Crossover operators are a crucial component of Genetic Algorithms and related approaches in Evolutionary Computation. Their purpose is to combine the genetic information of two parents to produce one or more offsprings that are “mixtures” of their parents. In this contribution we will be concerned with the specific setting of crossover operators for strings of fixed lengthover an alphabet of letters. Given two parental strings and one may for instance construct recombinant offsprings of the form and . The index serves as a breakpoint at which the two parents recombine. This so-called one-point crossover can be generalized to two or more breakpoints.
Given we say that is a -point crossover offspring of and if there are indices so that for all , , either for all or for all .
Note that this definition states that and are broken up into at most intervals that are alternately included into . This convention simplifies the mathematical treatment considerably and also conforms to the usual practice of including crossovers with fewer than the maximum number of breakpoints. Uniform crossover, where each letter is freely chosen from one of the two parents, is obtained by allowing breakpoints. We note, furthermore, that our definition ensures that the parental strings are also included in the set of possible offsprings.
Properties of -point crossover have been studied extensively in the past. [English(1997)] described key algebraic properties and isomorphisms between the search spaces induced by crossover and mutation with small populations have been analyzed by [Culberson(1995)]. A formal treatment of multi-point crossover with an emphasis on disruption analysis can be found in [DeJong and Spears(1992)]. A general review of genetic algorithms from the perspective of stochastic processes on populations can be found in [Schmitt(2001)]. In this context, crossover operators are represented by stochastic matrices. A similar matrix-based formalism is explored in [Stadler et al.(2000)Stadler, Seitz, and Wagner]. Coordinate transformations, more precisely the Walsh transform [Goldberg(1989)] and its generalizations to non-binary alphabets [Field(1995)] have played an important role in explaining the functioning of GAs in terms of building blocks and the Schema theorem [Holland(1975)]. As a generalization, an abstract treatment of crossover in terms of equivalence relations has been given by [Radcliffe(1994)].
[Gitchoff and Wagner(1996)] proposed to consider the function that assigns to each possible pair of parents the set of all possible recombinants. They asked which properties of could be used to characterize crossover operators in general and explored properties of -point crossover on strings. In particular, they noted the following four properties:
for all ,
for all ,
for all ,
Mulder introduced the concept of transit functions characterized by the axioms (T1), (T2), and (T3) as a unifying approach to intervals, convexities, and betweenness in graphs and posets in last decade of the 20th century. Available as preprint only but frequently cited for more than a decade, the seminal paper was published only recently [Mulder(2008)]. For example, given a connected graph , its geodetic intervals, i.e., the sets of vertices lying on shortest paths between a pair of prescribed endpoints form a transit function usually denoted by [Mulder(1980)] and referred as the interval function of a graph . Unequal crossover, where (T3) is violated, has been rarely explored in the context of evolutionary computation, which the exception of the work by [Shpak and Wagner(1999)]. In this contribution we restrict ourselves exclusively to the simpler case of homologous string recombination. Thus, from here on we will assume that satisfies (T1), (T2), and (T3).
A common interpretation of transit functions is to view as the subset of lying between and . Indeed, a transit function is a betweenness if it satisfies the two additional axioms
and implies .
It is natural, therefore, to regard a pair of distinct points and without other points between them as adjacent. The corresponding graph has as its vertex set and if and only if and . The graph is known as the underlying graph of .
[Moraglio and Poli(2004)] introduced the notion of geometric crossover operators relative to a connected reference graph with vertex set by requiring – in our notation – that for all . In the setting of [Moraglio and Poli(2004)], the reference graph was given externally in terms of a metric on . When studying crossover in its own right it seems natural to consider the transits sets of in relation to the intervals of itself. Hence we say that is MP-geometric if
for all .
Note the condition (MG) is an axiom for transit functions independent of any externally prescribed structure on . [Mulder(2008)] considered a different notion of “geometric” referring transit functions that satisfy (B2) and the axiom
and implies .
Mulder’s version of “geometric” is less pertinent for our purposes because crossover operators usually violate (B2).
Another interpretation of , which is just as useful in the context of crossover operators, is to regard as the set of offsprings reachable from the parents and in a single generation. It is natural then to associate with a function so that if and if only eventually can be generated from and and all their following generations of offsprings. Formally, if there is a finite sequence of pairs so that , for all , , and . By construction, for all . If is a transit function, then is also a transit function.
