GENETIC ALGORITHMS
These pages introduce some fundamentals of genetics
algorithms. Pages are intended to be used for learning
about genetics algorithms without any previous
knowledge from this area. Only some knowledge of
computer programming is assumed. You can find here
several interactive Java applets demonstrating work of
genetic algorithms.
Introduction to genetic algorithms with Java applets
Introduction to
Genetic Algorithms
GENETIC
ALGORITHMS
Main page
Introduction These pages introduce some fundamentals of genetics
algorithms. Pages are intended to be used for learning
Biological Background
about genetics algorithms without any previous
Search Space knowledge from this area. Only some knowledge of
Genetic Algorithm computer programming is assumed. You can find here
GA Operators several interactive Java applets demonstrating work of
GA Example (1D func.) genetic algorithms.
Parameters of GA
As the area of genetics algorithms is very wide, it is not
GA Example (2D func.) possible to cover everything in these pages. But you
Selection should get some idea, what the genetic algorithms are
Encoding and what they could be useful for. Do not expect any
Crossover and Mutation sophisticated mathematics theories here.
GA Example (TSP)
Now please choose next to continue or you can choose
Recommendations any topic from the menu on the left side. If you do not
Other Resources want to read all the introducing chapters, you can skip
Browser Requirements directly to genetic algorithms and return later.
FAQ You can also check recommendations for your browser.
About
Guest book (from 2/99) This site has also a Japanese translation.
(c) Marek Obitko, 1998
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(c) Marek Obitko, 1998
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About DNA
DNA (Deoxyribonucleic acid)
This is a part of DNA. More pictures are available.
(c) Marek Obitko, 1998
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About DNA
DNA
(Deoxyribonucleic acid)
Here you can see some pictures to get an idea how the DNA looks like. Some basic information about
biological background is also available.
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About DNA
(c) Marek Obitko, 1998
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Biological background
II. Biological Background
Chromosome
All living organisms consist of cells. In each cell there is the same set of chromosomes. Chromosomes
are strings of DNA and serves as a model for the whole organism. A chromosome consist of genes,
blocks of DNA. Each gene encodes a particular protein. Basically can be said, that each gene encodes a
trait, for example color of eyes. Possible settings for a trait (e.g. blue, brown) are called alleles. Each
gene has its own position in the chromosome. This position is called locus.
Complete set of genetic material (all chromosomes) is called genome. Particular set of genes in genome
is called genotype. The genotype is with later development after birth base for the organism's
phenotype, its physical and mental characteristics, such as eye color, intelligence etc.
Reproduction
During reproduction, first occurs recombination (or crossover). Genes from parents form in some way
the whole new chromosome. The new created offspring can then be mutated. Mutation means, that the
elements of DNA are a bit changed. This changes are mainly caused by errors in copying genes from
parents.
The fitness of an organism is measured by success of the organism in its life.
(c) Marek Obitko, 1998
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Introduction
I. Introduction
First Words
Genetic algorithms are a part of evolutionary computing, which is a rapidly growing area of artificial
intelligence.
As you can guess, genetic algorithms are inspired by Darwin's theory about evolution. Simply said,
solution to a problem solved by genetic algorithms is evolved.
History
Idea of evolutionary computing was introduced in the 1960s by I. Rechenberg in his work "Evolution
strategies" (Evolutionsstrategie in original). His idea was then developed by other researchers. Genetic
Algorithms (GAs) were invented by John Holland and developed by him and his students and
colleagues. This lead to Holland's book "Adaption in Natural and Artificial Systems" published in 1975.
In 1992 John Koza has used genetic algorithm to evolve programs to perform certain tasks. He called his
method "genetic programming" (GP). LISP programs were used, because programs in this language can
expressed in the form of a "parse tree", which is the object the GA works on.
(c) Marek Obitko, 1998
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Search Space
III. Search Space
Search Space
If we are solving some problem, we are usually looking for some solution, which will be the best among others. The space
of all feasible solutions (it means objects among those the desired solution is) is called search space (also state space).
Each point in the search space represent one feasible solution. Each feasible solution can be "marked" by its value or fitness
for the problem. We are looking for our solution, which is one point (or more) among feasible solutions - that is one point
in the search space.
