An Introduction to Genetic Algorithms - Boente

e defined likewise for string j, as illustrated in figure 4.3. Then |h| = |i| |i 2| + |j2|, and |k| = |j| |j 2| + |i2|. Vose
and Liepins simplify this with a nice trick of notation. First note that
where ^ denotes bitwise "and". Since 2 c 1 represents the string with l c zeros followed by c ones, |(2
^ i| returns the number of ones in the rightmost c bits of i. Likewise,
Let
Then |h| = |i| ³ i,j,c and |k| = |j| + ³i,j,c.
We can now write down a complete expression for ri,j(0). To simplify, let · = pm/(1 p m). Then, after some
algebra, we obtain
(4.10)
This gives the flavor of how this kind of analysis is done. With a clever use of logical operators and
permutations (which is beyond the scope of this discussion), Vose and Liepins were able to express the
general recombination operator in terms of M. (See Vose and Liepins 1991 for details.)
Let for vectors , where is the composition operator. Then, in the limit of an
infinite population,
Define Gp as
where denotes the sum of the components of vector Then, in the limit of an
infinite population,
Chapter 4: Theoretical Foundations of Genetic Algorithms
G and Gp act on different representations of the population, but one can be translated into the other by a
simple transformation.
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c 1)