On Functions with a Fixed Fitness Versus Distance-to-Optimum Relation
Leila Kallel CMAP – UMR CNRS 7641, Ecole Polytechnique, Palaiseau 91128, France
[email protected]
Bart Naudts Dept. of Maths & Comp. Sci. University of Antwerp, RUCA Groenenborgerlaan 171 B–2020 Belgium
[email protected]
Abstract Recent work stresses the limitations of fitness distance correlation (FDC) as an indicator of landscape difficulty for genetic algorithms (GAs). Realizing that the correlation value cannot be reliably related to landscape difficulty, we investigate in this paper whether an interpretation of the correlation plot can yield reliable information about the behavior of the GA. Our approach is as follows. We present a generic method for constructing fitness functions which share the same fitness versus distance-to-optimum relation. Special attention is given to relations which show no local optimum in the correlation plot, as is the case for the relation induced by Horn’s long path. We give an inventory of different types of GAbehavior found within a class of fitness functions with a common correlation plot. We also show that GA-behavior can be very sensitive to small modifications of the initial fitness–distance relation.
1 INTRODUCTION A number of summary statistics have been proposed to characterize classes of fitness functions exhibiting similar GA-behavior. We mention fitness distance correlation (FDC, [4]), correlation length and operator correlation [6], epistasis variance [2], schema variance [8] and hyperplane ranking [10]. They originated from empirical observations and their efficiency is widely discussed (e.g., [9, 1, 7]). These discussions have mostly been limited to the construction of examples and counterexamples. An interesting direction, scarcely taken, is to construct and study classes of fitness functions which share the same property or statistic value, and to understand the differences in GA-behavior of these functions. Exemplar for the latter approach are the NK-landscapes [5]
Marc Schoenauer CMAP – UMR CNRS 7641, Ecole Polytechnique, Palaiseau 91128, France
[email protected]
which provide a simple way of constructing classes of functions with a given amount of epistasis (order of interaction between genes). The study of NK-landscapes gives useful information about the relation between the structure of the landscape and the order of epistasis. Note that the notion of epistasis is not to be confused with Davidor’s epistasis variance, which cannot distinguish between higher orders of epistasis. Fitness distance correlation measures the degree to which the fitness values increase with the individuals approaching the optimum in Hamming distance. Recent work (among others [7]) shows that the values produced by this statistic not necessarily related to the behavior of the GA. In brief, unless one restricts the scope of the statistic to some narrow class of fitness functions, the only reliable information about the difficulty or structure of the fitness landscape one gains from FDC is — at most — the sign of the correlation. Of course, the whole fitness distance plot may give more information than the summary statistic (correlation value). But an interpretation of this plot can be dangerous, since only partial information about the structure of the fitness landscape is displayed. A local optimum in the fitness landscape is not necessarily visible in the fitness distance plot. For example, the point x in figure (a) is both a local optimum in the fitness landscape and in the correlation plot. The point y in figure (b) could or could not be a local optimum in the fitness landscape, dependent on the distribution of the neighbors of y. f
f
y
x
d
(a) peak
d
(b) no peak
Hence “continuous” lines in the plot do not necessarily define a feasible path to the optimum of the fitness function,
although in an intuitive way this may seem to be the case. They may even hide intractable local optima. In order to study the effects of invisible information in the plot (e.g., the presence of local optima, the number of genes represented by one point), we will focus on fitness versus distance-to-optimum relations which show no local optimum in the correlation plot, as is the case in figure (b). How much difference in GA-behavior can we expect for a fixed relation? This paper is organized as follows: the first section introduces some conventions and lists the main results. In subsequent sections we go deeper into the details. Section 3 presents a generic method to construct fitness functions which follow a given fitness versus distance-to-optimum relation, with Horn’s long path [3] serving as a basic example. This method is then used in Sect. 4 to construct a class of functions whose members share the same fitness versus distance-to-optimum relation. We make an inventory of the different types of GA-behavior observed in this class, and study the effects of small modifications in the relation. Finally, Sect. ?? addresses the same issues, but with a consequent modification of the outshape of the fitness-distance relation. It gives further evidence for the observation that the correlation plot can be completely unreliable to distinguish a straightforward problem from an intractable one.
