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Consider as an example parent permutation p1 = [0, 1, 2, 3, 4, 5, 6, 7] and parent permutation p2 = [1, 2, 0, 5, 6, 7, 4, 3]. Now consider that the random cross region begins at index 2 and ends at index 4, inclusive. Child c1 is initialized as a copy of p1, c1 = [0, 1, 2, 3, 4, 5, 6, 7], and we then swap the 2 with the 0 (the elements at index 2 in the parents) to get c1 = [2, 1, 0, 3, 4, 5, 6, 7]. Next, we swap the 3 with the 5 (the elements at index 3 in the parents) to get c1 = [2, 1, 0, 5, 4, 3, 6, 7]. Finally, we swap the 4 with the 6 (the elements at index 4 in the parents) to end up with c1 = [2, 1, 0, 5, 6, 3, 4, 7]. In a similar way, we initialize c2 as a copy of p2 and proceed with the designated swaps to end up with c2 = [1, 0, 2, 3, 4, 7, 6, 5].
PMX was introduced in the following paper:
Goldberg, D.E. and Lingle, R. Alleles, Loci, and the Traveling Salesman Problem. Proceedings of the 1st International Conference on Genetic Algorithms, 1985, pp. 154-159.
Although, we actually relied on the seminal book on genetic algorithms by one of PMX's authors
David E Goldberg:
Goldberg, D.E. Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, 1989.
Note that this implementation in Chips-n-Salsa is asymptotically faster than Goldberg and
Lingle's algorithmic description of PMX. In the original PMX description, the indexes of the
elements to swap were found with a linear search, and since on average there is a linear number
of these, PMX as originally described required O(n2) time. However, the implementation
here in Chips-n-Salsa computes the inverse of each permutation in linear time, which is then used
as a lookup table for the indexes of the elements to swap. Each swap is constant time, and also
involves a constant time update to the lookup table of indexes. Thus, this implementation, the
cross method, has an average case and worst case runtime of O(n), where n is
Constructor SummaryConstructorDescriptionConstructs a partially matched crossover (PMX) operator.
Method SummaryModifier and TypeMethodDescription
PermutationFullBinaryOperatorfor details of this method.
voidPerforms a crossover for an evolutionary algorithm, such that crossover forms two children from two parents.
split()Generates a functionally identical copy of this object, for use in multithreaded implementations of search algorithms.
PartiallyMatchedCrossoverpublic PartiallyMatchedCrossover()Constructs a partially matched crossover (PMX) operator.
crossDescription copied from interface:
CrossoverOperatorPerforms a crossover for an evolutionary algorithm, such that crossover forms two children from two parents. Implementations of this method modify the parameters, transforming the parents into the children.
splitpublic PartiallyMatchedCrossover split()Description copied from interface:
SplittableGenerates a functionally identical copy of this object, for use in multithreaded implementations of search algorithms. The state of the object that is returned may or may not be identical to that of the original. Thus, this is a distinct concept from the functionality of the
Copyableinterface. Classes that implement this interface must ensure that the object returned performs the same functionality, and that it does not share any state data that would be either unsafe or inefficient for concurrent access by multiple threads. The split method is allowed to simply return the this reference, provided that it is both safe and efficient for multiple threads to share a single copy of the Splittable object. The intention is to provide a multithreaded search with the capability to provide spawned threads with their own distinct search operators. Such multithreaded algorithms can call the split method for each thread it spawns to generate a functionally identical copy of the operator, but with independent state.
PermutationFullBinaryOperatorfor details of this method. This method is not intended for direct usage. Use the
cross(org.cicirello.permutations.Permutation, org.cicirello.permutations.Permutation)method instead.