The looking for a solution is then equal to a looking for some extreme (minimum or maximum) in the search space. The
search space can be whole known by the time of solving a problem, but usually we know only a few points from it and we
are generating other points as the process of finding solution continues.
Example of a search space
The problem is that the search can be very complicated. One does not know where to look for the solution and where to
start. There are many methods, how to find some suitable solution (ie. not necessarily the best solution), for example hill
climbing, tabu search, simulated annealing and genetic algorithm. The solution found by this methods is often
considered as a good solution, because it is not often possible to prove what is the real optimum.
NP-hard Problems
Example of difficult problems, which cannot be solved int "traditional" way, are NP problems.
There are many tasks for which we know fast (polynomial) algorithms. There are also some problems that are not possible
to be solved algorithmicaly. For some problems was proved that they are not solvable in polynomial time.
But there are many important tasks, for which it is very difficult to find a solution, but once we have it, it is easy to check
the solution. This fact led to NP-complete problems. NP stands for nondeterministic polynomial and it means that it is
possible to "guess" the solution (by some nondeterministic algorithm) and then check it, both in polynomial time. If we had
a machine that can guess, we would be able to find a solution in some reasonable time.
Studying of NP-complete problems is for simplicity restricted to the problems, where the answer can be yes or no. Because
there are tasks with complicated outputs, a class of problems called NP-hard problems has been introduced. This class is
not as limited as class of NP-complete problems.
For NP-problems is characteristic that some simple algorithm to find a solution is obvious at a first sight - just trying all
possible solutions. But this algorithm is very slow (usually O(2^n)) and even for a bit bigger instances of the problems it is
not usable at all.
Today nobody knows if some faster exact algorithm exists. Proving or disproving this remains as a big task for new
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Search Space
researchers (and maybe you! :-)). Today many people think, that such an algorithm does not exist and so they are looking
for some alternative methods - example of these methods are genetic algorithms.
Examples of the NP problems are satisfiability problem, travelling salesman problem or knapsack problem. Compendium
of NP problems is available.
(c) Marek Obitko, 1998
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About
About These Pages
About
These pages were developed during August and September 1998 at Hochschule für Technik und
Wirtschaft Dresden (FH) (University of Applied Sciences) by Marek Obitko, student of Czech Technical
University.
First versions of some applets were written during summer semester 1998 at Czech Technical University,
supervised by assoc. professor Pavel Slavík. During stay in Dresden the project was supervised by
professor Walter Pätzold from Hochschule für Technik und Wirtschaft Dresden.
Pages and Java Applets were all created by Marek Obitko, (c) 1998. If you have any comments,
questions or suggestions, you can send them to author.
Java is trademark of Sun Microsystems, Inc.
(c) Marek Obitko ([email protected]), 1998
http://cs.felk.cvut.cz/~xobitko/ga/about.html [7.5.2000 16:33:06]
Main page
GENETIC
ALGORITHMS
These pages introduce some fundamentals of genetics algorithms. Pages are
intended to be used for learning about genetics algorithms without any
previous knowledge from this area. Only some knowledge of computer
programming is assumed. You can find here several interactive Java applets
demonstrating work of genetic algorithms.
As the area of genetics algorithms is very wide, it is not possible to cover
everything in these pages. But you should get some idea, what the genetic
algorithms are and what they could be useful for. Do not expect any
sophisticated mathematics theories here.
Now please choose next to continue or you can choose any topic from the
menu on the left side. If you do not want to read all the introducing chapters,
you can skip directly to genetic algorithms and return later.
You can also check recommendations for your browser.
This site has also a Japanese translation.
[This page without frames] [This page with frames]
(c) Marek Obitko, 1998
http://cs.felk.cvut.cz/~xobitko/ga/main.html [7.5.2000 16:33:06]
Genetic algorithm
IV. Genetic Algorithm
Basic Description
Genetic algorithms are inspired by Darwin's theory about evolution. Solution to a problem solved by
genetic algorithms is evolved.