of f 1′ (convergence quality 3, vs. 2 for f 1 ). This is due to the upper path of f1′ (in the correlation plot), which hides an intractable local optimum. On the other hand, a small perturbation of the fitness values allows to to get rid of local constantness in the landscape. Applied to f 1 GA convergence quality (initially 2) becomes 1. Whereas GA behavior is still the same (quality 3) when the perturbation is applied to f 1′ . To sum up, GA convergence quality take values 1( f k , k >> 1), 2 ( f 1 ) and 3 ( f 1′ ), on the same correlation plot. A small translation of the fitness-distance relation along the distance axis, transforms f 1′ into an intractable problem (quality 4). Perturbing fitness values of the plot (to get rid of local constantness) does not change the convergence behavior with f 1′ , however, with f 1 convergence quality becomes 1. The second section considers a more general fitnessdistance relation, where the whole correlation plot is covered. It shows similar results to those of the first section: convergence quality 1, 3 and 4 can be observed with functions sharing the same correlation plot. A different rule is proposed, that completely changes GA trajectory in the correlation plot.
3 GENERAL CONSTRUCTION 2 MAIN RESULTS 2.1 Experimental conditions Unless explicitly stated, we use a generational GA with population size of 50, linear ranking selection, uniform crossover, and 1/ℓ-mutation with a rate of 0.5 per individual. The stopping criterion is set at 100,000 fitness evaluations without improvement of the best value, with a maximum of 1,000,000 fitness evaluations. All results are averaged over 50 independent runs.
In this section, we propose a way of constructing fitness functions which share the same fitness versus distance-tooptimum relation. The shape of this relation, depicted in Fig. 1, is similar to that induced by Horn’s long path [3]. Only successive points on Horn’s long path are within Hamming distance 1 from each other; they define a unique feasible path for a hill climber. Let us drop the requirement of proximity of neighboring points on the path, and constrain the relation between the fitness of a string and its distance to the optimum to functions of the form
2.2 Summary of the results Section 4 studies a class of fitness functions f k sharing the same fitness-distance relation of Eq. 8, sketched in Fig. 3. It shows that according to the choice of the rule (for instance k value in Eq. 9), GA convergence can be either stable and straightforward (convergence quality 1), or unstable (convergence quality 2). Moreover, GA trajectories in the correlation plot, varies according to the choice of the rule. However, changing the rule yields a consequent change of strings density in the correlation plot, which partially explains the observed differences. A mostly interesting result is obtained with the function f 1′ , which follows the same fitness-distance relation as f1 , and is equals the function f 1 on below average distances (to optimum) only. Yet, most GA runs fail to find the optimum
H (fitness) = distance-to-optimum,
(1)
H : [0, f max ] → {0, . . . , ℓ},
(2)
with H
−1
(0) = { f max }.
(3)
In this way, no local optimum is visible in the correlation plot, and we obtain a long-path shaped relation (each fitness value is associated with a unique distance). Note that the choice of optimum is arbitrary; unless otherwise mentioned, it is set at s ∗ = 11 . . . 11. All fitness function satisfying relation H can now be described by assigning values to a rule function, which maps the strings to natural numbers and obeys ∀s ∈ {0, 1}ℓ : rule(s) ∈ {1, . . . , #H −1(d(s, s ∗ ))}. (4)
by
Fitness
50 rule(s) =
i [ p( f i ) < pos(s) < p( f i+1 )],
(5)
i=1
30
where p( f i ) corresponds to the position in the path of the (unique) string at fitness fi . The brackets are used for the indicator function. The p( f i ) are determined recursively by
10 0
1 Distance to optimum (a) Correlation plot
Distance to optimum
K X
p( f 1 ) = 1, p( f i+1 ) = H ( f i ) − H ( f i+1 ) + p( f i ).
1
(6) (7)
4 CASE STUDY OF A FITNESS VERSUS DISTANCE-TO-OPTIMUM RELATION
0 10
30 Fitness (b) Relation H
50
Figure 1: (a) Correlation plot of Horn’s long path problem. (b) Relation H for Horn’s long path problem. The points marked on the fitness axis are the extrema f 1 , . . . , f K of the function H .
The first part of this section uses the method presented in the previous section to construct a class of fitness functions whose members share the same fitness versus distance-tooptimum relation. The properties of some of the fitness landscapes in this class are investigated. The second part of the section gives an inventory of the different types of GA-behavior found in this class. The effects of small modifications in the relation and different distributions of the individuals are studied in this second part. 4.1 The construction Let us take sinusoidal assignments of values for the function H , parametrized by the number of periods κ:
A fitness value can be assigned to each string by following the convention that lower rule values correspond to lower fitness values, i.e., if r ule(s) = 1 and d = d(s, s ∗ ), then the lowest fitness value of the set H −1(d) is assigned to s. If r ule(s) = k, with k ≤ #H −1(d), then the k-th greatest fitness value of the set H −1(d) is assigned to s.
