Genetic Algorithms vs. Simulated Annealing

16 Pages·2015·0.45 MB·English

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Computer Science and Engineering Research Computer Science and Engineering
5-1996
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Theodore W. Manikas
Southern Methodist University, [email protected]
James T. Cain
University of Pittsburgh - Main Campus, [email protected]
Follow this and additional works at: https://scholar.smu.edu/engineering_compsci_research
Part of the Digital Circuits Commons, Electrical and Electronics Commons, Hardware Systems
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Manikas, Theodore W. and Cain, James T., "Genetic Algorithms vs. Simulated Annealing: A Comparison of
Approaches for Solving the Circuit Partitioning Problem" (1996). Computer Science and Engineering
Research. 1.
https://scholar.smu.edu/engineering_compsci_research/1
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Genetic Algorithms vs(cid:2) Simulated Annealing(cid:3) A
Comparison of Approaches for Solving the Circuit
Partitioning Problem
by
Theodore W(cid:2) Manikas
James T(cid:2) Cain
Technical Report (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:5)
May (cid:5)(cid:2)(cid:2)(cid:3)
Department of Electrical Engineering
The University of Pittsburgh
Pittsburgh(cid:7) PA (cid:5)(cid:8)(cid:9)(cid:3)(cid:5)
Abstract
An important stage in circuit design is placement(cid:2) where components are assigned
to physical locations on a chip(cid:3) A popular contemporary method for placement is
the use of simulated annealing(cid:3) While this approach has been shown to produce good
placement solutions(cid:2) recent workin genetic algorithms has produced promising results(cid:3)
Thepurposeofthisstudyistodeterminewhichapproachwillresultinbetterplacement
solutions(cid:3)
A simpli(cid:4)ed model of the placement problem(cid:2) circuit partitioning(cid:2) was tested on
three circuits with both a genetic algorithm and a simulated annealing algorithm(cid:3)
When comparedwith simulated annealing(cid:2) the genetic algorithmwasfound toproduce
similar results for one circuit(cid:2) and better results for the other two circuits(cid:3) Based on
these results(cid:2) genetic algorithms mayalso yield better results thansimulated annealing
when applied to the placement problem(cid:3)
Group A Group B
Figure (cid:2)(cid:3) Graph representation of circuit partitioning(cid:4)
(cid:0) Introduction
Animportantstage incircuitdesignis placement(cid:5) wherecomponentsare assigned to physical
locations on a chip(cid:4) A popular contemporary method for placement is the use of simulated
annealing (cid:6)Sechen(cid:7)(cid:2)(cid:8)(cid:9)(cid:4) While this approach has produced good results(cid:5) recent work in ge(cid:2)
netic algorithms has also produced promising results (cid:6)Cohoon (cid:7)(cid:10)(cid:8)(cid:5) Shahookar (cid:7)(cid:11)(cid:8)(cid:5) Sait (cid:7)(cid:12)(cid:8)(cid:9)(cid:4)
The purpose of this study is to determine which approach(cid:5) genetic algorithms or simulated
annealing(cid:5) will result in better placement solutions(cid:4)
A simple model of the placement problem is the circuit partitioning problem(cid:4) A circuit
may be represented by a graph G(cid:13)(cid:6)V(cid:5)E(cid:9)(cid:5) where the vertex set V represents the components
of the circuit(cid:5) and edge set E represents the interconnections between components(cid:4) The par(cid:14)
titioningprocess splitsthe circuitintogroups ofrelativelyequalsizes(cid:4) Theobjectiveisassign
components to groups such that the number of interconnections between groups is minimal(cid:4)
An example of a circuit partition is shown in Figure (cid:2)(cid:4) The number of interconnections
