A Review of Tournament Selection

in Genetic Programming

Yongshen g Fa n g 1and Jun Li2

1Department of Finance, Anhui Polytechnic University,

Wuhu City, Anhui, P.R. China

2Department of Information and Computing Science, Anhui Polytechnic University,

Wuhu City, Anhui, P.R. China

fbmy@ahpu.edu.cn

Abstract. This paper provides a detailed review of tournament selec-

tion in genetic programming. It starts from introducing tournament se-

lection and genetic programming, followed by a brief explanation of the

popularity of the tournament selection in genetic programming. It then

reviews issues and drawbacks in tournament selection, followed by anal-

ysis of and solutions to these issues and drawbacks. It ﬁnally points out

some interesting directions for future work.

1 Motivation

Accurately and objectively identifying the economic situation of students is the

most important step of funding poor students in tertiary educational organiza-

tions. For eﬀectively implementing the funding policies, it requires a workable,

reasonable and scientiﬁcally sound assessment approach. Campus OneCard, the

platform of Digital Campus, accumulates large volume of complicated consump-

tion data from students. How to mine these consumption data in order to help ac-

curately and objectively identifying the economic situation of students becomes a

hot research topic. Genetic programming (GP) [1], one of the metaheuristic search

methods in Evolutionary Algorithms (EAs) [2], is based on the Darwinian natu-

ral selection theory. Its special characters make it a very attractive algorithm for

many real world problems, including data mining, ﬁnical prediction, image recog-

nition, and symbolic regression. Therefore, we intend to use GP in the student

consumption data mining project. To be able to apply GP in mining the student

consumption data successfully, it is necessary to understand the status of the cur-

rent research results of the most important operator — selection — in GP.

Selection is a key factor of aﬀecting the performance of EAs. Commonly used

parent selection schemes in EAs include ﬁtness proportionate selection [3], rank-

ing selection [4], and tournament selection [5]. The most popular parent selection

method in GP is tournament selection. Therefore, this paper focus on tournament

selection and gives a detailed review of tournament selection in GP in order to pro-

vide a thorough understanding of selecting behaviour, features and drawbacks in

tournament selection, as well as possible directions for our research work.

Z. Cai et al. (Eds.): ISICA 2010, LNCS 6382, pp. 181–192, 2010.

c

Springer-Verlag Berlin Heidelberg 2010

182 Y. Fang and J. Li

This paper is organised as follows: Section 2 brieﬂy introduces GP; Section 3

introduces standard tournament selection and explains why it is so population

in GP, as well as formal modellings which describe its sampling and selection

behaviour; Section 4 presents issues related to the sampling strategy in standard

tournament selection, together with corresponding analyses and clariﬁcations;

Section 5 discusses drawbacks related to selection pressure control in standard

tournament selection, followed by corresponding solutions; Section 6 concludes

the paper and shows possible directions for future work.

2 Genetic Programming

GP is a technique of enabling a Genetic Algorithm (GA) [3] to search a poten-

tially inﬁnite space of computer programs, rather than a space of ﬁxed-length

solutions to a combinatorial optimisation problem. These programs often take

the form of Lisp symbolic expressions, called S-expressions. The idea of applying

GAs to S-expressions rather than combinatorial structures is due originally to

Fujiki and Dickinson [6, 7], and was brought to prominence through the work of

Koza [1]. The S-expressions in GP correspond to programs which a user seeks to

adapt to perform some pre-speciﬁed tasks. The ﬁtness of an S-expression may

therefore be evaluated in terms of how eﬀectively it performs this task. GP with

individuals in S-expressions is referred to as tree-based GP.

GP also has other categorises based on the representations of individual, for

instance, linear structure GP [8–11] and graph-based GP [12–15]. Linear struc-

ture GP is based on the principle of register machines thus programs can be

linear sequences of instructions. Graph-based GP is suitable for the evolution of

highly parallel programs which eﬀectively reuse partial results [16].

Brieﬂy, to fulﬁll a certain task, GP starts with a randomly initialised pop-

ulation of programs. It evaluates each program’s performance using a ﬁtness

function, which generally compares the program’s outputs with the target out-

puts on a set of training data (“ﬁtness cases”). It assigns each program a ﬁtness

value, which in general represents the program’s degree of success in achieving

the given task. Based on the ﬁtness values, it then chooses some of the programs

using a stochastic selection mechanism, which consists of a selection scheme and

a selection pressure control strategy. After that, it produces a new population

of programs for the next generation from these chosen programs using crossover

(sexual recombination), mutation (asexual), and reproduction (copy) operators.