We say that is closed if . Equivalently, a transit set is closed if and only if holds for all , since in this case nothing can be generated from the children of and that is not accessible already from and itself. In particular, all singletons and all adjacencies, i.e., individual vertices and the edges of , are always closed. A transit function is called monotone if it satisfies
For all and implies ,
i.e., if all transit sets are closed. By construction, satisfies (M) for any transit function . A simple argument555(i) by definition, (ii) implies , (iii) if but either (i) or axiom (T1) is violated. shows that if and only if . Thus and have the same underlying graph . The sets , finally, generate a convexity consisting of all intersections of the (finitely many) transit sets .
One of the most fruitful lines of research in the field of transit functions is the search for axiomatic characterizations of a wide variety of different types of graphs and other discrete structures in terms of their transit functions. [Nebeský(2001)] showed that a function is the geodesic interval function of a connected graph if and only if satisfies a set of axioms that are phrased in terms of only. Later, [Mulder and Nebeský(2009)] improved the axiomatic characterization of by formulating a nice set of (minimal) axioms. The all-paths function of a connected graph (defined as lies on some -path in ) admits a similar axiomatic characterization [Changat et al.(2001)Changat, Klavžar, and Mulder]. These results immediately raise the question whether other types of transit functions can be characterized in terms of transit axioms only.
Since -point crossover on strings over a fixed alphabet forms a rather specialised class of recombination operators we ask here whether it can be defined completely in terms of properties of its transit function . Beyond the immediate interest in -point crossover operators we can hope in this manner to identify generic properties of crossover operators also on more general sets .
This contribution is organised as follows. In section 2 we consider transit functions whose underlying graphs are Hamming graphs since, as we show in section 3, -point crossover belongs to this class. We then investigate the properties of -point crossover in more detail from the point of view of transit functions. In section 4 we switch to a graph-theoretical perspective and derive a complete characterization of -point crossover on binary alphabets, making use of key properties of partial cubes. In order to generalize these results we consider topological aspects of -point crossover in section 6 and explore its relationship with oriented matroids. We conclude our presentation with several open questions.
2 Hamming Graphs and their Geodesic Intervals
In most applications, -point crossover will be applied to binary strings or, less frequently, to strings over a larger, fixed-size alphabet . In a population genetics context, however, the number of allels may be different for each locus, hence we consider the most general case here, where each sequence position is taken from a distinct alphabet with for . The Hamming graph is the Cartesian products of complete graphs with vertices; we refer to the book by [Hammack et al.(2011)Hammack, Imrich, and Klavžar] for more details on Hamming graphs and product graphs in general. The special case for all is usually called -dimensional hypercube . The shortest path distance on is the Hamming distance , which counts the number of sequence positions at which the string and differ.
Given a transit function and a point let , i.e., is the degree of in the underlying graph . We write for the maximal degree of the underlying graph.
The purpose of this section is to characterize transit functions whose underlying graphs are Hamming graphs. Our starting point is the following characterization of hypercubes, which follows from results by [Mulder(1980)] and [Laborde and Rao Hebbare(1982)]:
Suppose is connected and each pair of distinct adjacent edges lies in exactly one 4-cycle. Then is isomorphic to -dimensional hypercube if and only if the minimum degree of is finite and .
Graphs with the property that any pair of vertices has zero or exactly 2 common neighbours are called -graphs [Mulder(1980)]. We note that the condition in Proposition 1 is necessary as demonstrated by the example in Fig. 1. Proposition 1 can be translated into the language of transit functions as follows:
Let be a transit function on a set with a connected underlying graph. Then the underlying graph is isomorphic to -dimensional hypercube if and only if satisfies:
For every such that there exist unique such that ,
Proposition 2 ([Mollard(1991)])
Let be a simple connected graph such that two non-adjacent vertices in either have exactly common neighbors or none at all, and suppose has neither nor (Figure 2) as induced subgraph. Then is independent of and is isomorphic to the Hamming graph if and only if , where is the maximum integer such that is nonzero.
These results can again be translated into the language of transit functions:
Let be a transit function with a connected underlying graph. Then the underlying graph is isomorphic to Hamming graph if and only if satisfies:
For every such that there exists unique such that ,
There exist no such that and ,
There exist no such that
The representation of Hamming graphs as -fold Cartesian products of complete graphs implies a “coordinatization”, that is, a labeling of the vertices the reflects this product structure. The geodesic intervals in Hamming graphs then have very simple description:
where and are the coordinates of the vertices and . Thus is a subhypercube of dimension as shown in example in the book by [Hammack et al.(2011)Hammack, Imrich, and Klavžar]. The intervals of Hamming graphs have several properties that will be useful for our purposes. A graph is called antipodal if for every vertex there is a unique “antipodal vertex” with maximum distance from .
