F")))
2
(KfJ P
s
for a route:
1 - (I - (1 - 7T {1 -
F
lK,
F*))
' ( * -
K=-r
7T (1 - fl
K
?F
m
))l
2
K=-k
This equation is of limited use; however
reasonable
aps
rax nations can be made. First,
assume that k «• i and k •'• m, to yield (16a).
Then,
assume «. = m (meaning the two points to be
interconnected determine a square, to yield
(15b), (15c). and {15d}
k . k
(15a) 1 - (1 - (1 - 7T 0 - F*J) • {} - V
p»-k p'-k
(15b) 1 - (1 - (1 - 7T (1 - r)) • ( 1 - tr
p-fc p=-k
(1 - F )))
2
(15C) 1 - (1 - (I - 7T (1 -
F
E
))
2
)
2
p=-k
(15d) 1 - (1 - [1 - (1 - f
1
)
2
^
1
)
2
)
2
Equation (Mf) has the property that is
maximized if the two points being interconnected
are located at the corneis of a square, rather
than at the corners of an elongated rectangle,
and the value of the equation drops off rather
sharply if the two points farm a very elongated
rectangle. Equation (15; assures (optimistically)
that the point pairs alw.i/s determine a square,
rather than a rectangle. Jnd thus (15) represents
a maximum potential performance figure for the
router. The length of the path being routed does
clearly effect the P value for this router.
i