between groups is called a cutsize(cid:5) thus the goal is to minimize the cutsize(cid:4)
Partitioning was tested on three circuits using both genetic algorithm and simulated
annealing approaches(cid:4) This report describes the method used for this experiment(cid:5) and
discusses the results(cid:4)
(cid:2)
(cid:0) Method
Both a genetic algorithm and simulated annealing approach were tested on a set of circuits(cid:2)
This chapter explains both approaches(cid:3) and describes the method used for testing these
approaches(cid:2)
(cid:0)(cid:2)(cid:3) Genetic Algorithm
A genetic algorithm (cid:4)Holland(cid:5)(cid:6)(cid:7)(cid:8) is an iterative procedure that maintains a population of
individuals(cid:9) these individuals are candidate solutions to the problem being solved(cid:2) Each
iteration of the algorithm is called a generation(cid:2) During each generation(cid:3) the individuals
of the current population are rated for their e(cid:10)ectiveness as solutions(cid:2) Based on these
ratings(cid:3) a new population of candidate solutions is formed using speci(cid:11)c genetic operators(cid:2)
Each individual is represented by a string(cid:3) or chromosome(cid:9) each string consists of characters
(cid:4)genes(cid:8) which have speci(cid:11)c values (cid:4)alleles(cid:8)(cid:2) The ordering of characters on the string is
signi(cid:11)cant(cid:9) the speci(cid:11)c positions on the string are called loci(cid:2)
A genetic algorithm for partitioning(cid:3) based on Bui(cid:12)s approach(cid:5)(cid:13)(cid:7)(cid:3) was used for this study
(cid:4)Figure (cid:14)(cid:8)(cid:2) A graph partitioning solution is encoded as a binary string of C genes(cid:3) where C
(cid:15) total numberof components(cid:2) Each gene represents a component(cid:3) and the allele represents
the group (cid:4)(cid:16) or (cid:17)(cid:8)(cid:3) where the component is assigned(cid:2) For example(cid:3) the chromosome (cid:5)(cid:16)(cid:16)(cid:17)(cid:17)(cid:17)(cid:7)
represents a graph of (cid:11)ve components(cid:18) components (cid:17) and (cid:14) are in partition (cid:16)(cid:3) while compo(cid:19)
nents (cid:20)(cid:3) (cid:21) and (cid:6) are in partition (cid:17)(cid:2) The following sections explain the steps of the genetic
algorithm(cid:2)
Create Initial Population
A population of P chromosomes are randomly generated to create an initial population(cid:2)
C (cid:0)
Individuals are created by generating a random number in the range (cid:17) to (cid:14) (cid:14)(cid:9) each
individual must represent a valid partitioning solution(cid:2) A valid partitioning solution is
balanced(cid:18) each group has approximately the same number of components(cid:2)
Select Parents
Each individual has a (cid:2)tness value(cid:3) which is a measure of the quality of the solution
represented by the individual(cid:2) The formula from Bui(cid:5)(cid:13)(cid:7) is used to calculate the (cid:11)tness value
F for individual i(cid:18)
(cid:14)
GENETIC ALGORITHM
begin
create initial population of size P
repeat
select parent (cid:2) and parent (cid:3) from the population
o(cid:4)spring (cid:5) crossover(cid:6)parent (cid:2)(cid:7)parent (cid:3)(cid:8)
mutation(cid:6)o(cid:4)spring(cid:8)
update population
until stopping criteria met
report the best answer
end
Figure (cid:3)(cid:9) Genetic algorithm(cid:10)
Cw (cid:0)Cb
Fi (cid:5) (cid:6)Cw (cid:0)Ci(cid:8)(cid:11)
(cid:12)
where Cw is the largest cutsize in the population(cid:7) Cb is the smallest cutsize in the popu(cid:13)
lation(cid:7) and Ci is the cutsize of individual i(cid:10)
Each individual is considered for selection as a parent(cid:14) the probability of selection of
a particular individual is proportional to its (cid:15)tness value(cid:10) Bui(cid:16)(cid:17)(cid:18) recommends that the
