A: A DYNAMIC DISTRIBUTED DOUBLE GUIDED

GENETIC ALGORITHM FOR THE CASE OF THE

PROCESSORS CONFIGURATION PROBLEM

Sadok Bouamama, Khaled Ghédira

SOIE/University of Tunis

B. 204, departement of computer science

41, rue de la liberté, 2000 cité Bouchoucha. Tunisia

Keywords: Constraint satisfaction and optimization problems, multi-agent systems, genetic algorithms, Min-conflict-

heuristic, guidance operator.

Abstract: Within the framework of Constraint satisfaction and optimization problem (CSOP), we introduce a new

optimization distributed method based on Genetic Algorithms (GA). This method consists of agents

dynamically created and cooperating in order to solve the problem. Each agent performs its own GA on its

own sub-population. This GA is sometimes random and sometimes guided by both the template concept and

by the Min-conflict-heuristic. In addition with guidance, our approach is based on NEO-DARWINISM

theory and on the nature laws. In fact, by reference to their specificity the new algorithm will let the agents

able to count their own GA parameters. In order to show D

A advantages, experimental comparison with

GGA is provided by their application on the large processors configuration problem.

1 INTRODUCTION

CSP formalism (Tsang, 1993) consists of variables

associated with domains and constraints involving

subsets of these variables. A CSP solution is an

instantiation of all variables with values from their

respective domains. The instantiation must satisfy all

constraints.

In the realms of CSP, the instantiation of a

variable with a value from its domain is called a

label. A simultaneous instantiation of a set of

variables is called a compound label, which is a set

of labels. A complete compound label is one that

assigns values, from the respective domains, to all

the variables in the CSP.

A CSOP is a CSP with an objective function f that

maps every complete compound label to a numerical

value. The goal is to find a complete compound

label S such that f(S) gives an optimal value, and

that no constraint is violated. CSOPs make up the

framework to this paper.

CSOP which is NP-hard has been dealt with by

complete or incomplete methods. The first ones,

such as Branch and Bound (Tsang, 1993) are able to

provide an optimal solution. Unfortunately, the

combinatorial explosion thwarts this advantage. The

second ones, such as Guided Genetic Algorithms

(GGA) (Lau and Tsang, 1998) have the property to

avoid the trap of local optima. They also sacrifice

completeness for efficiency.

There is other distributed GAs known as Distributed

Guided Genetic Algorithms. These approaches have

been successfully applied to Max-CSP (Bouamama

and Ghedira, 2003, 2004). Basically these

distributed approaches outperform the Centralized

Genetic Algorithms (CGAs) (Lau, 1998), which are

especially known to be expensive in time. These

approaches give good results with the Max-CSPs, in

terms of both optimality and solution quality. Why

not to apply the same idea for CSOPs. This is the

aim of this paper. Our interest in GAs is also

motivated by their proven usefulness in many hard

optimization problems (DeJong, 1989)(Tsang,

1999), solving multiprocessor scheduling

problems(Tsujimura, 1993) (Michael et al., 1999).

2 CSOP FORMALISM

A constraint satisfaction and optimization problem,

or CSOP, is a quadruple (X, D, C, f); whose

components are defined as follows:

– X is a finite set of variables {x

, x

, ... x

242

Bouamama S. and Ghédira K. (2005).

D3G2A: A DYNAMIC DISTRIBUTED DOUBLE GUIDED GENETIC ALGORITHM FOR THE CASE OF THE PROCESSORS CONFIGURATION

PROBLEM.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics, pages 242-247

DOI: 10.5220/0001181102420247

 SciTePress

– D is a function which maps each variable in X to

its domain of possible values, of any type, and

is used to denote the set of objects mapped

from x

by D. D can be considered as D = {D

, …,D

};

– C is a finite, possibly empty, set of constraints on

an arbitrary subset of variables in X. these

constraints are represented in Extension or in

Intention.