Let be an induced sub-hypercube of a Hamming graph . Then for every there is a unique vertex so that .
This follows from a well known fact that hypercubes are antipodal graphs [Mulder(1980)]. It is well known that satisfies the monotone axiom (M) and thus also (B2).
Let and be two induced sub-hypercubes in a Hamming graph . Then is again an induced (possibly empty) sub-hypercube of .
For every coordinate , and contains at most two different letters from the alphabet . , and hence a hypercube. As an immediate consequence we note that satisfies even the stronger property
For all holds: if then there are so that .
The disadvantage of the results so far is that we have to require explicitly that is connected. In the light of condition (MG) above it seems natural to require connectedness of for recombination operators in general.
To-date, only sufficient conditions for connectedness of are known. Following ideas outlined by [van de Vel(1983)], we can show directly that the following property is sufficient:
For all : If , then if and only if .
As a technical device we will employ the partial order of defined, for given , by if and only if . As usual, we write if and . For we have the equivalence if and if only .
The underlying graph of a transit function is connected if satisfies axiom (CG).
Let be a transit function satisfying axiom (CG). Let be two distinct elements, and let be a maximal -chain between and , where the elements are labeled in increasing order .
We claim that, for any , , elements and form an edge in . To see this assume that, on the contrary, there is an element for some . Then (CG) implies , i.e., , contradicting maximality of the chain . Hence consists of consecutive edges whence is a connected graph.
However, property (CG) is much too strong for our purposes: Setting makes the condition in (CG) trivial, i.e., the axiom reduces to “ if and only if ”. Since implies we are simply left with axiom (B2), i.e., (CG) implies (B2). As we shall see below, however, string crossover in general does not satisfy (B2) and thus (CG) cannot not hold in general. Similarly, we cannot use Lemma 1 of [Changat et al.(2010)Changat, Mathew, and Mulder], which states that is connected whenever is a transit function satisfying (B1) and (B2).
Allowing conditions not only on but also on its closure we can make use of the fact that . Since satisfies the monotonicity axion (M) by construction, (B2) is also satisfied. Thus is connected if at least one of the following two conditions is satisfied: (i) satisfies (B1), or (ii) satisfies
and if and only if
The latter is equivalent to (CG) whenever satisfies (M). To see this observe that implies and by (M) .
So far, we lack a condition for the connectedness of that can be expressed by first order logic in terms of alone.
3 Basic Properties of -Point Crossover
We first show that the underlying graphs of -point crossover transit functions are Hamming graphs.
for all .
Since for by definition, it suffices to consider . By definition, if and only if and differ in a single coordinate, i.e., for which , i.e., and are adjacent in . Obviously, in this case. If there are two or more sequence positions that are different between the parents, then the crossover operator can “cut” between them to produce and generate an off-spring different from either parent so that .
Let be the -point crossover function. Then for all and all .
By construction agrees in each position with at least one of the parents, i.e., for , and thus . Conversely, choose an arbitrary . Find the first position in the coordinate representation in which disagrees with and form the recombinant that agrees with for and with for all . Then form the by recombining again after position . By construction, agrees with at least for all , i.e., in at least one position more than . Since we can repeat the argument at most time to find a sequence that agrees with in all positions. Since for all , we conclude that . As an immediate corollary we have:
-point crossover is MP-geometric for all .
MP-geometricity is a desirable property for crossover operators in general because it ensures that repeated application eventually produces the entire geodesic interval of the underlying graph structure.
Lemma 5 also implies a negative answer to one of the questions posed by [Mulder(2008)]: “Is the geodesic convexity uniquely determined by the geodesic interval function of a connected graph?”. More precisely, Lemma 5 shows that the -point crossover transit function also generates the geodesic convexity and hence that the geodesic convexity is not uniquely determined by the interval function as the , being the interval in a hypercube, is itself convex.
A trivial consequence of Lemma 5, furthermore, is the well known fact that the transit function of uniform crossover is the interval function on the Hamming graph:
for all .
For small distances, -point crossover also produces the full geodesic interval in a single step:
if and only if .