Example: Horn’s long path rule As an example, we give the rule function for Horn’s long path, assuming that we have already chosen an adequate function H , such that the obtained correlation plot is that of the long path (see Fig. 1). We denote the extrema of H by f 1 , f2 , . . . , f K . The optimum of the long path is reached in 110 . . . 0. Horn, Goldberg and Deb [3] give a recursive algorithm which either returns the position of a string in the long path or indicates that this string is off the path. Let pos(s) be the position of s in the path. It ranges from 1 to 3.2(ℓ−1)/2 − 1, which is the length of the path when ℓ is odd. The special value pos(s) = 0 indicates that s is off the path. Armed with this pos function, we define the rule function
H (x) =
ℓ/2 (x sin(x) + x max ) , x max − π
(8)
for x ∈ [0, x max], with x max = 2πκ − π/2. After a suitable restriction of its domain, H satisfies the conditions of Eqs. 2 and 3. For sake of symmetry, and to avoid a bias towards above average distances, the function H is slightly translated to left, which centers the plot around ℓ/2 (see Fig. 3). The fittest string s ∗ = 11 . . . 11 is at leftmost point. The strings represented by the rightmost point of the plot, which are in Hamming distance closest to 00 . . . 00, are collected in the set S0 . Having defined the relation H between fitness values and the distance of the strings to the optimum, we proceed by defining a parametrized class of rule functions, denoted {rk }, hereby obtaining a class of fitness functions { f k } satisfying the relation H . First some terminology. Given a string s, we denote by K 0 (s) the position of the k-th zero of s. The K 0 ’s range in {k, . . . , k + ℓ − d}. (K 0 (s) − k)/(ℓ − d + 1) is then the centred and normalized k-th zero position. We consider the rule rulek :
rulek (s) =
K 0 (s) − k |H −1(d)|, ℓ−d +1
(9)
This rule takes its values in {1..|H −1(d)|} if |H −1(d)| ≤ ℓ− d + 1. It then follows the conditions of eq. 4, and therefore, defines a fitness function f k with a fitness-distance relation H (eq. 8):
• Keeping k fixed, we observe that there are more lowfit strings at distance ℓ/2 + a than at distance ℓ/2 − a. Similarly, there are less high-fit strings at distance ℓ/2 + a than at distance ℓ/2 − a. • If k = 1 then, at some fixed distance from optimum, the number of strings decreases when fitness increases (Fig 2(a)). • As k increases, the number of strings increases at high fitness values, and decreases at low fitness values (Fig 2-b,c). • If k > (d + 1)/2, there are more strings at top than at bottom fitness values (Fig 2-b with d=5 or 2-c with d=15). Note that strings at distances d > ℓ/2, and with minimal k-th zero position (K 0 (s) = k), define an easy path up to s 0 . In fact, within these points, the fitness is a Hamming fitness to s 0 . The same thing happens symmetricly: strings at distances d < ℓ/2, define an easy path (with a Hamming fitness) to s ∗ . These paths will be called right and left paths in the following. 4.2 Hill-climber behavior on the function class { fk } This landscape contains many local optima for a 1-bit mutation HC, but experiments show almost none for the GA (defined in 2). Figure 3-c presents the main kinds of HC trajectories: For k = 1, the 1-bit mutation HC, the 1/l-mutation (1+1), (1+50) and (50+50) evolution strategies show the same behavior: either they take the easy left-most path to the optimum, or get stuck in some local optimum. As k increases most HC runs (ex. 70% for k=5) get stuck in a local optimum within 100 fitness evaluations. But
Number of strings
d=15 d=25 d=35
0
1 k-th zero position (a) k = 1
1013 1011 109 107 105 103 101
d=15 d=25 d=35 d=45 d=5
Number of strings
Figure 2 shows the distribution of strings at distance ℓ/2 from the optimum in function of the k-th zero relative position: (K 0 (s) − k)/(l − d + 1). This relative position is directly correlated to fitness values. Hence, the plot also characterizes the distribution of strings in the fitness– distance plot:
1013 1011 109 107 105 103 101
0
1 k-th zero position (b) k = 5
1013 1011 109 107 105 103 101
Number of strings
Fitness values at distance d are allocated within the set {H −1(d)} such that the further the kth zero is in its feasible range, the better the fitness is.
d-15 d=25 d=35
0
1 k-th zero position (c) k = 9
Figure 2: Number of strings in logarithmic scale versus (normalized) k-th zero position in [k, ℓ+k−d], with ℓ = 50.
the search becomes more efficient for GA and evolution strategies.