probabilitythat thebest individualischosenshould be (cid:19) timestheprobabilitythatthe worst
individual is chosen(cid:10) Thus(cid:7) the P chromosomes are sorted in ascending order according to
their (cid:15)tness values(cid:7) and a probability distribution function is created(cid:10) The probability factor
r is found by
(cid:0)
r (cid:5) (cid:19)P(cid:0)(cid:0)
Assume that the probabilities assigned to each individual is a geometric progression(cid:7)
where the sum of all these probabilities S is given by
(cid:0) P(cid:0)(cid:2) (cid:2)(cid:0)rP
S (cid:5) (cid:2)(cid:11)r(cid:11)r (cid:11)(cid:2)(cid:2)(cid:2)(cid:11)r (cid:5) (cid:2)(cid:0)r
Therefore(cid:7) the probability that chromosome i is selected(cid:7) Prfig(cid:7) is found by
(cid:12)
Parent 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 (cid:2)(cid:3) Crossover example(cid:4)
i(cid:0)(cid:0)
r
Prfig (cid:5)
S
Crossover
After two parents are selected(cid:6) crossover is performed on the parents to create two o(cid:2)(cid:3)
spring(cid:4) A chromosome split point is randomly selected(cid:6) and is used to split each parent
chromosome in half(cid:4) The (cid:7)rst o(cid:8)spring is created by concatenating the left half of the (cid:7)rst
parent and the right half of the second parent(cid:6) while the second o(cid:8)spring is created by con(cid:9)
catenating the left half of the (cid:7)rst parent and the complement of the right half of the second
parent(cid:4) An example of crossover is shown in Figure (cid:2)(cid:4)
Mutation
Each o(cid:8)spring must meet the same constraints as its parents(cid:3) the number of ones and
zeroes in the bit pattern should be nearly equal(cid:4) However(cid:6) the crossover operation may
produce an o(cid:8)spring that do not meetthis requirement(cid:4) An o(cid:8)spring is alteredvia mutation(cid:6)
which randomly adjusts bits in the o(cid:8)spring so that its bit pattern is valid(cid:4) The mutation
procedure determinesthe value b(cid:6) which is the absolute value of the di(cid:8)erence in the number
of ones and zeroes(cid:4) A bit location on the o(cid:8)spring is randomly selected(cid:6)then starting at that
location(cid:6) b bits are complemented(cid:10)zeroes becomeones(cid:6) ones become zeroes(cid:11)(cid:4) This operation
results in o(cid:8)spring that represent valid partitions(cid:4)
Update Population
(cid:12)
The creation of two o(cid:2)spring increases the size of the population to P (cid:3)(cid:4)(cid:5) Since we want
to maintain a constaint population size of P(cid:6) two individuals will need to be eliminatedfrom
the population(cid:5) The goal of the algorithm is to converge to the best quality solution(cid:6) thus
the two individuals with the lowest (cid:7)tness values are removed from the population(cid:5)
Stopping Criteria
Bui(cid:8)(cid:9)(cid:10) uses a swing value W to determine when the algorithm stops(cid:5) If there is no
improvement after W generations(cid:6) then the algorithm stops(cid:5) No improvement means that
there are no changes in the maximum (cid:7)tness value of the population(cid:5) The (cid:7)nal solution is
the individual with the highest (cid:7)tness value(cid:5)
(cid:0)(cid:2)(cid:0) Simulated Annealing
Simulated annealing (cid:11)Kirkpatrick(cid:8)(cid:12)(cid:10)(cid:13) is an iterative procedure that continuously updates
one candidate solution until a termination condition is reached(cid:5) A simulated annealing
algorithmfor circuitpartitioningwas created(cid:6) and isshown inFigure(cid:14)(cid:5) Acandidate solution
is randomly generated(cid:6) and the algorithm starts at a high starting temperature T(cid:0)(cid:5) The
following sections explain the steps of the simulated annealing algorithm(cid:5)
