# Class CycleAlphaMutation

java.lang.Object
org.cicirello.search.operators.permutations.CycleAlphaMutation
All Implemented Interfaces:
`Splittable<MutationOperator<Permutation>>`, `MutationOperator<Permutation>`, `UndoableMutationOperator<Permutation>`

public final class CycleAlphaMutation extends Object implements UndoableMutationOperator<Permutation>

This class implements the Cycle(α) form of cycle mutation on permutations, where one mutation generates a random permutation cycle. Given the original parent permutation and its mutant, a permutation cycle can be defined as follows. Imagine a graph with n vertexes, where n is the permutation length. Now consider that for each index i, we define an edge in that graph between vertex parent[i] and vertex mutant[i]. A permutation cycle consists of all of the elements from one of the cycles in that graph. The length of a cycle is the number of elements in it. Consider an example permutation, p1 = [0, 1, 2, 3, 4], and another permutation, p2 = [0, 3, 2, 1, 4]. This pair of permutations has a 2-cycle (i.e., a cycle of length 2) consisting of elements 1 and 3. Consider a second example, p1 = [0, 1, 2, 3, 4], and p2 = [0, 4, 2, 1, 3]. This example has a 3-cycle consisting of elements 1, 3, and 4. Notice that position 1 has elements 1 and 4, position 4 has elements 4 and 3, and position 3 has elements 3 and 1, so in the hypothetical graph described above, there would be edges from 1 to 4, 4 to 3, and 3 to 1, a cycle of length 3.

The Cycle(α) version of cycle mutation chooses the cycle size randomly from {2, 3, ..., n} where cycle length k is chosen with probability proportional to αk-2. It then generates a random permutation cycle of length k. The combination of k elements is chosen uniformly at random from all possible combinations of k elements. Note that a 2-cycle is simply a swap.

The worst case runtime of a single call to the `mutate` method is O(n), which occurs when the randomly chosen cycle length is n. However, this is a very low probability event. Lower cycle lengths are given significantly higher probability. The average case runtime of a single call to the `mutate` method is O(min(n, ((2-α)/(1-α))2)). Thus, provided α is not close to 1, the average runtime is a constant depending upon the value of α.

Cycle mutation in both of its forms, including Cycle(α), was introduced in the following article:

Vincent A. Cicirello. 2022. Cycle Mutation: Evolving Permutations via Cycle Induction, Applied Sciences, 12(11), Article 5506 (June 2022). doi:10.3390/app12115506

• ## Constructor Summary

Constructors
Constructor
Description
`CycleAlphaMutation(double alpha)`
Constructs an CycleAlphaMutation mutation operator.
• ## Method Summary

Modifier and Type
Method
Description
`final void`
`mutate(Permutation c)`
Mutates a candidate solution to a problem, by randomly modifying its state.
`CycleAlphaMutation`
`split()`
Generates a functionally identical copy of this object, for use in multithreaded implementations of search algorithms.
`final void`
`undo(Permutation c)`
Returns a candidate solution to its previous state prior to the most recent mutation performed.

### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`
• ## Constructor Details

• ### CycleAlphaMutation

public CycleAlphaMutation(double alpha)
Constructs an CycleAlphaMutation mutation operator.
Parameters:
`alpha` - The alpha parameter of the mutation operator (see class documentation).
Throws:
`IllegalArgumentException` - if alpha is less than or equal to 0 or greater than or equal to 1.
• ## Method Details

• ### mutate

public final void mutate(Permutation c)
Description copied from interface: `MutationOperator`
Mutates a candidate solution to a problem, by randomly modifying its state. The mutant that is produced is in the local neighborhood of the original candidate solution.
Specified by:
`mutate` in interface `MutationOperator<Permutation>`
Parameters:
`c` - The candidate solution subject to the mutation. This method changes the state of c.
• ### undo

public final void undo(Permutation c)
Description copied from interface: `UndoableMutationOperator`

Returns a candidate solution to its previous state prior to the most recent mutation performed.

For example, consider the following. Let c' be the current state of c. Let c'' be the state of c after mutate(c); If we then call undo(c), the state of c should revert back to c'.

The behavior of undo is undefined if c is altered by some other process between the calls to mutate and undo. The behavior is also undefined if a different candidate is given to undo then the last given to mutate. For example, if the following two statements are executed, mutate(c); undo(d);, the effect on d is undefined as it wasn't the most recently mutated candidate solution.

Specified by:
`undo` in interface `UndoableMutationOperator<Permutation>`
Parameters:
`c` - The candidate solution to revert.
• ### split

public CycleAlphaMutation split()
Description copied from interface: `Splittable`
Generates 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 `Copyable` interface. 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.
Specified by:
`split` in interface `Splittable<MutationOperator<Permutation>>`
Specified by:
`split` in interface `UndoableMutationOperator<Permutation>`
Returns:
A functionally identical copy of the object, or a reference to this if it is both safe and efficient for multiple threads to share a single instance of this Splittable object.