– f an objective function which maps every

instantiation to a numerical value.

3 DYNAMIC DISTRIBUTED

DOUBLE GUIDED GENETIC

ALGORITHM FOR CSOP

3.1 Basic principles

Our approach draws basically on the concept of both

species and ecological niches. The species consists

of several organisms having common characteristics

whereas the ecological niche represents the task

performed by a given species. Goldberg sets that the

sexual differentiation based on specialization via

both the building of species and the exploitation of

ecological niches provides good results (Goldberg,

1989). A certain number of methods have been

settled in order to favorite the building of ecological

niches (Ghedira, 2002) in GAs.

So, the idea here is to partition the initial

population into sub-populations and to assign each

one of them to an agent called Species agent. A

given sub-population consists of chromosomes

having their fitness values in the same range. This

range, said FVR, is called the specificity of the

Species agent Species

FVR

. Species agents are in

interaction, in order to reach an optimal solution for

the problem. For this reason, each Species agent

performs its own GA. The latter is guided by both

template (Tsang, 1999) concept and min-conflict

heuristic (Minton, 1992). An intermediary agent is

necessary between the society of Species agents and

the user, essentially to detect the best partial solution

reached during the dialogue between the Species.

This agent, called Interface, may also possibly create

new Species agents.

3.2 Min-Conflict-Heuristic and the

Template Concept

Each chromosome is attached to a template (Tsang,

1999) that is made up of weights referred to as

template

i,j

. Each one of them corresponds to gene

i,j

where i refers to the chromosome and j to the

position. δ

i,j

represents the sum of costs of violated

constraints by gene

i,j

. These weights are updated by

means of the penalty operator (see sub-section 3.7).

Templates will be used by GA in replacement. As

we use the min-conflict-heuristic, replacement have

to be elitist, i.e a chromosome is replaced by a better

chromosome. For this, heavier templates genes have

more probability to be replaced.

3.3 Preparing CSOP

Relationship between both genetic and CSOP

formalisms is outlined as below; each chromosome

(respectively gene) is equivalent to a CSOP potential

solution (respectively variable). Moreover, each

allele corresponds to a value.

Given an objective function f, we define an

fitness function (FF) g which will be used by the

optimization process (Lau, 1998).

g(ps) = f(ps) +

(CP

* I

(ps)) (1)

Where ps is a potential solution, λ is a parameter

to the algorithm called Regularization parameter. It

is a parameter that determines the proportion of

contribution that penalties have in the fitness

function.

is the penalty for gene

(all CP

are initialized

to 0) and I

is an indication of whether ps satisfies all

constraints or not:

(ps) = 1 if ps satisfies all constraints;

0 otherwise. (2)

Let us mention here that I

is specific for every

genei. for this, we sum over the index i the I

values

in order to express the contribution of every gene in

the solution.

3.4 Agent Structure

Each agent has a simple structure: its acquaintances

(the agents it knows and with which it can

communicate), a local knowledge composed of its

static and dynamic knowledge, and a mailbox where

it stores the received messages to be later processed

one by one.

3.4.1 Species Agent

A Specie agent has got as acquaintances the other

Specie agents and the Interface agent. Its static

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knowledge consists of the CSOP data (i.e. the

variables, their domains of values, the constraints

and the objective function), the specificity (i.e. the

fitness function range) and its local GA parameters

(mutation probability, cross-over probability,

number of generations, etc.). Its dynamic knowledge

takes components as the population pool, which

varies from one generation to another

(chromosomes, population size).

3.4.2 Interface Agent

An Interface agent has as acquaintances all the

Specie agents. Its static knowledge consists of the

∑CSP data. Its dynamic knowledge includes the best

chromosome (i.e. the chromosome having the best

fitness function value).

