Eight Queens with Evolutionary Computing

Eight Queens with Evolutionary Computing
Ergun Çoruh, B.S. in E.E., Software Developer
November 2010
Abstract
The objective of this paper is to study and verify fundamental evolutionary
concepts by means of a simple computational example.
1 Introduction
Evolutionary computing mimics biological evolution to solve problems on a computer.
The author was inspired by The Eight Queens Problem given in the book
Introduction to Evolutionary Computing [1].
In this paper the author shall develop a computer program named evocom
to solve The Eight Queens Problem and then he shall quantitively analyse and
demonstrate fundamental aspects of biological evolution.
2 What is Evolution
Evolution (also known as biological, genetic or organic evolution) is the change in
the inherited traits of a population of organisms through successive generations
[2].
Figure 1 shows a basic flow diagram of evolution. Population is a pool of
individuals who own certain set of traits (phenotypes) which enable them to
adapt to certain environmental conditions. These traits are made from the
individual’s genetic material (genotypes, or chromosomes). Through a parent
selection mechanism fit parents are selected to reproduce.
The genetic material of offsprings are constructed when both parents’ chromosomes
are paired and recombined (shuffled like a deck of cards). During
recombination copy errors may occur too with certain probability. Such errors
cause mutation of recombined genes.
Recombination and mutation make new individuals with unique genetic material.
As a result these offsprings are either better or worse fitted for survival in
the environment they were born into. Through a process called survivor selection
(aka natural selection) fit offsprings who adapt better to the environment
would survive and be added into the population. Unfit ones shall not survive.
This cycle repeats with more individuals who are better at being adapted to the
environment joining the population in an increasing rate.
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Therefore in certain sense each species is a solution to fix a certain group
of environmental challenges.
Initialise
Termination
parent selection
Population
survivor selection
Figure 1: Evolution flow diagram
3 Evolutionary Algorithms
Parents
Offspring
recombination
mutation
Evolutionary algorithms follow the model given in Figure 1. The aspects of
evolutionary algorithms are:
• Representation (definition of individuals)
• Fitness function (evaluation function)
• Population (unit of evolution)
• Parent selection (pushes quality)
• Variation operators (recombination and mutation)
• Survival selection (replacement)
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Parent selection and survivor selection are selection operators which operate
at population level whereas recombination and mutation are variation operators,
and they operate on individuals.
3.1 Representation
Representation defines the mapping between the genotype space and the phenotype
space (see Figure 2). The entire genotype space of a species is called a
genome.
3.2 Fitness function
Figure 2: Evolution flow diagram
Fitness function is a function or procedure that assigns a quality measure to
genotypes. Essentially fitness function defines what improvement means. Fitness
function needs to be applied upon the population during parent selection
and survivor selection in order to determine the top (best) and the bottom
(worst) end of a population [1].
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3.3 Population
A population is a multi-set of genotypes (i.e. we should assume that there
may be individuals who are identical or very close in their fitness score). The
population forms the unit of evolution (i.e. individuals do not evolve, but the
population does) [1].
3.4 Parent selection
The role of parent selection is to select individuals based on their fitness score.
In nature too parents mate based on their fitness score. In biology this is often
broadly called sexual selection. Together with survival selection parent selection
is one of two forces that pushes quality.
3.5 Variation operators
There are two fundamental forces that form the basis of evolutionary systems:
• Variation operators
• Selection
Variation operators (recombination and mutation) create the necessary diversity
and thereby facilitate novelty to deal with environmental challenges.
Selection on the other hand acts as a force to push quality [1].
3.6 Purpose
In evolutionary computing algorithms perform stochastically, i.e. for a given set
of inputs number of iterations reaching a solution varies in each run.
There are also other limitations such as it is possible that the program may
climb a local hill rather than a global one, i.e. the program may not be able to
find the best set of solutions in a given run.
Nevertheless these known limitations are irrelevant for our purpose. The objective
of this study is not to maximise performance of evolutionary algorithms
(which may not necessarily perform better against deterministic algorithms) but
to demonstrate certain aspects of evolution, these are:
• Evolutionary process makes a given population increasingly better at being
adapted to the environment.
• Effects of mutation on diversity.
• Genetic drift in small populations.
