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this paper deals with selective pressure and diversity in cellular genetic algorithms ( cgas ) which are a subclass of genetic algorithms where the population is embedded in a grid .
these concepts are closely related to the exploration / exploitation trade - off .
previous studies on cgas selected the size and the shape of neighborhoods @xcite , or the shape of the grid @xcite as basic parameters to tune the search process .
altering these structural parameters entails a deep change in the way we deal with the problem .
for instance , there is no built - in mean to swap from a rectangular grid to a square grid without misshaping the neighborhood relation .
we suggest using _ anisotropic selection _ ( as ) to promote diversity and to control accurately the selective pressure in genetic search . the main advantage of the _ anisotropic selection _
scheme is that it allows to control the exploration / exploitation trade - off without affecting neither the grid topology nor the neighborhood shape ; so the cellular genetic algorithm we propose merely works on a square grid and a simple von neumann neighborhood shape .
+ the paper is divided in 6 sections .
section [ section1 ] gives a brief definition of cgas and an overview of existing techniques used to control the exploration / exploitation tradeoff .
section [ section2 ] introduces the as scheme .
section [ section3 ] studies the influence of as on the selective pressure . in section [ section4 ]
we compare as and rectangular grids topologies influence on the selective pressure . in section [ section5 ]
we show how as promotes the emergence of niches . in section [ section6 ]
we use a cga on a quadratic assignment problem to measure the correlation between anisotropy and performance .
finally we tie together the results of the previous sections and suggest directions for further research .
this section presents a brief overview on cellular genetic algorithms and a standard technique to measure the selective pressure .
[ section1 ]      cellular genetic algorithms are a subclass of genetic algorithms ( gas ) in which exploration and population diversity are enhanced thanks to the existence of small overlapped neighborhoods @xcite . such algorithms are specially well suited for complex problems @xcite .
we assume a two - dimensional toroidal grid as a spatial population structure .
each grid cell contains one individual of the population .
the overlapping neighborhoods provide an implicit mechanism for migration of genetic material throughout the grid .
a genetic algorithm is assumed to be running simultaneously on each grid cell , continuously selecting parents from the neighborhood of that grid cell in order to produce an offspring which replaces the current individual .      a standard technique to study the induced selection pressure without introducing the perturbing effect of variation operators is to let selection be the only active operator , and then monitor the number of best individual copies @xmath0 in the population @xcite .
the takeover time is the time it takes for the single best individual to conquer the whole population .
the grid is initialized with one cell having the best fitness and all the other having a null fitness . since no other evolution mechanism but
selection takes place , we can observe the way the best individual spreads over the grid by counting generation after generation the number of copies of this one .
a shorter takeover time thus means a higher selective pressure .
it has been shown that when we move from a panmictic population , as in standard ga , to a spatially structured one of the same size with synchronous updating of the cells , the global selection pressure induced on the entire population is weaker @xcite .
links have been established between takeover time and neighborhood size and shape or grid topology .
neighborhood size and shape in a cga are parameters that have some influence on the takeover time
. a larger overlap of local neighborhoods of the same shape speeds up the best individual s spreading over the grid .
the influence of the shape is given by sarma and de jong through a measure on the neighborhood which represents the spatial dispersion of a cell pattern @xcite .
rather than the size of the neighborhood in terms of individuals , the key particularity of a local neighborhood is its radius .
the takeover time decreases while the radius increases in a spatially structured population .
[ sec - effectgridtopo ] we measure the relation between grid topology and selective pressure for rectangular grids where the population size is fixed to @xmath1 .
we use the following grid shapes : @xmath2 ; @xmath3 ; @xmath4 ; @xmath5 ; @xmath6 and @xmath7 individuals .
the selection strategy is a binary tournament .
for each cell we randomly choose two individuals in its neighborhood .
the best one then replaces the individual of the cell on the grid if it is fitter or with probability 0.5 if fitnesses are equal .
figure [ fig - crois - tournoi ] shows the average of @xmath8 independant runs of growth of @xmath0 against generations ; the takeover time is reached when @xmath0 is equal to the size of the grid ( see table i ) . the average growth rate @xmath9 , that is the number of new best individual copies per time unit , of these curves for four rectangular grid shapes ( @xmath2 , @xmath3 , @xmath4,@xmath5 ) is plotted in figure [ acrois_tournoi ] .
this figure helps us to understand the growth of @xmath0 .
the growth rate @xmath9 is the same for all grids for the first generations .
then , the spreading speeds down to reach a constant speed for rectangular grids .
this constant is @xmath10 where @xmath11 is the probability of selecting the best individual when there is one copy of it in the neighborhood and @xmath12 the shortest side of the grid @xcite .
more accurately , @xmath9 decreases when @xmath12 is filled by copies of the best individual ( see figure [ display_rec](b ) ) .
then , the speed becomes constant until the best individual has spread over to the other side ( see figure [ display_rec](c ) ) .
this explains why the @xmath2 grid curve has no constant period : the two sides are filled at the same time .
finally , the growth rate falls down to zero as the best individual finishes conquering the grid .
the results of the experiments we conducted are in agreement with e.alba and j.troya observations that narrow grid shapes induce low selective pressure @xcite .
we will see in the next sections that this behavior can be observed with the as too .
[ cols="^ " , ]     [ display_two ]
we experiment a cga using anisotropic selection on a quadratic assignment problem ( qap ) : nug30 .
our aim here is not to obtain better results with respect to other optimization methods , but rather to observe the behavior of a cga with as . in particular , we seek an optimal value for the anisotropy degree .
the qap is an important problem in both theory and practice .
it was introduced by koopmans and beckmann in 1957 and is a modal for many practical problems @xcite .