Let us now concentrate on the behavior of the GA on the functions { f k }. Two types of GA-trajectories have been observed corresponding to the plots of figure 3: 1. Some runs go inside the fitness–distance plot, as in Fig. 3(b). They always find the optimum within 100 generations.
Fitness
4.3 GA-behavior on the function class { fk }
2. Some runs take the upper-right path, as in Fig. 3(a). In this case, two different situations have been observed:
2. With k increasing, the convergence time on the upperright path drops below 400 generations, but the number of runs which follow this path decreases.
0 0
2. With k increasing, the number of high-fit strings increases, and the search gets easier on the upper path. s0
On the other hand, the difficulty of the upper path (from to s ∗ ) is due to isolation. For instance, consider a point s on the upper path, at distance d > ℓ/2. The next better string s ′ is still on the upper path. But s ′ is at a smaller distance from optimum, hence, K 0 (s ′ ) does not belong to the same range than K 0 (s) (K 0 (s ′ ) is likely to be higher than K 0 (s ′ )). Hence s and s ′ are likely to be at a big hamming distance from each other. Two facts explain this phenomenon: when d(x, s ∗ ) decreases, first, the upper bound of K 0 (x), x ∈ E (l + k − d(x, s ∗ )) increases; second, because of the shape of H , the number of fitness values also may increase. These facts result respectively in a higher and smaller range for K 0 (s ′ ) than for K 0 (s) (see. Eq. 9). Therefore, many points on the upper path are at big hamming distance from each other. Moreover, there are flat or low fit areas in between.
20
0
The distribution of strings in the fitness–distance plot partially explains the observed GA-behavior, since the GA is mainly attracted to dense and high-fit areas. We distinguish again between k = 1 and k increasing:
0
1 Distance to optimum (b) 15% GA runs
Fitness
1. For k = 1, the dense areas define an easy right path to s 0 because of the conic shape of H . Therefore, most GA runs take the right path to s 0 . Then, due to elitism, the GA necessarily follows the top line from s 0 to s ∗ .
1 Distance to optimum (a) 85% GA runs
Fitness
1. When k = 1, most GA-runs follow this path. It takes the GA a very long time (1000 up to 5000 generations) to move from the right to the left side on the top line of the upper path (from s 0 to s ∗ ). Convergence is quicker on the left half of the top line.
20
20
0 0
1 Distance to optimum (c) 3 HC behaviors
Figure 3: Feasible 1-st zero position rule. GA and HC trajectory (of so far best individual at each generation) is plotted on the correlation plot of the whole space. (a) GA trajectory go straightforward up to s 0 , then follows the difficult upper path to converge at One string. (b) GA don’t go through the difficult upper path, and converges more or less quickly. (c) Observed 1-bit flip Hill Climber behaviors. 50 % runs take the leftmost path to s ∗ , 25% take the rightmost path to s0 , and 25% get rapidly stuck in a local optima inside the plot.
Fitness
fitness values of any two strings at different fitness-distance positions. The landscape of f 1+ still have local optima in hamming space, for instance, the 1-bit-flip HC shows the same three kinds of trajectories as with f1 , plotted in Fig. 3-c.
20
On the other hand, evolution strategies behave differently: the upper path intractable for a (1+1)-ES with the function f 1 , gets straightforward with the f 1+ landscape (it converges within 6000 fitness evaluations). Moreover, only 30% (vs. 50% with f 1 ) of the runs take the right+upper path (Figure 4).