Calculate Gain
The gain of a partitioning solution is calculated by use of the ratio cut formula (cid:11)Wei(cid:8)(cid:15)(cid:10)(cid:13)(cid:16)
cutsize
Gain (cid:17)
jAj(cid:2)jBj
where jAj (cid:17) the number of vertices in group A(cid:6) and jBj (cid:17) the number of vertices in
group B(cid:5)
Accepting Vertex Moves
M is the number of move states per iteration(cid:5) For each move state(cid:6) a vertex is randomly
selected as a candidate to movefrom its original group to the other group(cid:5) When a vertexV
is randomly selected for movementfrom one partition to another(cid:6) its score(cid:6) or acceptance of
(cid:18)
begin
T (cid:0) T(cid:0)
tstop (cid:0) ts
Current Gain (cid:0) Calculate Gain(cid:2)(cid:3)
while tstop (cid:2) (cid:4) do
Accept Move (cid:0) FALSE
for i (cid:0) (cid:5) to M do
randomly select vertex V to move from one partition to another
New Gain (cid:0) Calculate Gain(cid:2)(cid:3)
(cid:0)
(cid:6)Gain (cid:0) New Gain Current Gain
if Accept Gain Change(cid:2)(cid:6)Gain(cid:7)T(cid:3) then
Current Gain (cid:0) New Gain
Accept Move (cid:0) TRUE
else
return V to original partition
if Accept Move then
tstop (cid:0) ts
else
(cid:0)
tstop (cid:0) tstop (cid:5)
T (cid:0) T (cid:0)(cid:3)
end
Figure (cid:8)(cid:9) Simulated annealing algorithm(cid:10)
(cid:11)
Accept Gain Change(cid:2)(cid:0)Gain(cid:3)T(cid:4)
begin
if move results in unbalanced partition then
reject move
else if (cid:0)Gain (cid:2) (cid:2) then
accept move
else
R (cid:3) random number (cid:4)(cid:2) (cid:2) R (cid:2) (cid:5)(cid:6)
(cid:0)(cid:0)Gain
Y (cid:3) e T
if R (cid:2) Y then
accept move
else
reject move
end
Figure (cid:7)(cid:8) Simulated annealing scoring function
move(cid:9) is evaluated according to the function shown in Figure (cid:7)(cid:10) A move is always rejected
if it will result in an unbalanced partition(cid:9) while a move is always accepted if it will improve
the solution(cid:10) Otherwise(cid:9) a move is randomly accepted(cid:9) with the probability of acceptance
dependent on the system temperature T(cid:10) The higher the temperature(cid:9) the greater the prob(cid:11)
ability that an inferior move will be selected(cid:10) This process allows the candidate solution to
exploremoreregions of the solution space at the earlystages of the algorithm(cid:10) The objective
is to keep the solution from converging to a local optimum(cid:10)
Stopping Criteria
After each iteration(cid:9) the temperature T is scaled by a cooling factor (cid:3)(cid:9) where (cid:2) (cid:2) (cid:3) (cid:2) (cid:5)(cid:10)
The algorithm stops if there have been no changes to the solution after ts iterations(cid:10)
(cid:0) Experiment and Results
Three circuits were selected for data sets(cid:12) the graphical representations of these circuits are
shown in Figures (cid:13)(cid:9) (cid:14)(cid:9) and (cid:15)(cid:10) For the genetic algorithm(cid:9)the population size P and swing
value W were varied during testing(cid:10) For simulated annealing(cid:9) the starting temperature T(cid:0)(cid:9)
cooling factor (cid:3)(cid:9) number of movestate M(cid:9) and stopping value ts(cid:10) were varied during testing(cid:10)
Each set of parameter combinations forms a treatment(cid:12) there were approximately (cid:16)(cid:2) trials
(cid:13)
Circuit P W
(cid:2) f(cid:3)(cid:4)(cid:2)(cid:5)(cid:4)(cid:2)(cid:3)(cid:4)(cid:6)(cid:5)g f(cid:6)(cid:4)(cid:3)(cid:4)(cid:2)(cid:5)g
(cid:6) f(cid:2)(cid:3)(cid:4)(cid:7)(cid:5)(cid:4)(cid:3)(cid:5)(cid:4)(cid:2)(cid:5)(cid:5)g f(cid:6)(cid:4)(cid:3)(cid:4)(cid:2)(cid:5)g
(cid:7) f(cid:2)(cid:3)(cid:4)(cid:7)(cid:5)g f(cid:6)(cid:4)(cid:3)(cid:4)(cid:2)(cid:5)g
Table (cid:2)(cid:8) Experimental parameter ranges for the genetic algorithm(cid:9)