3.5 Global Dynamic

The Interface agent randomly generates the initial

population and then partitions it into sub-populations

accordingly to their specificities i.e. the fitness value

range FVR. After that the former creates Species

agents to which it assigns the corresponding sub-

populations. Then the Interface agent asks these

Species to perform their optimization processes. So,

before starting its own optimization process, i.e. its

own behaviour, each Specie agent, Species

FVR

initializes all templates and penalties counters

corresponding to its chromosomes. After that it

carries out its genetic process on its initial sub-

population, i.e. the sub-population that the Interface

agent has associated to it at the beginning. This

process, which will be detailed in the algorithms,

returns a sub-population “pop” that has been

submitted to the crossing and mutating steps only

once, i.e. corresponding to one generation. For each

chromosome of pop, Specie

FVR

computes their

fitness function values FV according to formula (1).

Consequently, two cases may occur. The first one

corresponds to a chromosome having an FV in the

same range as its parents. In this case, the

chromosome replaces one of the latter randomly

chosen. In the second case, this value (FV) is not in

the same range (FVR), i.e, the specificity of the

corresponding Species

FVR

. Then the chromosome is

sent to another Species

if such agent already

exists, otherwise it is sent to the Interface agent. The

latter creates a new agent having FV as specificity

and transmits the quoted chromosome to it.

Whenever a new Species agent is created, the

Interface agent informs all the other agents about

this creation and then asks the new Species to

perform its optimization process. Note that message

processing is given a priority. So, whenever an agent

receives a message, it stops its behaviour, saves the

context, updates its local knowledge, and restores

the context before resuming its behaviour.

3.6 Guided Cross-over and Guided

Mutation

Out of each pair of chromosomes, the cross-over

operator produces a new child. The child inherits the

best genes, i.e. the “lighter” ones, from its parents.

The probability, for a parent chromosome

(i=i

), where sum = template

i1,j

+ template

i2,j

propagate its gene

i,j

to its child chromosome is equal

to 1-template

i,j

/ sum. This confirms the fact that the

“lighter” genes, i.e. having the best FV, are more

likely than the other to be passed to the child.

For each one of its chromosomes selected

according to the mutation probability P

mut

Species

FVR

uses the min-conflict-heuristic first to

determine the gene (variable) involved in the worst

FV, secondly to select from this gene domain the

value that violates the minimal number of

constraints and finally to instantiate this gene with

this value. If all the Species agents did not meet any

better chromosome at the end of their behaviour or

they attain the stopping criterion, they successively

transmit one of their randomly chosen

chromosomes, linked to its specificity to the

Interface agent. The latter determines and displays

the best chromosome namely the one which have the

best FV.

3.7 Penalty Operator and Local

Optima Detector

To enhance the approach, the agents are given more

autonomy and more dynamicity. In fact, we add an

other GA’s parameter that we call LOD for local

optima detector. The latter represents the number of

generation in which the neighboring does not give

improvement, i.e. if the FV of the best chromosome

remains unchanged for a specific number of

generations; and so we can conclude that the agent

optimization sub-process is trapped in a local

optimum. In fact if the unchanged FV is lesser than

the last stationary FV then automatically LOD have

to be equal to one. Otherwise the LOD will remain

unchanged i.e. LOD is a parameter to the whole

optimization process and it will be dynamically

updated by every agent.

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Let us mention that every Specie

FVR

have to save

its best FV for the next generations. This will be

very useful not only in the case of stationary fitness

values, but also to select the best chromosome in the

species. In fact, if the best FV remain unchanged for

LOD

generation, the process will be considered as

trapped in a local optimum. Thus for all the

chromosomes having this FV, the related penalty

counter PC

of all its genes is incremented by one.

As we are in an optimization case, every

Specie

AFR

have to send its best chromosome to the

Interface Agent. The latter updates its local

knowledge by this information. This must be done

once after every generation. The Interface Agent

will, at every attempt, compare the best chromosome

he has with the best one sent by the species agents.

Only those having the best FV will be maintained.