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3.7 The Algorithm
The pseudo code of the evolutionary algorithm is given as follows:
BEGIN
INITIALISE population with random candidate solutions
EVALUATE each candidate
REPEAT UNTIL ( TERMINATION CONDITION ) DO
1 SELECT parents
2 RECOMBINE pairs of parents
3 MUTATE the resulting offspring
4 EVALUATE new candidates
5 SELECT individuals for the next generation
OD
END
4 The Eight Queens Problem
Problem Definition: Consider a regular 8 X 8 chess board. Place eight
queens such that none of them is able to check (capture) each other using
standard chess queen’s moves. Thus a solution requires that no two queens
share the same row, column, or diagonal.
In chess a queen can check every other piece diagonally, vertically
and horizontally.
4.1 Solutions
There are 4426165368 combinations eight queens can be placed on a chess board
(choose 8 from 64), and 92 of them are solutions [3].
C = n!/k!(n − k)! = 64!/8!(64 − 8)! = 4426165368
Let S be the solution set and C be all possible combinations.
Then:
S ⊂ C
Vast majority of combinations can be eliminated upfront as no two queens
can be placed vertically or horizontally.
Hence we can represent each solution candidate as a vector V :
V = 〈c1, c2, . . . , c8〉
Note that each position in vector V corresponds to a row number i, and each
value ci represents a column number at which a queen is placed. Hence if c2 is
equal to 5 then it follows that a queen is placed at 2 nd row 5 th column.
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Example: Some valid solutions are given as follows:
V = 〈8, 3, 1, 6, 2, 5, 7, 4〉
V = 〈4, 2, 5, 8, 6, 1, 3, 7〉
It should also be noted that in this scheme number of solution candidates is
reduced to permutations P of 8:
where
and
4.2 Fitness function
#{P} = 8! = 40320
S ⊂ P ⊂ C
#{S} = 92
In Section 4.1 we deduced a formula to represent each distinct solution as a
vector:
V = 〈c1, c2, . . . , c8〉
Each vector can be thought of a complete genotype (genome) to solve the
Eight Queens Problem. Hence a genome in this sense is equivalent to a solution.
Each element of the vector can be thought as a gene expressing a queen being
present at a given square. The overall genetic combination shall determine the
final fitness score such that no two queens can check each other.
Let number of checks in a given vector be:
0

where s is the interpolated fitness score (the larger the better). Since minimum
number of checks is when k is 0, the fittest individual shall have the score
of 7.
For example the following vector is perfectly fit (has score of 7) as no two
queens can check each other with this solution:
V = 〈8, 3, 1, 6, 2, 5, 7, 4〉
We should now develop a function to evaluate the fitness for a given vector.
The function must calculate the number of checks i.e. the value of k.
Let ci is the column index at i th row position where a queen is placed. Then
the check condition for ci must be when the row distance from current position
is equal to the column distance, i.e. when two queens are situated in the corners
of an equilateral right triangle.
Let d be the row distance between cj and ci. It follows that:
Then:
d = j − i : j > i
increment k ⇐⇒ d = cj ± ci
The following code listing shows the fitness function implemented in Python:
def fitness(c):
score = 0
score_min = 7
for i in range(0,7):
for j in range(i+1,8):
d = j-i
if c[j] > c[i]:
if c[i]+d == c[j]:
score += 1
break
else:
if c[i]-d == c[j]:
score += 1
break
return score_min-score
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Eight Queens Algorithm Description
Representation Permutations
Recombination ”Cut-and-crossfill” crossover
Recombination probability 100 pc
Mutation Swap
Mutation probability 80 pc
Parent selection Best 2 out of random 5
Survival selection Replace worst
Population size 100
Number of offspring 2
Initialisation Random
Termination condition Solution of 10,000 fitness evaluation
4.3 Evocom
The author has written a computer program called evocom and implemented
the evolutionary algorithm given in Section 3.7. Evocom is a Python command
line application that has various options to adjust configuration parameters.
coruh-imac:evocom ergun$ python evocom.py -h
evocom started.
Usage: evocom.py [options]
Options:
-h, --help show this help message and exit
-p POPULATION, --population=POPULATION
-t TRIALS, --trials=TRIALS
-r RUNS, --runs=RUNS
-b BESTFITS, --bestfits=BESTFITS
-m MUTATION, --mutation=MUTATION
-s SELECTFROM, --selectfrom=SELECTFROM
-v, --verbose
Population: Initial size of the population. Default is 100.
Trials: Maximum iterations to terminate the program in case search fails.
Default is 10000.
Runs: Number of runs. This parameter is needed to find the average performance
as each run finishes in different number of iterations. Default is 10.
Best fits: Number of fittest individuals to find before breaking the iteration.
Default is 5.
Select from: Number of parents randomly chosen from the population to pick
the fittest two to mate. Default is 5.