Circuit T(cid:0) (cid:2) M ts
(cid:2) f(cid:2)(cid:5)(cid:5)(cid:5)g f(cid:5)(cid:9)(cid:10)(cid:4)(cid:5)(cid:9)(cid:11)(cid:4)(cid:5)(cid:9)(cid:11)(cid:11)(cid:3)g f(cid:3)(cid:4)(cid:2)(cid:5)(cid:4)(cid:6)(cid:5)g f(cid:7)(cid:4)(cid:3)(cid:4)(cid:2)(cid:5)g
(cid:6) f(cid:2)(cid:5)(cid:5)(cid:5)(cid:4)(cid:3)(cid:5)(cid:5)(cid:5)(cid:4)(cid:2)(cid:5)(cid:5)(cid:5)(cid:5)g f(cid:5)(cid:9)(cid:10)(cid:4)(cid:5)(cid:9)(cid:11)(cid:4)(cid:5)(cid:9)(cid:11)(cid:11)(cid:3)g f(cid:3)(cid:4)(cid:2)(cid:5)(cid:4)(cid:6)(cid:5)g f(cid:7)(cid:4)(cid:3)(cid:4)(cid:2)(cid:5)g
(cid:7) f(cid:2)(cid:5)(cid:5)(cid:5)g f(cid:5)(cid:9)(cid:11)(cid:11)(cid:3)g f(cid:6)(cid:5)g f(cid:7)(cid:4)(cid:3)g
Table (cid:6)(cid:8) Experimental parameter ranges for simulated annealing(cid:9)
per treatment(cid:9) The parameter ranges used for each circuit are shown in Table (cid:2) for the
genetic algorithm(cid:4) and in Table (cid:6) for the simulated annealing algorithm(cid:9)
For each graph(cid:4) the mean cutsizes of the genetic algorithm and simulated annealing are
compared(cid:9) We want to estimate the di(cid:12)erences between the means with a (cid:11)(cid:3)(cid:13) degree of
con(cid:14)dence(cid:9) According to Freund(cid:15)(cid:11)(cid:16)(cid:4) if x(cid:17)(cid:0) and x(cid:17)(cid:2) are the values of the means of independent
(cid:2)
random samples of size n(cid:0) and n(cid:2) from the normal populations with known variances (cid:3)(cid:0) and
(cid:2)
(cid:3)(cid:2)(cid:4) then
(cid:2) (cid:2) (cid:2) (cid:2)
(cid:3)(cid:0) (cid:3)(cid:2) (cid:3)(cid:0) (cid:3)(cid:2)
(cid:18)x(cid:17)(cid:0) (cid:2)x(cid:17)(cid:2)(cid:19)(cid:2)z(cid:0)(cid:2)(cid:2)(cid:3) (cid:20) (cid:4) (cid:5)(cid:0) (cid:2)(cid:5)(cid:2) (cid:4) (cid:18)x(cid:17)(cid:0) (cid:2)x(cid:17)(cid:2)(cid:19)(cid:20)z(cid:0)(cid:2)(cid:2)(cid:3) (cid:20)
sn(cid:0) n(cid:2) sn(cid:0) n(cid:2)
is a (cid:18)(cid:2)(cid:2)(cid:2)(cid:19)(cid:2)(cid:5)(cid:5)(cid:13) con(cid:14)dence interval for the di(cid:12)erence between the population means(cid:9)
For a (cid:11)(cid:3)(cid:13) con(cid:14)dence interval(cid:4) (cid:18)(cid:2) (cid:2) (cid:2)(cid:19) (cid:21) (cid:5)(cid:6)(cid:11)(cid:3)(cid:4) so (cid:2) (cid:21) (cid:5)(cid:6)(cid:5)(cid:3)(cid:4) and (cid:2)(cid:7)(cid:6) (cid:21) (cid:5)(cid:6)(cid:5)(cid:6)(cid:3)(cid:9) From
the z(cid:22)tables for standard normal distribution (cid:18)Table III in Freund(cid:15)(cid:11)(cid:16)(cid:19)(cid:4) z(cid:3)(cid:3)(cid:3)(cid:2)(cid:4) (cid:21) (cid:2)(cid:6)(cid:11)(cid:23)(cid:9) For this
study(cid:4) index(cid:2) refers to the geneticalgorithm(cid:4) whileindex (cid:6) refers to the simulatedannealing
method(cid:9) Table (cid:7) shows the results(cid:4) which are used to calculate the con(cid:14)dence intervals(cid:9) A
bar graph that compares the mean cutsizes is shown in Figure (cid:23)(cid:9)
For data set (cid:2)(cid:4) the (cid:11)(cid:3)(cid:13) con(cid:14)dence interval is
Circuit x(cid:17)(cid:0) (cid:3)(cid:0) n(cid:0) x(cid:17)(cid:2) (cid:3)(cid:2) n(cid:2)
(cid:2) (cid:7)(cid:9)(cid:5)(cid:5)(cid:24) (cid:5)(cid:9)(cid:5)(cid:23)(cid:3) (cid:6)(cid:24)(cid:5) (cid:24)(cid:9)(cid:10)(cid:23)(cid:5) (cid:5)(cid:9)(cid:23)(cid:2)(cid:10) (cid:24)(cid:5)(cid:5)
(cid:6) (cid:3)(cid:9)(cid:7)(cid:7)(cid:7) (cid:2)(cid:9)(cid:2)(cid:6)(cid:25) (cid:6)(cid:24)(cid:5) (cid:24)(cid:9)(cid:11)(cid:25)(cid:10) (cid:5)(cid:9)(cid:6)(cid:25)(cid:25) (cid:2)(cid:23)(cid:6)(cid:5)
(cid:7) (cid:23)(cid:9)(cid:23)(cid:24)(cid:5) (cid:2)(cid:9)(cid:2)(cid:3)(cid:11) (cid:2)(cid:5)(cid:5) (cid:10)(cid:9)(cid:10)(cid:25)(cid:3) (cid:5)(cid:9)(cid:3)(cid:23)(cid:7) (cid:24)(cid:5)
Table (cid:7)(cid:8) Table of results(cid:9)
(cid:10)