The search algorithm repeats until it ﬁnds an optimal or acceptable solution,

or runs out of resources. A much more comprehensive ﬁeld guide to GP can be

found in [16].

3 Tournament Selection

The standard tournament selection randomly samples kindividuals with re-

placement from the current population of size Ninto a tournament of size k

and selects the one with the best ﬁtness from the tournament. Therefore, the

A Review of Tournament Selection in Genetic Programming 183

selection process of the tournament selection consists of two steps: sample and

then select. Commonly used tournament sizes are 2, 4 and 7. In general, since

the standard breeding process in GP produces one oﬀspring by applying mu-

tation to one parent and produces two oﬀspring by applying crossover to two

parents, the total number of tournaments needed is Nat the end of generating

all individuals in the next generation.

Tournament selection has the following features compared with other selection

schemes:

–Its selection pressure can be adjusted easily.

–It is simple to code, eﬃcient for both non-parallel and parallel architecture

[17].

–It does not require sorting the whole population ﬁrst. It has the time com-

plexity O(N)[18].

The last two features make tournament selection very population in GP: 1) GP

is very computationally intensive, requiring a parallel architecture to improve its

eﬃciency; 2) It is common to have millions of individuals in a population when

solving complex problems [19], thus sorting a whole population is really time

consuming. The linear time complexity in tournament selection is therefore very

attractive for GP.

As the selection process of standard tournament selection consists of sampling

and selecting, there are a large number of research focusing on diﬀerent sampling

and selecting strategies [20–24]. In addition to these practical studies, there are

many theoretical studies that model and compare the selection behaviour of a

variety of selection schemes [18, 25–29], as well as many dedicated theoretical

studies on standard tournament selection [17, 30, 31].

Based on the concept of takeover time [18], B¨ack [25] compared several selec-

tion schemes, including tournament selection. He presented the selection prob-

ability of an individual of rank jin one tournament for a minimisation task1,

with an implicit assumption that the population is wholly diverse (i.e., every

individual has distinct ﬁtness value), as:

N−k((N−j+1)

k−(N−j)k)(1)

In order to model the expected ﬁtness distribution after performing tournament

selection in a population with a more general form, Blickle and Thiele extended

the selection probability model in [25] to describe the selection probability of

individuals with the same ﬁtness. The model is quite abstract although it is quite

elegant. They deﬁned the worst individual to be ranked 1st and introduced the

cumulative ﬁtness distribution,S(fj), which denotes the number of individuals

with ﬁtness value fjor worse. They then calculated the selection probability of

individuals with rank jas:

S(fj)

Nk

−S(fj−1)

Nk

(2)

1Therefore the best individual is ranked 1st.

184 Y. Fang and J. Li

In order to demonstrate the computational savings in backward-chaining evolu-

tionary algorithms, Poli and Langdon [31] calculated the probability that one

individual is not sampled in one tournament as 1 −1

N, then consequently the

expected number of individuals not sampled in any tournament as:

NN

N−1−ky

(3)

where yis the total number of tournaments required to form an entire new

generation.

In order to illustrate that selection pressure in standard tournament selection

is insensitive to population size in general for populations with a more general

situation (i.e., some programs have the same ﬁtness value and therefore have

thesamerank),Xieet al. [32] presented a sampling probability model that any

program pis sampled at least once in y∈{1, ..., N }tournaments as:

1−N−1

NNy

Nk

(4)

and a selection probability model that a program pof rank jis selected at least

once in y∈{1, ..., N }tournaments as:

1−⎛

⎜

⎝1−j

i=1 |Si|

Nk

−j−1

i=1 |Si|

Nk

|Sj|⎞

⎟

⎠

y

(5)

where |Sj|is the number of programs of the same rank j.

4 Issues Related to Sampling Strategy

There are two commonly recognised issues in standard tournament selection.

These two issues are closely related to each other because they are both caused

by the sampling with replacement scheme in standard tournament selection.