When the optimization process settles on a local

optimum, the penalty of potential solution associated

to this local optimum is increased. This helps the

search process to escape from local optima, and

drives it towards other candidate solutions. It is

worth pointing out that a slight variation in the way

that penalties are managed could make all the

difference to the effectiveness of our approach. This

is done by incrementing its penalty value by 1:

= CP

+ 1 (3)

3.8 Mutation and Guidance

Probability

The approach, as decribed until now, can not be

considered as a classic GA. In fact, in classic GAs

the mutation aims to diversify a considered

population and then to avoid the population

degeneration

(Goldberg, 1989). In this approach,

mutation operator is used improperly since it is

considered as a betterment operator of the

considered chromosome. However, if a gene value

was inexistent in the population there is no way to

obtain it by cross-over process. Thus, it is sometimes

necessary to have, a random mutation in order to

generate the possibly missing gene values. Our

approach is a local search method. The first known

improvement mechanism of local search is the

diversification of the search process in order to

escape from local optima

(Schiex, 1995). No doubt,

the simplest mechanism to diversify the search is to

consider a noise part during the process. Otherwise

the search process executes a random movement

with probability p and follows the normal

process with a probability 1-p (Schiex, 1995).

The mutating sub-process will change; for each

selected chromosome following mutation probability

Pmut, the mutation will be random with a

probability 1-Pguid and guided with a probability

Pguid. So that in the proposed mutating sub-process

it’s possible to destroy a given solution in order to

enhance exploration.

3.9 Dynamic Approach

The main interest of the second improvement is

based on the NEO-DARWINISM theory (Darwin,

1859) and on the laws of nature « The Preservation

of favoured races in the struggle for life ». This

phenomenon can be described, in nature, by an

animal society in which the strongest members are

luckier to be multiplied (so their crossing–over

probability is high). The power of these elements

allows them not to be infected by illnesses (mutation

is then at a lower rate). On the contrary case the

weakest limbs of these animals are frequently ill or

unable to combat illnesses (mutation is frequent),

usually this kind of animals can’t attract females

(reproduction is limited). In fact to cross-over a

strong species and to give more mutation possibility

for a weak species can be very worthy.

So, from now on Pcross and Pmut will be

function of fitness and of a newer operator called ε.

This operator is a weight having values from 0 to 1.

In the new optimization process, each species

agent proceeds with its own genetic process. Indeed

before starting the optimization process agents have

to count their parameters Pcross and Pmut on the

basis of their fitness values. For a given Species

agent three cases are possible as described by the

new genetic process

4 EXPERIMENTATION

4.1 Experimental design

The goal of our experimentation is to compare a

distributed implementation with a centralized one of

genetic algorithm enriched by both template concept

and min-conflict-heuristic. The first implementation

is referred to as Distributed Guided Genetic

Algorithm (D

A) whereas the second one as

Guided Genetic Algorithm (GGA). The

implementation has been done with ACTALK (BRI,

89), a concurrent object language implemented

above the Object Oriented language SMALLTALK-

80.

This choice of Actalk is justified by its convivial

aspect, its reflexive character, the possibility of

D3G2A: A DYNAMIC DISTRIBUTED DOUBLE GUIDED GENETIC ALGORITHM FOR THE CASE OF THE

PROCESSORS CONFIGURATION PROBLEM

245

carrying out functional programming as well as

object oriented programming , and the simulation of

parallelism. In our experiments we uses the case of

the processor large configuration problem as

described and formulated in (Lau, 1998). In our

experiments we carry out 30 times the algorithms

and we take the average without considering

outliers. Concerning the GA parameters, all the

experimentations employ a number of generations

(NG) equal to 10, a size of initial population equal to

1000, a cross-over probability equal to 0,5, a

mutation probability equal to 0,2, a probability of

guidance equal to 0.5, LOD is equal to 3, λ equal to

10, ε is equal to 0.6. The performance is assessed by

the two following measures:

• Run time: the CPU time requested for solving a

problem instance,

• Satisfaction: the number of satisfied

constraints.

