Genetic Algorithms vs Sim ulated Annealing A
Comparison of Approac hes or f Solving the C ircuit
P artitioning roblem P
b y
Theo dore W Manik as
James T C ain
T ec hnical Rep ort
y
Departmen t f o E lectrical Engineering
The niv U ersit y o f P ittsburgh
Pittsburgh P A
Abstract
An imp ortan t s tage in circuit design is plac ement here w comp onen ts are assigned
to ph ysical lo cations on a c hip A p opular con temp orary metho d for placemen tis
the use of simulate daenn aling While this approac h has b een sho wn to pro duce o g o d
placemen t solutions recen tw in genetic lgorithms a has pro duced promising results
The purp ose of this study is to determine whic h approac h ill w result in b tter e placemen t
solutions
A implid s mo del of the placemen t problem cir cuit p artitioning as tested on
three circuits with b th o a g enetic lgorithm a and a sim ulated annealing lgorithm a
When compared with sim ulated annealing the enetic g algorithm w as found o t pro duce
similar results for one circuit and b e tter results for the ther o t w o circuits Based on
these results genetic algorithms ma y lso a y ield b tter e results than sim ulated annealing
when applied to the placemen t problem
ork
MaGroup A
Group B
Figure Graph represen tation of circuit artitioning p
In tro duction
An imp o rtan t stage in circuit design is plac ement where comp onen ts are a ssigned to ph ysical
lo cations on a c hip A p opular con temp orary metho d for placemen t is the use of simulate d
anne aling ec hen While t his approac h as h pro duced go o d results recen tw ork in ge
netic algorithms has also pro duced promising results Coho on Shaho o k ar Sait
The purp ose of this study is to determine whic h approac h genetic lgorithms a or sim ulated
annealing will result in b etter placemen t solutions
A simple mo del of the placemen t roblem p is the cuit p artitioning problem A circuit
ma y b e represen ted b y a graph GV where the v ertex set V represen ts the comp onen ts
of the circuit and edge set E represen ts the in terconnections b et w een c omp nen o ts The par
titioning pro cess splits the circuit i n to groups o f r elativ ely equal sizes he T ob jectiv e is assign
comp onen ts to groups s uc htatth he n um ber of in terconnections b et w een groups is minim al
An example of a c ircuit partition is sho wn in Figure The n um ber fo in terconnections
bet w een groups is called a cutsize h us thegalo istomiiinm ze the c utsize
P artitioning w as tested on three circuits u sing b oth genetic a lgorithm nd a sim ulated
annealing approac hes This rep ort describ es the metho d u sed for t his exp erimen t
discusses the results
and
cir Metho d
Both a genetic algorithm and sim ulated annealing approac hw ere tested on a set of circuits
This c hapter explains b o th approac hes and d escrib es the metho d used for testing these
approac hes
Genetic Algorithm
A g enetic algorithm olland is an iterativ e pro cedure that main tains a p o pulation of
individuals these individuals are candidate solutions to the problem b eing solv ed Eac h
iteration of the algorithm is called a gener ation During eac h generation he t individuals
of the curren t p opulation are rated for their ectiv eness as solutions Based on these
ratings a new p o pulation of candidate solutions is formed using p s eci genetic op erators
Eac h individual is represen ted b y a string or chr omosome ca h string consists of c haracters
genes whic hha v e sp e ci v alues al leles The rdering o o f c haracters on the string is
signian t the p s eci p sitions o on the string are called lo ci
A genetic algorithm for artitioning p ased b on Bui pproac a h w as used for this study
igure A raph g partitioning solution is enco ded s a a binary string of C genes where C
total n um b er of omp c onen ts Eac h gene represen ts a comp nen o t and the allele represen ts
the group or where the comp onen t is ssigned a F or example the c hromosome
represen ts a raph g of e comp nen o ts comp onen ts nd a re a in partition while comp o
nen ts and a re in partition The f ollo wing sections explain the steps of the genetic
algorithm
Create Initial P opulation
A p opulation of P c hromosomes are randomly generated to create an initial p opulation
C
Individuals are created b y enerating g a random n um b er n i t h
individual m ust represen ta valid partitioning solution Av alid partitioning solution is
b alanc e dac h g roup has appro ximately the same n ber of pocom enn ts
Select P aren ts
Eac h individual has a ness value ihc h s i a measure of the qualit y of the solution
represen ted b y the individual The form ula from B ui s i u sed to calculate the ness v alue