4.1 Multi-Sampled Issue

One issue is that because individuals are sampled with replacement, it is possible

to have the same individual sampled multiple times in a tournament, which is

referred as multi-sampled issue [33].

To address this issue, Xie et al. [33] analysed the other form of tournament

selection described in [5] and termed it as no-replacement tournament selection.

The no-replacement tournament selection samples individuals into a tournament

without replacement, that is, it will not return a sampled individual back to the

population immediately thus no individual can be sampled multiple times into

A Review of Tournament Selection in Genetic Programming 185

the same tournament. After the winner is determined, it then returns all individ-

uals of the tournament to the population. Therefore, no-replacement tournament

selection is a solution to the multi-sampled issue.

Xie et al. [33] gave mathematical models describing the sampling and selection

behaviour in no-replacement tournament selection as follows: if Dis the event

that an arbitrary program is drawn or sampled in a tournament of size k,the

probability of Dis:

P(D)= k

N(6)

For a particular program p∈Sj,ifEj,y is the event that pis selected at least

once in y∈{1, ..., N }tournaments, the probability of Ej,y is:

P(Ej,y )=1−⎛

⎜

⎜

⎜

⎝

1−1

|Sj|⎛

⎜

⎜

⎜

⎝

j

i=1 |Si|

k

N

k−j−1

i=1 |Si|

k

N

k⎞

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎠

y

(7)

They then used three selection pressure measures, namely loss of program di-

versity [29], selection frequency [27], and their own measure selection probability

distribution, to compare no-replacement tournament selection with standard one

based on simulations of four populations with diﬀerent ﬁtness rank distributions.

Their simulation results showed that there exist few diﬀerence between the two

tournament selection schemes. To further investigate underlying reasons, they

presented a model to describe the relationship between population size, tourna-

ment size, and conﬁdence level in order to answer the following question:

“ for a given population of size N, if we keep sampling in-

dividuals with replacement, then after how many sampling

events, will we have a certain level of conﬁdence that there

will be duplicates amongst the sampled individuals?”

They ﬁnally showed that for common tournament sizes 4 or less, it is not

expected to see any duplicates in anything except very small populations. Even

for tournament size 7, it is not expected to see duplicates for populations less

than 200. For most common and reasonable settings of tournament sizes and

population sizes, the multi-sampled issue seldom occurs in standard tourna-

ment selection. Further, since duplicated individuals do not necessarily inﬂuence

the result of a tournament when the duplicates have worse ﬁtness values than

other sampled individuals, the probability of signiﬁcant diﬀerence between stan-

dard tournament selection and no-replacement tournament selection will be even

smaller. Therefore eliminating the multi-sampled issue in standard tournament

selection is very unlikely to signiﬁcantly change the selection performance. They

concluded that the multi-sampled issue generally is not crucial to the selection

behaviour in standard tournament selection.

186 Y. Fang and J. Li

4.2 Not-Sampled Issue

The other issue is that because individuals are sampled with replacement in

standard tournament selection, it is also possible to have some individuals not

sampled at all when using small tournament sizes, which is referred as not-

sampled issue [34].

This issue was illustrated initially through an experimental work by Gather-

cole [35]. He showed the selection frequency of each individual and the likelihoods

of not-selected and not-sampled individuals in tournament selection of diﬀerent

tournament sizes through 1000 simulations on a sample population of size 50.

In his simulation, only one child is produced by crossover or mutation, thus the

total number of tournaments required to generate the next entire population

is a function of the crossover rate, the mutation rate and the population size,

instead of being just the same as the population size. His experimental results

are interesting and useful, however it is not clear whether the sample population

was fully diverse or not. This issue was then theoretically described by [31].

Two earliest known attempts to address the not-sampled issue are unbiased

tournament selection by Sokolov and Whitley [24] and fully covered tournament

selection by Xie [36]. A further detailed analysis of the issue was given recently

by Xie et al. [34].