The first one shows the complexity whereas the

second recalls the quality. In order to have a quick

and clear comparison of the relative performance of

the two approaches. It’s tempting to note that

experimentation has been done and have the same

tendency of graphics for these cases 8,12,16,20 and

32 processors. This is why we choose to present

only the case of 16 and 32 processors.

4.2 Experimental results

The performance of the two approaches will be

compared as follow:

• For the same value of fitness, we compute the

time elapsed by each approch to attain it.

• For the same CPU time, we compute the fitness

value attained by the two approches.

Figures 1 and 2 illustrates the CPU time point of view,

and shows that the results provided by D

A are better

compared to those given by AGG. We mention that the

difference between two times becomes more significant

when one increases the value of the fitness function . This

proves that

A requires less time to solve a given

problem.

We have come to these results thanks to

agent interaction reducing GA temporal complexity.

In fact, the comunication between agents helps them

to in the solution investigation.

50000

100000

150000

200000

250000

300000

350000

400000

450000

500000

1000 1500 2000 2500 3000

fitness Values

CPU tim

D3G2A

GGA

Figure 1: Generated CPU Times for fixed fitness values in

the case of 16 processors

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

1000

1500

200

2500

30000

350

4000

CPU time

Fi t ness val ue

D3G2A

GGA

Figure 2: Generated fitness values for fixed CPU Times

in the case of 16 processors

500000

1000000

1500000

2000000

2500000

1000 1500 2000 2500 3000

Fitness values

CPU t i me

D3G2A

GGA

Figure 3: Generated CPU Times for fixed fitness values in

the case of 32 processors

The soltion quality (the fitness function value)

point of view is expressed in the figures 3 and 4. The

A always reaches, for a fixed time CPU, a

better value of the function fitness. This value is

more significant for more significant times CPU.

This clearly expresses the contribution in term of

quality of the found solution. this is the result of the

diversification and the intensification used in our

approach.

500

1000

1500

2000

2500

3000

3500

4000

4500

200000

0000

750000

100000

0000

175000

0000

CPU ti m e s

Fi t ness val ue

D3G2A

GGA

Figure 4: Generated fitness values for fixed CPU Times in

the case of 32 processors

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246

5 CONCLUSION AND

PERSPECTIVES

We have developed a newer approach called D

This approach is a dynamic distributed double

guided genetic algorithm enhanced by three new

parameters called guidance probability P

guid

, the

local optima detector LOD and the weight ε,. The

latter is a weight used by Species agents to

determine their own genetic process parameters on

the basis of their chromosomes Fitness values.

Compared to the centralized guided genetic

algorithm and applied to LPCP, our new approaches

have been experimentally shown to be better in

terms

of fitness value and CPU time.

The improvement is due to both diversification

and guidance. The first increases the algorithm

convergence by escaping from local optima

attraction basin. The latter helps the algorithm to

attain optima. Consequently D

A gives more

chance to the optimization process to visit all the

search space. We have come to this conclusion

thanks to the proposed mutation sub-process. The

latter is sometimes random, aiming to diversify the

search process, and sometimes guided in order to

increase the best of the fitness fonction value. The

genetic sub-process of D

A Species agents will no

longer be the same depending on their fitness values.

This operation is based on the species typology. The

sub-population of a species agent can be considered

as strong or weak with reference to its fitness value.

For a strong species, it’s better to increase cross-over

probability and to decrease mutation probability.

However, when dealing with a weak species, cross-

over probability is decreased and mutation

probability is increased. The occurrence of these

measures not only diversifies the search but also

explore wholly its space.

No doubt further refinement of this approach

would allow its performance to be improved. Further

works could be focused on applying these

approaches to solve real hard CSOPs like the radio

link frequency allocation problem

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