+ the qap can be described as the problem of assigning a set of facilities to a set of locations with given distances between the locations and given flows between the facilities .
the goal is to place the facilities on locations in such a way that the sum of the products between flows and distances is minimal .
+ given @xmath13 facilities and @xmath13 locations , two @xmath14 matrices @xmath15 $ ] and @xmath16 $ ] where @xmath17 is the distance between locations @xmath18 and @xmath12 and @xmath19 the flow between facilities @xmath20 and @xmath21 , the objective function is : + @xmath22 where @xmath23 gives the location of facility @xmath20 in the current permutation @xmath11 .
+ nugent , vollman and ruml suggested a set of problem instances of different sizes noted for their difficulty @xcite .
the instances they suggested are known to have multiple local optima , so they are difficult for a genetic algortihm .
we experiment our algorithm on their 30 variables instance called nug30 .
we consider a population of 400 individuals placed on a square grid .
each individual represents a permutation of @xmath24 .
we need a special crossover that preserves the permutations :    * select two individuals @xmath25 and @xmath26 as genitors . *
choose a random position @xmath20 .
* find @xmath21 and @xmath18 so that @xmath27 and @xmath28 . *
swap positions @xmath20 and @xmath21 from @xmath25 and positions @xmath20 and @xmath18 from @xmath26 .
* repeat @xmath29 times this procedure where @xmath13 is the length of an individual .
this crossover is an extended version of the upmx crossover proposed in @xcite .
the mutation operator consists in randomly selecting two positions from the individual and exchanging these positions .
the crossover rate is 1 and we perform one mutation per individual in average . + we consider 500 runs for each anisotropy degree .
each run stops after 1500 generations .
figure [ alpha - perf ] shows the average performance of the algorithm towards @xmath30 : for each value of @xmath30 we average the best solution of each run .
performances are growing with @xmath30 and then fall down as @xmath30 is getting closer to its limit value .
this curve shows the influence of the selective pressure on the performances and how important it is to control it accurately .
the best average performance is observed for @xmath31 , which corresponds to a good exploration / exploitation tradeoff . in the neighborhood of this optimal value
the algorithm favors propagation of good solutions in the vertical direction with few interactions on the left or right sides .
this kind of dynamics is well adapted to multi - modal problems as we can reach local optima on each columns of the grid and then migrate them horizontally to find new solutions .
performances would probably improve if the selective pressure did not remain static during the search process .
as in @xcite , we can define some criteria to self - adjust the anisotropy degree along generations .
furthermore , we can assign a different anisotropy degree to each cell of the grid , so that we can determinate criteria to self - adjust selective pressure locally upon measures on neighborhoods .
this paper presents a new selection scheme in cellular genetic algorithms .
the main objective is to control the exploration / exploitation tradeoff in a flexible way .
we propose to exploit the cellular ga characteristics to promote diversity during a genetic search process .
previous studies on cgas selected structural parameters , as neighborhood or grid shape , to tune the selective pressure .
the main drawback of these techniques is that altering a structural parameter entails a deep change in the way we deal with the problem .
the new selection scheme we suggest is based on _ fuzzy neighborhood _ where a cell is chosen according to different probabilities . in order to favor one direction rather than the other one , _
anisotropic selection _ chooses individuals in fuzzy neighborhood .
experiments performed in order to establish relation between the takeover time and the degree of anisotropy are consistent with our expectation that selection pressure decreases with the degree of anisotropy .
analysis of the growth curves allows to distinguish three different phases in the diffusion process .
experimental results establish linear correlation in takeover between as and cga using rectangular grid .
then we point out capabilities of as to promote the emergence of niches . finally , using a cga with as on a qap
we have shown the existence of an anisotropic optimal value of @xmath30 such that the best average performance is observed .
this paper is a preliminary investigation and a more extensive analysis must be made to confirm that equilibrium between exploration and exploitation makes as a good technique for complex problems in static or dynamic environments .
future work should address the following issues : comparison between as and changes in the neighborhood shape and size , measuring as effects with cga using mutation and crossover , change the balance of directions dynamically .
the latter point is an important feature : by tuning the control parameter @xmath30 , it would be possible to make the algorithm to self - adjust the selective pressure , depending on global or local measures .
this adaptive ability has two important advantages : first , parameter @xmath30 may vary in a continuous way , second , variations of this parameter have affect neither on the grid topology nor on the neighborhood shape .
such self - adaptive algorithms have been studied in previous works , but they need to change the grid topology to control the selective pressure @xcite , which means it is uniform in the grid . as allows different propagation speeds on each area of the grid , promoting diversification and intensification ( exploitation ) at the same time on different spots . in general , we have to continue investigation of _ anisotropic selection _ to assess its validity and generality .
m.  giacobini , a.  tettamanzi , and m.  tomassini .
modelling selection intensity for linear cellular evolutionary algorithms . in p.
l. et  al . ,
editor , _ artificial evolution , sixth international conference , evolution artificielle , ea  2003 _ , lecture notes in computer science , pages 345356 , marseille , france , october 2003 . springer .
in this paper we introduce a new selection scheme in cellular genetic algorithms ( cgas ) 
 . _ anisotropic selection _ ( as ) promotes diversity and allows accurate control of the selective pressure . 
 first we compare this new scheme with the classical rectangular grid shapes solution according to the selective pressure : we can obtain the same takeover time with the two techniques although the spreading of the best individual is different . 
 we then give experimental results that show to what extent as promotes the emergence of niches that support low coupling and high cohesion . 
 finally , using a cga with anisotropic selection on a quadratic assignment problem we show the existence of an anisotropic optimal value for which the best average performance is observed . 
 further work will focus on the selective pressure self - adjustment ability provided by this new selection scheme .    
 [ heuristic methods ]