F for individual i
um
eac to range he
GENETIC ALGORITHM
b egin
create initial p opulation of size P
rep eat
select paren t nd a paren t from the p opulation
opring crosso v eraren t paren t
m utationpring
up date p opulation
un til stopping criteria met
rep ort the b est answ er
Figure Genetic a lgorithm
C C
w b
F C C
i w i
where C is the largest cutsize in the p opulation C is the smallest utsize c in the p opu
w b
lation and C is the cutsize f o ndividual i i
i
Eac h individual is considered for selection as a p ar the probabilit y of selection of
a particular individual is prop ortional t o ts i ness v alue Bui recomme nds that the
probabilit y hat t the b est individual is c hosen should b e times the probabilit y that the w orst
individual is c hosen Th us the P c hromosomes are sorted in ascending order according to
their ness v alues and a probabilit y distribution function i s created The probabilit y factor
r is found b y
P
r
Assume that the probabilities assigned to eac h i ndividual is a eometric g progression
where the sum of all these probabilities S en b y
P
r
P
S r r r
r
Therefore the probabilit y hatt c hromosome i is selected Pr f i g sfondu b y
giv is
ent
endParent 1
0 1 1 0 1 0 1
Parent 2
1 1 0 1 0 1 1
Offspring 1 0 1 1 1 0 1 1
Offspring 2 0 1 1 0 1 0 0
Figure Crosso v er example
i
r
Pr f i g
S
Crosso v er
After t w o aren p ts are selected ossover is p erformed on the aren p ts to create t w o o
spring Ac hromosome split p in o t is randomly selected nd a is used to split eac h paren t
c hromosome in half The st opring i s created b y concatenating the left h alf f o the st
paren t and the righ t alf h of the second p aren t while the second opring i s created b y con
catenating the left half of the rst aren p t a nd the c omplement of the righ t h of t
paren t An example of crosso v er is sho wn in Figure
Mutation
Eac h opring m ust meet the same constrain ts as its aren p ts the n um b r e o f ones and
zero es in the bit attern p s hould b e n early equal Ho w ev er the crosso v er op eration ma y
pro duce an opring that do not meet this requiremen t An opring is altered via mutation
whic h randomly adjusts bits in the opring so that i ts bit p attern is v T m utation
pro cedure determines the v alue b whic h s i the absolute v alue of the d irence i n the n um ber
of ones and zero es A bit lo cation on the opring is randomly elected s then starting at that
lo cation b bits are compleme n ted ero es b ecome ones ones b ecome zero es This op eration
results in opring that r epresen tv alid partitions
Up date P opulation
he alid
second he alf
crThe creation of t w o pring o increases the size o f the p pulation o to P S ince w ew an t
to main tain a c onstain t p opulation size f o P w o individuals will need to b e eliminated from
the p opulation The goal of the algorithm i s to con v erge to the b est qualit y solution th us
the t w o individuals with the lo w est ness v alues are remo v ed from the p opulation
Stopping riteria C
Bui uses a swing value W o t d etermine when the algorithm stops If there is n o
impro v em en t after W generations then the algorithm stops No impr ovement means that
there are no c hanges in the maxim um ness v alue of the p opulation The al solution is
the individual with the ighest h ness v alue
Sim ulated Annealing
Sim ulated annealing irkpatric k is an iterativ e ro p cedure that con tin uously up dates
one candidate solution un til a termination condition s i reac hed Asim ulated annealing
algorithm for circuit partitioning w as created nd a is sho wn in Figure A andidate c solution
is randomly generated and the algorithm starts a t a high starting t emp erature T The
follo wing sections explain the steps of the sim ulated annealing algorithm
Calculate Gain
The gain of a p artitioning solution is calculated b y use of the r atio cut formula ei
cutsiz e
Gain
j A j j B j
where j A j eh n ber of v ertices in group A nd a j B j the n um ber of v ertices in
group B
Accepting V ertex Mo v es
M is the n um ber of move states p er iteration F or eac hmo v e state a v ertex is randomly
selected as a candidate o t mo v e from its original group to the other group When a v ertex V
is randomly selected for m o v em en t f rom one p artition to another its or e or a cceptance of
sc
um
b egin
T T
t t
stop s
Curren t Gain C alculate Gain
while t do
stop
Accept Mo v e F ALSE
for i to M do
randomly select v ertex V to mo v e f rom one artition p to another
New Gain Calculate Gain
Gain New Gain C ur r Gain
if Accept Gain Change Gain then
Curren t Gain ew N
Accept Mo v e TR UE