In [24], Sokolov and Whitley believed that the potential of better individuals

not getting chosen for recombination due to the random sampling is the bias

presented in standard tournament selection. Therefore, they developed their un-

biased tournament selection that “lines up two diﬀerent permutations of the

population and performs a pairwise comparison” with a constraint, which forces

compared individuals to be distinct. Consequently, their method can ensure that

every individual is sampled at least once. Tournament size 2 was used to test the

unbiased tournament selection on three problems: one with permutation-based

solution representation and two under bit encodings. Although the advantage of

a generational GA using the unbiased tournament selection varied for diﬀerent

population sizes on the three problems, they concluded that the impact of the

bias is signiﬁcant, and the unbiased tournament selection provides better perfor-

mance than other selection methods, including standard tournament selection,

a rank based selection and ﬁtness proportionate selection.

In [36], Xie also mentioned that since the sampling behaviour in standard tour-

nament selection was random, the individual with bad ﬁtness could be selected

multiple times and the individual with good ﬁtness could never be selected. He

presented fully covered tournament selection which excludes the individuals that

have been selected for next selection to ensure every individual have an equal

chance to participate tournaments. He tested the fully covered tournament se-

lection on two symbolic regression problems and concluded that the method is

eﬀective, implying that not-sample issue is worth of addressing.

Later, Xie et al. provided a detailed algorithm termed round-replacement tour-

nament selection which seems based on the fully covered tournament selection

and extended the analysis of the not-sampled issue. The algorithm is as follows:

A Review of Tournament Selection in Genetic Programming 187

1: Initialise an empty population T

2: while need to generate more oﬀspring do

3: if population size <k then

4: Reﬁll: move all individuals from the temporary population Tto the

population S

5: end if

6: Sampling kindividuals without replacement from the population S

7: Select the winner from the tournament

8: Move the ksampled individuals into the temporary population T

9: return the winner

10: end while

They described that in the round-replacement tournament selection any program

will be sampled exactly ktimes during the selection phase thus there is no need

to model the sampling probability. They gave a model to describe the selection

probability as follows: for a particular program p∈Sj,ifWjis the event that p

wins or is selected in a tournament of size k, the probability of Wjis:

P(Wj)= k

n=1 1

n|Sj|−1

n−1j−1

i=1 |Si|

k−n

N

k(8)

They used two measures, namely loss of program diversity and selection prob-

ability distribution, to compare round-replacement tournament selection with

standard one based on simulations of three populations with diﬀerent ﬁtness

rank distributions. Their simulation results showed that there exist some diﬀer-

ence between the two tournament selection schemes. They then tested the ef-

fectiveness of the round-replacement tournament selection on the even-6-parity,

a symbolic regression, and a binary classiﬁcation problems using three diﬀer-

ent tournament sizes, namely 2, 4 and 7. However, the experimental results

showed that the improvement of the round-replacement tournament selection

is statistically signiﬁcant only when the tournament size is 2 for the symbolic

regression and the binary classiﬁcation problems but practically the diﬀerences

are small. They ﬁnally concluded that although there are some diﬀerent selection

behaviour in the round-replacement tournament selection comparing with the

standard tournament selection, the diﬀerent selection behaviour leads to better

GP search results only when tournament size 2 is used for some problems; overall

solving the not-sampled issue does not appear to signiﬁcantly improve a GP sys-

tem for the given tasks; and the not-sampled issue in the standard tournament

selection is not critical.

The clariﬁcations from the literature are very useful. The results show us that

in order to improve tournament selection, simply tackling diﬀerent sampling

with or without replacement strategies is not suﬃcient and should not be our

research focus.

188 Y. Fang and J. Li

5 Drawbacks Related to Selection Pressure Control

5.1 Finer Level Selection Pressure Control

Controlling selection pressure in tournament selection could be done by chang-

ing tournament size. However, tournament size can only be an integer number.

This drawback limits the ability of tournament selection to inﬂuence selection

pressure at a coarse level. In order to make tournament selection be able to tune

selection pressure at a ﬁne level, Goldberg and Deb [18] presented an alterna-

tive tournament selection, which uses an extra probability p. When conducting

a tournament between two individuals, the individual with higher ﬁtness value

can be selected as a parent with the probability p, while the other is with the

probability 1 −p. By setting pbetween 0.5 and 1, it is possible to tune the se-

lection pressure continuously between the random selection and the tournament

selection with tournament size two. Recently, Hingee and Hutter [37] showed

that every probabilistic tournament is equivalent to a unique polynomial rank-

ing selection scheme.

Another attempt to address this drawback is a ﬁne grained tournament selec-

tion by Filipovi´cet al. [21] in the context of GAs for a plant location problem.