Algorithm is started with a set of solutions (represented by chromosomes) called population. Solutions
from one population are taken and used to form a new population. This is motivated by a hope, that the
new population will be better than the old one. Solutions which are selected to form new solutions
(offspring) are selected according to their fitness - the more suitable they are the more chances they have
to reproduce.
This is repeated until some condition (for example number of populations or improvement of the best
solution) is satisfied.
Example
As you already know from the chapter about search space, problem solving can be often
expressed as looking for extreme of a function. This is exactly what the problem shown here
is. Some function is given and GA tries to find minimum of the function.
You can try to run genetic algorithm at the following applet by pressing button Start. Graph
represents some search space and vertical lines represent solutions (points in search space).
The red line is the best solution, green lines are the other ones.
Button Start starts the algorithm, Step performs one step (i.e. forming one new generation),
Stop stops the algorithm and Reset resets the population.
Here is applet, but your browser does not support Java. If you want to see applets, please check browser
requirements.
Outline of the Basic Genetic Algorithm
1. [Start] Generate random population of n chromosomes (suitable solutions for the problem)
2. [Fitness] Evaluate the fitness f(x) of each chromosome x in the population
3. [New population] Create a new population by repeating following steps until the new population
is complete
1. [Selection] Select two parent chromosomes from a population according to their fitness (the
better fitness, the bigger chance to be selected)
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Genetic algorithm
2. [Crossover] With a crossover probability cross over the parents to form a new offspring
(children). If no crossover was performed, offspring is an exact copy of parents.
3. [Mutation] With a mutation probability mutate new offspring at each locus (position in
chromosome).
4. [Accepting] Place new offspring in a new population
4. [Replace] Use new generated population for a further run of algorithm
5. [Test] If the end condition is satisfied, stop, and return the best solution in current population
6. [Loop] Go to step 2
Some Comments
As you can see, the outline of Basic GA is very general. There are many things that can be implemented
differently in various problems.
First question is how to create chromosomes, what type of encoding choose. With this is connected
crossover and mutation, the two basic operators of GA. Encoding, crossover and mutation are introduced
in next chapter.
Next questions is how to select parents for crossover. This can be done in many ways, but the main idea
is to select the better parents (in hope that the better parents will produce better offspring). Also you may
think, that making new population only by new offspring can cause lost of the best chromosome from the
last population. This is true, so so called elitism is often used. This means, that at least one best solution
is copied without changes to a new population, so the best solution found can survive to end of run.
Some of the concerning questions will be discussed later.
Maybe you are wandering, why genetic algorithms do work. It can be partially explained by Schema
Theorem (Holland), however, this theorem has been criticised in recent time. If you want to know more,
check other resources.
(c) Marek Obitko, 1998
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Browser requirements
Browser Requirements
For best viewing of these pages you need a browser with support of frames, JavaScript and Java 1.1 (if
you see errors instead of applets, your browser supports Java 1.0). Recommended is Netscape Navigator
from version 4.07. You can also use Microsoft Internet Explorer from version 4.0, but support of Java is
strange in this browser (you may experience problems with redrawing and controlling applet).
Get Netscape
However, if you do not need to see Java Applets, any older browser (even without frames) can be used.
Netscape and Netscape Navigator are registered trademarks of Netscape Communications Corporation.
Microsoft Internet Explorer is trademark of Microsoft Corporation.
Java is trademark of Sun Microsystems, Inc.
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Other resources
Appendix: Other Resources
At this page are some selected links to web sites or ftps, where you can find more information about
genetic algorithms and concerning stuff.
ENCORE, the EvolutioNary COmputation REpository network
ftp://alife.santafe.edu/pub/USER-AREA/EC/ (there are also some others nodes)
FAQ - The Hitch-Hiker's Guide to Evolutionary Computation
ftp://alife.santafe.edu/pub/USER-AREA/EC/FAQ/www/index.html
FAQ - Genetic programming
http://www-dept.cs.ucl.ac.uk/research/genprog/gp2faq/gp2faq.html
The Genetic Algorithms Archive - many links, information about mailing list, some fun stuff
http://www.aic.nrl.navy.mil:80/galist/
Artificial Life Online - links, if you are looking for some introductory materials, look here
http://alife.santafe.edu/
Yahoo! Science:Computer Science:Algorithms:Genetic Algorithms - directory of other links
http://www.yahoo.com/Science/Computer_Science/Algorithms/Genetic_Algorithms/
Usenet groups comp.ai.genetic and comp.ai.alife
Note: All links were checked at the time of creating. If you find any broken link, please inform me.