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5 Use Cases
5.1 Adaptation Rate
Evolution makes a given population of species increasingly better at being
adapted to their environment unless abrupt catastrophic conditions emerge.
Adaptation is a gradual process since evolution by natural selection takes place
over many generations (iterations).
Using evocom number of iterations vs. best-fits was plotted. When number
of fittest individuals reached the value of best-fits parameter configured with -b
command line option data were printed and the program terminated.
The data in Figure 3 clearly demonstrates how adaptation rate increases over
time. The X axis shows number of generations, the Y axis shows the number
of fittest individuals in a population of 100. It takes fewer generations to adapt
to the environment as number of fit individuals increases.
Adaptation Data Series
Best fits (BF) Generations (G)
20 507
40 661
60 748
80 837
100 909
Wolfram input:
Figure 3: Adaptation rate graph
plot{{0,0},{507,20},{661,40},{748,60},{837,80},{909,100}}
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Sample evocom output:
coruh-imac:evocom ergun$ python evocom.py -r 100 -b 20
evocom started.
===============================
Average iterations = 507
Total failures = 0
population = 100
trials = 10000
runs = 100
best fits = 20
mutation probability= 80%
select from = 5
===============================
5.2 Effects of Mutations
Mutations are necessary to drive diversity in a given population. Recombinations
(crossing-over) of parent genes would not be enough for better adaptation.
It should be noted that the graph given in Figure 4 should not be taken as
a universal benchmark as mutations are complex phenomena with many nonlinear
parameters in effect.
Using evocom we recorded number of iterations vs. mutation probabilities
while searching for 10 bets-fit individuals in a population of 100. The mutation
probability was specified with -m command line option (-m 80 means 80 percent
mutation probability).
Figure 4: Mutation probability graph
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Mutation Probability Data Series
Probability (P) Generations (G)
0 1081
5 444
10 308
20 248
40 236
60 268
80 351
100 599
Wolfram input:
Plot{{0,1081},{5,444},{10,308},{20,248},{40,236},{60,268},{80,351},{100,599}}
In Fifure 4 the X axis shows probability of applying mutation on offspring
genes. The Y axis shows generations required to reach to a constant number of
fit individuals (10) in a given population of 100.
It is interesting to note that very low and very high probability of mutations
have adverse effects on adaptation, the optimum range lies in between.
Note also from the command output (see below) when mutation probability
is zero (i.e. no mutation is applied to crossed-over genes) then 8 out of 100 runs
actually failed, i.e. unable to produce 10 fit individuals in 10000 iterations.
This demonstrates the necessity to have small non-zero probability of mutations
to drive diversity in populations.
Samle evocom output:
coruh-imac:evocom ergun$ python evocom.py -r 100 -b 10 -m 0
evocom started.
===============================
Average iterations = 1081
Total failures = 8
population = 100
trials = 10000
runs = 100
best fits = 10
mutation probability= 0%
select from = 5
===============================
5.2.1 The mutation function
In evocom the mutation function simply swaps two genes at random positions
as shown below:
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’’’ mutate
Mutate an offspring genome
Simply swap two bits in random location
’’’
def mutate(c):
g = [0,1,2,3,4,5,6,7]
# find two random positions and swap
g = random.sample(g,8)
i = g[0]
j = g[1]
m = copy.copy(c)
m[i],m[j] = m[j],m[i]
return m
In the main algorithm mutation itself is applied with random probability (80
percent default):
# mutate with probability 80%
mp = random.randint(1,100)
if mp

evocom started.
===============================
Average iterations = 1337
Total failures = 11
population = 10
trials = 10000
runs = 100
best fits = 5
mutation probability= 80%
select from = 5
===============================
Genetic Drift Data Series
Population (P) Generations (G)
10 1337
20 271
40 181
60 211
80 219
100 260
Wolfram input:
Figure 5: Adaptation rate graph
ListPlot{{10,1337},{20,271},{40,181},{60,211},{80,219},{100,260}}
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References
[1] J.E. Smith, 2008. Introduction to Evolutionary Computing, Springer.
[2] Wikipedia, The Free Encyclopedia, Evolution,
http://en.wikipedia.org/wiki/Evolution
[3] Wikipedia, The Free Encyclopedia, Eight Queens Puzzle,
http://en.wikipedia.org/wiki/Eight_queens_puzzle
[4] Wikipedia, The Free Encyclopedia, Genetic drift,
http://en.wikipedia.org/wiki/Genetic_drift
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