They argued that standard tournament selection does not allow precise setting

of the balance between exploration and exploitation [30]. In their ﬁne grained

tournament selection method, the tournament size is not ﬁxed but close to a

pre-set value. They claimed that the ﬁne grained tournament selection makes

the ratio between exploration and exploitation be able to be set very precise,

and that the method solves the plant location problem successfully.

5.2 Automatic and Dynamic Selection Pressure Control

In general, the larger the tournament size, the higher the selection pressure;

by using diﬀerent tournament sizes, the selection pressure can be changed to

inﬂuence the convergence of the genetic search process. However, it will not work

as we expected due to two existing drawbacks during population convergence.

One drawback is when groups of programs having the same or similar ﬁtness

values, the selection pressure between groups increases regardless of the given

tournament size conﬁguration, resulting in “better” groups dominating the next

population and possibly causing premature convergence [38]. The other draw-

back is when most of programs in population have the same ﬁtness value, the

selection behaviour eﬀectively becomes random [39]. Therefore, tournament size

itself alone is not adequate for controlling selection pressure.

In fact, according to [38], these drawbacks are part of a more general issue: the

evolutionary learning process itself is very dynamic. At some stages, it requires

a fast convergence rate (i.e., high parent selection pressure) to ﬁnd a solution

quickly; at other stages, it requires a slow convergence rate (i.e., low parent

selection pressure) to avoid being conﬁned to a local maximum. These require-

ments could be achieved by changing tournament size dynamically in standard

tournament selection. However, standard tournament selection is not aware of

A Review of Tournament Selection in Genetic Programming 189

the dynamic requests. In order to pick correct tournament size, it should col-

laborate with an extra component that can reveal the underlying dynamics and

determine the requests.

The known attempts to implicitly and partially address the drawbacks in-

clude bucket tournament selection by Luke and Panait [22] and clustering tour-

nament selection by Xie et al. [40]. They are implicit and partial solutions to the

drawbacks because the original goals of the alternatives were not to solve the

drawbacks. Instead, the buckets tournament selection is used to apply lexico-

graphic parsimony pressure on parent selection for problem domains where few

individuals have the same ﬁtness, and the clustering tournament selection is a

consequence of a ﬁtness evaluation saving algorithm which clusters a population

using a heuristic called ﬁtness-case-equivalence.

Fortunately, later on, Xie et al. re-analysed and re-test their clustering tour-

nament selection explicitly for the drawbacks and adopted two additional pop-

ulation clustering methods, including phenotype-based2and genotype-based3

[38]. They concluded that diﬀerent population clustering methods have diﬀerent

impact on diﬀerent problems but the clustering tournament selection can auto-

matically and dynamically adjust selection pressure along evolution as long as

the population is clustered properly. They further presented mathematical mod-

els and simulation results to explain why the clustering tournament selection is

eﬀective [32].

From these research results, we think that clustering tournament selection

is a promising direction to improve tournament selection as it is an automati-

cally biased parent selection scheme that is needed by the dynamic evolutionary

process: when most of the population are of worse ﬁtness ranks and evolution

encounters a danger of missing good individuals, it tends to increase selection

bias to better individuals, hoping to drive the population to promising regions

quickly; when the population tends to converge to local optima and evolution

encounters a danger of losing genetic material, it tends to decrease selection bias

to better ones, hoping to keep the population diverse. We also think it would be

challenging to choose or develop an appropriate population clustering method

for a given problem but it is certainly an interesting research topic.

6 Conclusions

This paper reviewed the current status of research of tournament selection in GP.

It clearly showed that diﬀerent sampling with or without replacement strategies

have limited impact on the selection behaviour in standard tournament selection.

It also pointed out a promising research direction that is about population cluster-

ing and clustering tournament selection. As the student consumption data is very

complex, we should follow these research results to implement and/or to improve

the tournament selection operator in order to obtain good mining results.

2A population is clustered based on ﬁtness value.

3A population is clustered based on exact program structure and content.