(c) Marek Obitko, 1998
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Recommendations
XIII. Recommendations
Parameters of GA
This chapter should give you some basic recommendations if you have decided to implement your
genetic algorithm. These recommendations are very general. Probably you will want to experiment with
your own GA for specific problem, because today there is no general theory which would describe
parameters of GA for any problem.
Recommendations are often results of some empiric studies of GAs, which were often performed only on
binary encoding.
q Crossover rate
Crossover rate generally should be high, about 80%-95%. (However some results show that for
some problems crossover rate about 60% is the best.)
q Mutation rate
On the other side, mutation rate should be very low. Best rates reported are about 0.5%-1%.
q Population size
It may be surprising, that very big population size usually does not improve performance of GA (in
meaning of speed of finding solution). Good population size is about 20-30, however sometimes
sizes 50-100 are reported as best. Some research also shows, that best population size depends on
encoding, on size of encoded string. It means, if you have chromosome with 32 bits, the
population should be say 32, but surely two times more than the best population size for
chromosome with 16 bits.
q Selection
Basic roulette wheel selection can be used, but sometimes rank selection can be better. Check
chapter about selection for advantages and disadvantages. There are also some more sophisticated
method, which changes parameters of selection during run of GA. Basically they behaves like
simulated annealing. But surely elitism should be used (if you do not use other method for saving
the best found solution). You can also try steady state selection.
q Encoding
Encoding depends on the problem and also on the size of instance of the problem. Check chapter
about encoding for some suggestions or look to other resources.
q Crossover and mutation type
Operators depend on encoding and on the problem. Check chapter about operators for some
suggestions. You can also check other sites.
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Recommendations
Applications of GA
Genetic algorithms has been used for difficult problems (such as NP-hard problems), for machine
learning and also for evolving simple programs. They have been also used for some art, for evolving
pictures and music.
Advantage of GAs is in their parallelism. GA is travelling in a search space with more individuals (and
with genotype rather than phenotype) so they are less likely to get stuck in a local extreme like some
other methods.
They are also easy to implement. Once you have some GA, you just have to write new chromosome (just
one object) to solve another problem. With the same encoding you just change the fitness function and it
is all.On the other hand, choosing encoding and fitness function can be difficult.
Disadvantage of GAs is in their computational time. They can be slower than some other methods. But
with todays computers it is not so big problem.
To get an idea about problems solved by GA, here is a short list of some applications:
q Nonlinear dynamical systems - predicting, data analysis
q Designing neural networks, both architecture and weights
q Robot trajectory
q Evolving LISP programs (genetic programming)
q Strategy planning
q Finding shape of protein molecules
q TSP and sequence scheduling
q Functions for creating images
More information can be found through links in the appendix.
(c) Marek Obitko, 1998
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Operators of GA
V. Operators of GA
Overview
As you can see from the genetic algorithm outline, the crossover and mutation are the most important
part of the genetic algorithm. The performance is influenced mainly by these two operators. Before we
can explain more about crossover and mutation, some information about chromosomes will be given.
Encoding of a Chromosome
The chromosome should in some way contain information about solution which it represents. The most
used way of encoding is a binary string. The chromosome then could look like this:
Chromosome 1 1101100100110110
Chromosome 2 1101111000011110
Each chromosome has one binary string. Each bit in this string can represent some characteristic of the
solution. Or the whole string can represent a number - this has been used in the basic GA applet.
Of course, there are many other ways of encoding. This depends mainly on the solved problem. For
example, one can encode directly integer or real numbers, sometimes it is useful to encode some
permutations and so on.
Crossover
After we have decided what encoding we will use, we can make a step to crossover. Crossover selects
genes from parent chromosomes and creates a new offspring. The simplest way how to do this is to
choose randomly some crossover point and everything before this point point copy from a first parent
and then everything after a crossover point copy from the second parent.