0 0
1 Distance to optimum
Figure 4: Typical (1+1)-ES trajectories shown on the fitness–distance plot of the function f 1+ . This function is straightforwardly optimized by the (1+1)-ES. The picture is inversed at below average distances d < ℓ/2: The number of fitness values either decreases or is constant. Each time it decreases, the required range of K 0 for strings on the upper path gets a lot wider, and search gets easier. In light of the fitness landscape properties, we investigated GA behavior on different functions ( f k )k∈N , sharing the same fitness-distance relation. However, the density of strings in the plot varies for each k value. Hence, we compared functions that have different densities on the correlation plot. In the following two sections, we stick to the case of k = 1, and consider two modifications of the rule of Eq. 9. A first modification removes the flat areas in the fitness landscape that are responsible for isolation. 2-to symmetrize the distribution of individuals w.r.t. d = ℓ/2. 4.4 Removing the flat areas with the function f1+ Thereafter is presented a caricatural modification of the fitness function, that should point out obvious limitations of any statistical measure of the fitness landscape, based on fitness values. It also shows the sensitivity of GA behavior (in terms of trajectory) to local features of the fitness landscape. The fitness of a string s is now given by: X f 1+ (s) = f 1 (s) + (J (s)/ℓ) ∗ 10− j +1),
(10)
j =2..ℓ/5
where f 1 is the fitness defined above, based on the 1-st zero position. And J (s) is the position of the j-th zero in s. f 1+ , gets rid of a significant number of flat areas present in the f 1 landscape, by adding a tiny shadow (of less than 10−1 ) around each fitness-distance point. Yet, the overall fitness-distance relation is the same: strings density at each fitness level ( f + / − 10−1) are the same, as well as relative
The same thing happens for the GA: it converges within 300 generations on this right+upper path of f 1 + (vs. 1000 to 5000 generations with f 1 ). Moreover, the GA easily escapes the rightmost absorbing path of f 1 , and goes inside the fitness-distance plot. A noticeable feature is that any statistical measure of the fitness landscape f 1+ based on fitness values, would give roughly the same result as with f 1 . Though, convergence speed is completely different for the GA. The top path, initially intractable for (1+1)-ES, gets obvious to solve. Observed ES and GA trajectories are also different: they easily go inside the fitness-distance plot with f 1+ , rather than taking the absorbing rightmost path as with f1 . 4.5 Adding symmetry to the distribution with f1′ In the following, the same fitness-distance relation H of Eq. 1 is considered. A modification of the rule defined in Eq. 9 for above average distances only, transforms the (easy with f 1+ , difficult with f 1 ) upper path into a GA-intractable one: Recall that with the rule rulek of Eq. 9, the number of high fit candidates with at distances d + nbi ts/2 is much smaller than the number of high fit ones at distance d − nbi ts/2 (cf. Fig. 2). To symmetrize the distribution of individuals in the plot with respect to d = ℓ/2, we propose an alternative rule for above average distances. Let’s note K 1 (s) the k-th one position in s. Then, we define: K 0 (s) − k |H −1(d)|, i f d ≤ ℓ/2 ℓ − d + 1 rule′k (s) = K (s) − k 1 |H −1(d)|, i f d > ℓ/2 ℓ−d +1 (11) This yields a new fitness function f k′ , that follows the same fitness-distance relation (H ) as f k . Both functions are equal for strings at distance d < ℓ/2 from optimum. Due to the conic shape of H , f01 still defines an easy path to s 0 , but via strings with minimal k-th one position
(K 1 (s) = k).
In fact, strings on the upper path at d ≤ ℓ/2, have their k − th zero at highest range, and are very close to s10 = (111..1000..0). Whereas points at d > ℓ/2, have their k − th one at highest range, and are close to the bitwise complement of s10 ((000..0111..1)). Hence, it is not surprising that all runs get stuck in some intractable local optimum around d = ℓ/2 on the upper path.
Fitness
In the same way as for f 1 , experiments show that 70% GA runs take this easy path to s 0 , then, start walking on the upper path towards the optimum. But this time, the search stops around d = ℓ/2.
20
0 Remarks:
0
• Note that s01 is the most strong local optima possible in hamming space. Yet, it lies in a locally ideal fitness distance correlation (τ = −1). Note also that a GA involving complementary crossover would easily get out of this local optimum.
30
20 Fitness
• Note that shaping flat areas (as in section 4.4) of f k transforms it in a (1+1)-ES easy-function. This is not the case with f k′ . There is no feasible path that links (000..0111..1) to (111..1000..0) via points with either a high k-th zero or a high k-th one position. Shaping flat areas does not necessarily make landscapes easier.
10
0 0
The latter point is addressed in next section. A new function is designed, such that its outshape gives no useful information for the search.
5 DISCUSSION We presented a case study of GA behavior on a number of functions, which correlation plot shows no local optimum in the fitness-distance space. The overall conclusion is that
1
H 35 30 25
Fitness
20 15 10 5 0 0
• The conic shape of H yields two attractors for the GA that often starts at middle distances from s ∗ : rightmost and leftmost paths. One can expect a different shape for the relation H , to completely change GA behavior.
