1. Topological Interpretation of Crossover Alberto Moraglio & Riccardo Poli {amoragn,rpoli}@essex.ac.uk GECCO 2004

2. Contents I.Topological Interpretation & Generalization of Crossover & Mutation II.Geometric Interpretation & Formalization III.Implications IV.Current & Future Work

3. I. Topological Interpretation & Generalization

4. What is crossover? Crossover Is there any common aspect ? Is it possible to give a representation- independent definition of crossover and mutation? 100000011101000 100111100011100 100110011101000 100001100011100

5. Genetic operators & Neighbourhood structure Forget the representation and consider the neighbourhood structure (= search space structure) Mutation: offspring are “close to” their parent  in the direct neighbourhood

6. Direct Neighbour Mutation 000001 010011 100101 111110 Representation: Binary String Move: Bit Flip Neighbourhood: Hamming Representation + Move = Neighbourhood ? Mutation: Offspring in the direct neighbourhood What is crossover?

7. Neighbourhood and Crossover Crossover idea: combining parents genotypes to get children genotypes “somewhere in between” them Topologically speaking, “somewhere in between” = somewhere on a shortest path Why on a shortest path?

8. Shortest Path Crossover 011001 010001011101011011 010101011111 010011 010111 D0 : P1 D2 : P2 D1 Parent1: 011101 Parent2: 010111 Children: 01*1*1 Children are on shortest paths More than one shortest path in general

9. Interpretation & Generalization Traditional mutation & crossover have a natural interpretation in the neighbourhood structure in terms of closeness and betweenness Given any representation plus a notion of neighbourhood (move), mutation & crossover operators are well-defined

10. II. Geometric Interpretation & Formalization

11. From graphs to geometry Forget the neighbourhood structure and consider the metric space (= space with a notion of distance) The distance in the neighbourhood is the length of the shortest path connecting two solutions Mutation  Direct neighbourhood  Ball Crossover  All shortest paths  Line Segment

12. Balls & Segments In a metric space (S, d) the closed ball is the set of the form where x belongs to S and r is a positive real number called the radius of the ball. In a metric space (S, d) the line segment or closed interval is the set of the form where x and y belong to S and are called extremes of the segment and identify the segment.

13. Squared balls & Chunky segments 3 3 000001 010 011 100 101 111 110 B(000; 1) Hamming space 3 B((3, 3); 1) Euclidean space 3 B((3, 3); 1) Manhattan space Balls 1 2 1 2 000001 010011 100101 111 110 [000; 011] = [001; 010] 2 geodesics Hamming space 1 3 [(1, 1); (3, 2)] 1 geodesic Euclidean space 1 3 [(1, 1); (3, 2)] = [(1, 2); (3, 1)] infinitely many geodesics Manhattan space Line segments

14. Uniform Mutation & Uniform Crossover Uniform topological crossover: Uniform topological ε-mutation: Genetic operators have a geometric nature

15. Representation independent and rigorous definition of crossover and mutation in the neighbourhood seen as a geometric space

16. III. Implications: - Crossover Principled Design - Simplification & Clarification - Unification & General Theory

17. I - Crossover Principled Design Domain specific solution representation is effective Problem: for non-standard representations it is not clear how crossover should look like But: given a combinatorial problem you may know already a good neighbourhood structure Topological Interpretation of Crossover  Give me your neighbourhood definition and I give you a crossover definition

18. += ? Crossover Design Example

19. Non-labelled graph neighbourhood MOVE: Insert/remove an edge Fixed number of nodes 0 1 2 1 2 3

20. + Offspring

21. II - Simplification & Clarification Other theories: –Recombination spaces based on hyper- neighbourhoods –Crossover & mutation are seen as completely independent operators using different search spaces Topological crossover: –Crossover interpreted naturally in the classical neighbourhood –Crossover and mutation in the same space (direct comparison with other search methods (local search))

22. Clarification: Equivalences Theorem Space Structure Topological Crossover Topological Mutation Distance Neighbourhood Function Neighbourhood Graph Topological Crossover & Topological Mutation Isomorphism One Distance, One Mutation, One Crossover One Representation, Various Edit Distances, One Crossover for each Distance

23. III – Unification & General theory One EC theory problem: –EC theory is fragmented. There is not a unified way to deal with different representations. Topological framework: –Topological genetic operators are rigorously defined without any reference to the representation. These definitions are a promising starting point for a general and rigorous theory of EC.

24. IV. Current & Future Work

25. Work in progress EAs Unification: Existing crossovers and mutations fit the topological definitions Preliminary work on important representations: –Binary strings (genetic algorithms) –Real-valued vectors (evolutionary strategy) –Permutations (ga for comb. optimisation) –Parse trees (genetic programming) –DNA strands (nature)

26. Future work THEORY: Generalizing and accommodating pre-existent theories into topological framework (schema theorem, fitness landscapes, representation theories…) PRACTICE: Testing crossover principled design on important problems with non- standard representation (problem domain representation)

27. Questions?