else
return V to original partition
if Accept v e then
t t
stop s
else
t t
stop stop
T T
Figure Sim ulated annealing algorithm
end
Mo
Gain
entA c c ept Gain Change Gain
b egin
if mo v e r esults in un balanced partition then
reject mo v e
else if Gain then
accept mo v e
else
R random n um ber
Gain
T
Y e
R Y then
accept mo v e
else
reject mo v e
Figure Sim ulated annealing scoring function
mo v e is ev aluated according to t he function sho wn in Figure A o m v e s i alw a ys rejected
if it will result in an un balanced partition while a o m v e s i alw a ys accepted if it will impro v e
the solution Otherwise a mo v e is randomly a ccepted with the probabilit y f o acceptance
dep enden t o n the system temp erature T The higher the temp erature the greater the prob
abilit y that an inferior mo v e w ill b e selected This ro p cess allo ws the candidate solution to
explore more regions of the solution space at the early stages of the a lgorithm The o b jectiv e
is to k eep the solution from con v erging to a lo cal ptim o um
Stopping riteria C
After eac h iteration the temp erature T i s scaled b ya c o oling factor w here
The algorithm stops if there ha v e b een no c hanges to the solution a fter t iterations
s
Exp erimen t nd a Results
Three circuits w ere elected s for data sets the g raphical represen tations o f these circuits are
sho wn in Figures nd a F or the genetic lgorithmhe a p pulation o size P and swing
v alue W w v aried during testing F or sim ulated annealing the starting temp erature T
co oling factor um ber of mo v e state M and stopping v alue t ere v aried d uring testing
s
Eac h set of parameter com binations f orms a tr e atment there w ere appro ximately trials
ere
end
if
Circuit P W
f g f g
f g f g
f g f g
T able Exp erimen tal parameter ranges for t he genetic a lgorithm
Circuit T M t
s
f g f g f g f g
f g f g f g f g
f g f g f g f g
T able Exp erimen tal parameter ranges for sim ulated annealing
p er treatmen t The parameter ranges used for eac h circuit are sho wn in T able for the
genetic algorithm and in T able for the sim ulated annealing algorithm
F or eac h raph g the ean m cutsizes of the g enetic a lgorithm and sim ulated annealing are
compared W ew t o t estimate the dirences b et w een the m eans with a degree of
conence According o t F reund i f x and x t v alues f o the means of i ndep enden t
random samples of size n and n from the n ormal p opulations with kno wn v ariances
then
s s
x x z x x z
n n n n
is a conence in terv al for the dirence b et w een he t p pulation o means
F or a conence in terv al s o a nd F rom
the zables for standard ormal n distribution able I I I in F reund z F or this
study index refers to the genetic algorithm while ndex i refers to the sim ulated annealing
metho d T able s ho ws the results whic h re a used to calculate the conence in terv als A
bar graph that compares the mean cutsizes i s sho wn in Figure
F or data set the conence in terv al is
Circuit x n x n
T able T able of results
and
he are
an
Mean Cutsizes
9
8
7
6
5
4
3
2
1
0
Circuit 1 Circuit 2 Circuit 3
Genetic Algorithm
Simulated Annealing
Figure Comparison f o ean m cutsizes
Since b oth limits are negativ e w e can conclude t hat with onence c the genetic
algorithm pro uces d a solution ith w a smaller a v erage cutsize than sim ulated annealing
F or data set the conence in terv al is
Both limits are p o sitiv e but the irence d is less than one Since cutsizes are in teger
v alues no signian t dirence can b e found b e t w een the genetic algorithm a nd sim ulated
annealing
F or data set the conence in terv al is
Since b oth limits are negativ e w e can conclude t hat with onence c the genetic
algorithm pro uces d a solution ith w a smaller a v erage cutsize than sim ulated annealing
Th us the genetic a lgorithm pro duced a smaller a v erage cutsize than sim ulated annealing
for circuits and while no s ignian t dirence w as found b et w een the metho ds when
applied to circuit
1 3
2 4
5
Figure raph G
1
9
2
10
3
11
4
12
5 13
6
14
7
15
8
16
Figure raph G
Conclusion
Based on the results of the tudy s the genetic algorithm w as sho wn to pro duce solutions equal
to or b etter than sim ulated annealing when applied to the circuit partitioning problem
Recall that the circuit partitioning problem w as used to mo del the placemen t roblem p
Sim ulated annealing is a p opular con temp orary placemen t m etho d ho w er t he results of
this study indicate that genetic algorithms ma y lead to b etter results
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Figure raph G
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