Crossover can then look like this ( | is the crossover point):
Chromosome 1 11011 | 00100110110
Chromosome 2 11011 | 11000011110
Offspring 1 11011 | 11000011110
Offspring 2 11011 | 00100110110
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Operators of GA
There are other ways how to make crossover, for example we can choose more crossover points.
Crossover can be rather complicated and very depends on encoding of the encoding of chromosome.
Specific crossover made for a specific problem can improve performance of the genetic algorithm.
Mutation
After a crossover is performed, mutation take place. This is to prevent falling all solutions in population
into a local optimum of solved problem. Mutation changes randomly the new offspring. For binary
encoding we can switch a few randomly chosen bits from 1 to 0 or from 0 to 1. Mutation can then be
following:
Original offspring 1 1101111000011110
Original offspring 2 1101100100110110
Mutated offspring 1 1100111000011110
Mutated offspring 2 1101101100110110
The mutation depends on the encoding as well as the crossover. For example when we are encoding
permutations, mutation could be exchanging two genes.
(c) Marek Obitko, 1998
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Example of GA - Minimum of Function
VI. GA Example
Minimum of Function
About the Problem
As you already know from the chapter about search space, problem solving can be often expressed as
looking for extreme of a function. This is exactly what the problem shown here is.
Some function is given and GA tries to find minimum of the function. For other problems we just have to
define search space and the fitness function which means to define the function, which we want to find
extreme for.
Example
You can try to run genetic algorithm at the following applet by pressing button Start. Graph represents
some search space and vertical lines represent solutions (points in search space). The red line is the best
solution, green lines are the other ones. Above the graph are displayed old and new population. Each
population consists of binary chromosomes - red and blue point means zeros and ones. On the applet you
can see process of forming the new population in steps.
Button Start starts the algorithm, Step performs one step (i.e. forming one new generation), Stop stops
the algorithm and Reset resets the population.
We suggest you to start with pressing button Step and watching how GA works in details. The outline of
GA has been introduced in one of the previous chapters. First you can see elitism and then forming new
offspring by crossover and mutation until a new population is completed.
Here is applet, but your browser does not support Java. If you want to see applets, please check browser
requirements.
(c) Marek Obitko, 1998
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Parameters of GA
VII. Parameters of GA
Crossover and Mutation Probability
There are two basic parameters of GA - crossover probability and mutation probability.
Crossover probability says how often will be crossover performed. If there is no crossover, offspring is
exact copy of parents. If there is a crossover, offspring is made from parts of parents' chromosome. If
crossover probability is 100%, then all offspring is made by crossover. If it is 0%, whole new generation
is made from exact copies of chromosomes from old population (but this does not mean that the new
generation is the same!).
Crossover is made in hope that new chromosomes will have good parts of old chromosomes and maybe
the new chromosomes will be better. However it is good to leave some part of population survive to next
generation.
Mutation probability says how often will be parts of chromosome mutated. If there is no mutation,
offspring is taken after crossover (or copy) without any change. If mutation is performed, part of
chromosome is changed. If mutation probability is 100%, whole chromosome is changed, if it is 0%,
nothing is changed.
Mutation is made to prevent falling GA into local extreme, but it should not occur very often, because
then GA will in fact change to random search.
Other Parameters
There are also some other parameters of GA. One also important parameter is population size.
Population size says how many chromosomes are in population (in one generation). If there are too few
chromosomes, GA have a few possibilities to perform crossover and only a small part of search space is
explored. On the other hand, if there are too many chromosomes, GA slows down. Research shows that
after some limit (which depends mainly on encoding and the problem) it is not useful to increase
population size, because it does not make solving the problem faster.
Some recommendations for all parameters can be found in one of the following chapters.
Example
Here you can see example similar to previous one. But here you can try to change crossover
and mutation probability. You can also control elitism.
On the graph below you can see performance of GA. Red is the best solution, blue is
average value (fitness) of all population.
Try to change parameters and look how GA behaves.
Here is applet, but your browser does not support Java. If you want to see applets, please check browser
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