0.5 Distance to optimum
• With H , most (64%) runs take the right path, and get stuck in (00..11). Experiments show that a symmetric function to H with respect to d = ℓ/2 (fig. 5), give the opposite results: few runs (40%) take the right path to s10. • Strings distribution with f k is asymmetric with respect to d = ℓ/2: there are more low fit strings at above average distances. Hence, it is not surprising that H sym scarcely changes f k runs behavior: With H 70% runs take the right path, versus 50% with H sym .
1 Distance to optimum H0
0.5
1
Distance to optimum
H sym Figure 5: GA behavior on fitness function f 1 : With H0, only 2 runs out of 50 follow the right+upper path. function H is centered around d = ℓ/2: Most GA-runs take the right+upper path. The inverse happens with H sym: only few runs take the right+upper paths.
convergence quality varies from 1 to 4 according to (the invisible) strings distributions in the plot.
ume I of SFI studies, pages 619–712. Addison Wesley, 1989.
The fact is that two neighboring points in the correlation plot are not necessarily close in Hamming distance to each other: The extreme case of function f 1′ shows how a locally linear relation between fitness and distance to optimum (the upper path) can hide an intractable local optimum in the Hamming space. On the other hand correlation plots that cover the entire fitness-distance space, may contain easy paths to the optimum, eventually resulting in a onemax like behavior (g1′ ).
[6] B. Manderick, M. de Weger, and P. Spiessens. The genetic algorithm and the structure of the fitness landscape. In R. K. Belew and L. B. Booker, editors, Proceedings of the 4th International Conference on Genetic Algorithms, pages 143–150. Morgan Kaufmann Publishers, 1991.
However, the path that is chosen by the GA is heavily dependent on the position of the initial population in the correlation plot. For instance, a small translation of the fitness-distance relation along the distance axis completely changes the dynamic path of f 1′ for example.
[8] N. J. Radcliffe and P. D. Surry. Fitness variance of formae and performance prediction. In L. D. Whitley and M. D. Vose, editors, Foundations of Genetic Algorithms 3, pages 51–72. Morgan Kaufmann, 1995.
[7] Bart Naudts and Leila Kallel. Some facts about the so called GA-hardness measures. Submitted.
Hence, the uniform bitstring initialization is sometimes responsible for the failure of the GA (it causes the GA to take the difficult right path in the case of f 1′ , and concentrate the search within strings having an alternate number of blocks of ones and zeros in the case of g2′ ).
[9] S. Rochet, G. Venturini, M. Slimane, and E. M. El Kharoubi. A critical and empirical study of epistasis measures for predicting GA performances: a summary. In J.-K. Hao, E. Lutton, E. Ronald, M. Schoenauer, and D. Snyers, editors, Artificial Evolution 97, LNCS. Springer Verlag, 1998.
This shows how the commonly used uniform initialization can present serious drawbacks. Next section proposes alternative initialization procedures and gives some guidelines to choose, in each problem case, the adequate initialization.
[10] D. Whitley and ?? ?? In L. J. Eshelman, editor, Proceedings of the 6th International Conference on Genetic Algorithms, pages ?–? Morgan Kaufmann, 1995.
References [1] L. Altenberg. Fitness distance correlation analysis: an instructive counterexample. In Th. Bäck, editor, Proceedings of the 7th International Conference on Genetic Algorithms, pages 57–64. Morgan Kaufmann Publishers, 1997. [2] Y. Davidor. Epistasis variance: a viewpoint on GAhardness. In G. J. E. Rawlins, editor, Foundations of Genetic Algorithms, pages 23–35. Morgan Kaufmann Publishers, 1991. [3] J. Horn and D. E. Goldberg. Genetic algorithms difficulty and the modality of fitness landscapes. In L. D. Whitley and M. D. Vose, editors, Foundations of Genetic Algorithms 3, pages 243–269. Morgan Kaufmann Publishers, 1995. [4] Terry Jones and Stephanie Forrest. Fitness distance correlation as a measure of problem difficulty for genetic algorithms. In L. J. Eshelman, editor, Proceedings of the 6th International Conference on Genetic Algorithms, pages 184–192. Morgan Kaufmann Publishers, 1995. [5] S. A. Kauffman. Adaptation on rugged fitness landscapes. In Lectures in the Sciences of Complexity, vol-