190 Y. Fang and J. Li

References

1. Koza, J.R.: Genetic Programming — On the Programming of Computers by Means

of Natural Selection. MIT Press, Cambridge (1992)

2. Eiben, A.E., Smith, J.E.: Introduction to Evolutionary Computing. Springer,

Heidelberg (2003)

3. Holland, J.H.: Adaptation in Natural and Artiﬁcial Systems. University of Michi-

gan Press, Ann Arbor (1975)

4. Grefenstette, J.J., Baker, J.E.: How genetic algorithms work: A critical look at

implicit parallelism. In: Schaﬀer, J.D. (ed.) Proceedings of the 3rd International

Conference on Genetic Algorithms, pp. 20–27. Morgan Kaufmann Publishers,

San Francisco (1989)

5. Brindle, A.: Genetic algorithms for function optimisation. PhD thesis, Department

of Computing Science, University of Alberta (1981)

6. Fujiko, C.: An evaluation of holland’s genetic operators applied to a program gen-

erator. Master’s thesis, University of Idaho (1986)

7. Fujiko, C., Dickinson, J.: Using the genetic algorithm to generate lisp source code

to solve the prisoner’s dilemma. In: Proceedings of the Second International Con-

ference on Genetic Algorithms on Genetic algorithms and their application, pp.

236–240. Lawrence Erlbaum Associates, Inc., Mahwah (1987)

8. Banzhaf, W., Nordin, P., Keller, R., Francone, F.D.: Genetic Programming – An

Introduction. In: On the Automatic Evolution of Computer Programs and its Ap-

plications. Morgan Kaufmann, San Francisco (1998)

9. Ferreira, C.: Gene expression programming: a new adaptive algorithm for solving

problems. Complex Systems 13, 87 (2001)

10. Oltean, M.: Multi-expression programming. Technical report, Babes-Bolyai Univ.,

Romania (2006)

11. Oltean, M., Grosan, C.: Evolving evolutionary algorithms using multi expression

programming. In: Banzhaf, W., Ziegler, J., Christaller, T., Dittrich, P., Kim, J.T.

(eds.) ECAL 2003. LNCS (LNAI), vol. 2801, pp. 651–658. Springer, Heidelberg

(2003)

12. Handley, S.: On the use of a directed acyclic graph to represent a population of

computer programs. In: Proceedings of the 1994 IEEE World Congress on Com-

putational Intelligence, Orlando, Florida, USA, vol. 1, pp. 154–159. IEEE Press,

Los Alamitos (1994)

13. Hirasawa, K., Okubo, M., Katagiri, H., Hu, J., Murata, J.: Comparison between

Genetic Network Programming (GNP) and Genetic Programming (GP). In: Pro-

ceedings of the 2001 Congress on Evolutionary Computation, vol. 2, pp. 1276–1282

(2001)

14. Miller, J.F., Job, D., Thomson, P.: Cartesian genetic programming. In: Poli, R.,

Banzhaf, W., Langdon, W.B., Miller, J., Nordin, P., Fogarty, T.C. (eds.) EuroGP

2000. LNCS, vol. 1802, pp. 131–132. Springer, Heidelberg (2000)

15. Poli, R.: Parallel distributed genetic programming. Technical report, School of

Computer Science, University of Birmingham (1996)

16. Poli, R., Langdon, W.B., McPhee, N.F.: A ﬁeld guide to genetic programming

(2008) (With contributions by J. R. Koza), http://lulu.com, and freely available

at http://lulu.com

17. Miller, B.L., Goldberg, D.E.: Genetic algorithms, tournament selection, and the ef-

fects of noise. Technical Report 95006, University of Illinois at Urbana-Champaign

(1995)

A Review of Tournament Selection in Genetic Programming 191

18. Goldberg, D.E., Deb, K.: A comparative analysis of selection schemes used in

genetic algorithms. Foundations of Genetic Algorithms, 69–93 (1991)

19. Koza, J.R., Keane, M.A., Streeter, M.J., Mydlowec, W., Yu, J., Lanza, G.: Ge-

netic programming IV: Routine Human-Competitive Machine Intelligence. Kluwer

Academic, Dordrecht (2003)

20. Harik, G.R.: Finding multimodal solutions using restricted tournament selection.

In: Proceedings of the Sixth International Conference on Genetic Algorithms, pp.