If we can place one cross-over cut between any two position at which and differ. In this way we obtain all possible recombinations, i.e., is a subhypercube of dimension in the underlying graph . Conversely if then there is such that requires more than cross-over cuts between positions at which and , which completes the proof.
Next, we observe that the transit sets of -point crossover can be constructed recursively.
W.l.o.g we can assume that and . Let and without loss of generality we can assume that ends with 0. Let denote the coordinate, with the last appearance of 1 in . Let be an element with for and otherwise. It follows that and moreover .
A key property in the theory of transit functions is the so-called Pasch axiom
For , and implies that .
satisfies the Pasch axiom (Pa).
Consider three arbitrary strings , , and . Then is a concatenation of a prefix of with the corresponding suffix of , or vice versa. Each has an analogous representation, leading to four cases depending on whether is a prefix or a suffix of and , resp., see Fig. 3. In case 1, if has a shorter -suffix than . Otherwise . In case 2, has a -prefix up to and has a -suffix starting at . If the two parts of overlap, i.e., then . If then a common crossover product is obtained by recombining both with and with at position . Case 3, has a -suffix and has an -prefix, can be treated analogously. Case 4, in which matches a prefix of both and can be treated as in case 1. In summary, thus for any choice of and , i.e., satisfies (Pa).
satisfies axiom (Pa) for all .
which in turn implies (B1).
Furthermore, also satisfies (M) and therefore in particular (B2). As an immediate consequence we conclude that is geometric in the sense of Nebeský. Note that this is not true for itself since (B2) is violated for all for all pairs of vertices with distance . Lemma 1 of [Changat et al.(2010)Changat, Mathew, and Mulder], furthermore, implies that is connected since is a transit function satisfying (B1) and (B2).
The main result of [Nebeský(1994)], see also [Mulder and Nebeský(2009)], states that a geometric transit function equals the interval function of its underlying graph, , if and only if satisfies in addition the two axioms
, , and , implies .
, , , implies .
Again we need (S1) and (S2) to hold for rather than itself.
The 1-point crossover operator satisfies the (S1) axiom.
Let be 1-point crossover operator. Since , it follows that and as well as and differ in only a single coordinate. Writing and assuming we must have either
W.l.o.g., suppose is of the form (1), therefore
. Let .
Since is of the form (1), we have
. Let .
since is of the form (1), we have
can be written as
, which implies . Thus the axiom (S1) follows.
The 1-point crossover operator satisfies axiom (S2).
From , it follows that and differ in only one coordinate, say , and and differ in a single coordinate, say . W.l.o.g., let . Since , and differ only in position , we conclude that
From and (*) we obtain . Hence . Therefore . This implies and axiom (S2) follows.
As shown in [Mulder and Nebeský(2009)], the axiom
implies both (S1) and (S2).
On hypercubes, i.e., assuming an alphabet with just two letters, uniform crossover satisfies . The unique median is defined coordinate-wise by majority voting of , see [Mulder(1980)]. On hypercubes, thus satisfy (MO). This argument fails, however, for general Hamming graphs. The reason is that axiom (MO) fails for each position at which the three sequences are pairwise distinct: .
For let denote the set of indices from definition 1 such that is a -point crossover offspring of and . If is an offspring such that is placed before in the definition we denote this by and otherwise.
Let . If ,
and , then
and holds for all .
Since it follows that and , we have .
[Gitchoff and Wagner(1996)] conjectured that for each transit set there is a unique pair of parents from which it is generated unless is a hypercube. We settle this conjecture affirmatively:
If then if and only if .
The implication from right to left is trivial. For other direction we use Lemma 10. Assume, for contradiction, that and . Then and . From it follows also that , which in turn implies . Therefore, there exists a set of indices , , such that and . From and Lemma 10 we conclude that there exist such that . Hence . This contradiction completes the proof of the theorem.
For the special case , Theorem 3 for implies the following statement.
For every , , , , implies that either or .
For with , the transit set induces a cycle of size , an hence the only other transit sets that are included in are singletons and edges.
4 Graph theoretical approach for -point crossover operators
Transit sets inherit a natural graph structure as an induced subgraph of the underlying graph . In the case of crossover operators and their corresponding transit sets , the distance in the underlying graph plays a crucial role in their characterization.
Recall that -dimensional hypercubes are antipodal graphs, i.e., for any vertex there is a unique antipodal vertex with , where