24–31. Morgan Kaufmann, San Francisco (1995)

21. Filipovi´c, V., Kratica, J., To˘si´c, D., Ljubi´c, I.: Fine grained tournament selection

for the simple plant location problem. In: 5th Online World Conference on Soft

Computing Methods in Industrial Applications, pp. 152–158 (2000)

22. Luke, S., Panait, L.: Lexicographic parsimony pressure. In: Proceedings of the

Genetic and Evolutionary Computation Conference, pp. 829–836 (2002)

23. Matsui, K.: New selection method to improve the population diversity in genetic

algorithms. In: Proceedings of 1999 IEEE International Conference on Systems,

Man, and Cybernetics, pp. 625–630. IEEE, Los Alamitos (1999)

24. Sokolov, A., Whitley, D.: Unbiased tournament selection. In: Proceedings of Ge-

netic and Evolutionary Computation Conference, pp. 1131–1138. ACM Press, New

York (2005)

25. Back, T.: Selective pressure in evolutionary algorithms: A characterization of se-

lection mechanisms. In: Proceedings of the First IEEE Conference on Evolutionary

Computation, pp. 57–62 (1994)

26. Blickle, T., Thiele, L.: A comparison of selection schemes used in evolutionary

algorithms. Evolutionary Computation 4(4), 361–394 (1997)

27. Branke, J., Andersen, H.C., Schmeck, H.: Global selection methods for SIMD com-

puters. In: Fogarty, T.C. (ed.) AISB-WS 1996. LNCS, vol. 1143, pp. 6–17. Springer,

Heidelberg (1996)

28. Miller, B.L., Goldberg, D.E.: Genetic algorithms, selection schemes, and the vary-

ing eﬀects of noise. Evolutionary Computation 4(2), 113–131 (1996)

29. Motoki, T.: Calculating the expected loss of diversity of selection schemes. Evolu-

tionary Computation 10(4), 397–422 (2002)

30. Blickle, T., Thiele, L.: A mathematical analysis of tournament selection. In: Pro-

ceedings of the Sixth International Conference on Genetic Algorithms, pp. 9–16

(1995)

31. Poli, R., Langdon, W.B.: Backward-chaining evolutionary algorithms. Artiﬁcial

Intelligence 170(11), 953–982 (2006)

32. Xie, H., Zhang, M., Andreae, P.: Another investigation on tournament selection:

modelling and visualisation. In: Proceedings of Genetic and Evolutionary Compu-

tation Conference, pp. 1468–1475 (2007)

33. Xie, H., Zhang, M., Andreae, P., Johnston, M.: An analysis of multi-sampled issue

and no-replacement tournament selection. In: Proceedings of Genetic and Evolu-

tionary Computation Conference, pp. 1323–1330. ACM Press, New York (2008)

34. Xie, H., Zhang, M., Andreae, P., Johnston, M.: Is the not-sampled issue in tourna-

ment selection critical? In: Proceedings of IEEE Congress on Evolutionary Com-

putation, pp. 3711–3718. IEEE Press, Los Alamitos (2008)

35. Gathercole, C.: An Investigation of Supervised Learning in Genetic Programming.

PhD thesis, University of Edinburgh (1998)

36. Xie, H.: Diversity control in GP with ADFs for regression tasks. In: Zhang, S.,

Jarvis, R.A. (eds.) AI 2005. LNCS (LNAI), vol. 3809, pp. 1253–1257. Springer,

Heidelberg (2005)

192 Y. Fang and J. Li

37. Hingee, K., Hutter, M.: Equivalence of probabilistic tournament and polynomial

ranking selection. In: Proceedings of IEEE Congress on Evolutionary Computation,

pp. 564–571 (2008)

38. Xie, H., Zhang, M., Andreae, P.: Automatic selection pressure control in genetic

programming. In: Proceedings of the Sixth International conference on Intelligent

Systems Design and Applications, pp. 435–440. IEEE Computer Society Press, Los

Alamitos (2006)

39. Gustafson, S.M.: An Analysis of Diversity in Genetic Programming. PhD thesis,

University of Nottingham (2004)

40. Xie, H., Zhang, M., Andreae, P.: Population clustering in genetic programming.

In: Collet, P., Tomassini, M., Ebner, M., Gustafson, S., Ek´art, A. (eds.) EuroGP

2006. LNCS, vol. 3905, pp. 190–201. Springer, Heidelberg (2006)