Topological Interpretation of
Crossover
Alberto Moraglio & Riccardo Poli
{amoragn,rpoli}@essex.ac.uk
GECCO 2004
Contents
I. Topological Interpretation & Generalization of Crossover & Mutation
II. Geometric Interpretation & Formalization
III. Implications
IV. Current & Future Work
What is crossover?
CrossoverIs there any
commonaspect ?
Is it possible to give arepresentation-
independent definitionof crossover and
mutation?
100000011101000
100111100011100
100110011101000
100001100011100
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
Direct Neighbour Mutation
000
001
010
011
100
101
111
110
Representation: Binary String
Move: Bit Flip
Neighbourhood: Hamming
Representation + Move = Neighbourhood
?
Mutation: Offspring in the direct neighbourhoodWhat is crossover?
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?
Shortest Path Crossover
011001
010001 011101 011011
010101 011111
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
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
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
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.
}),(|{);( ryxdSyrxB
)},(),(),(|{];[ yxdyzdzxdSzyx
Squared balls & Chunky segments
33
000 001
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
000 001
010 011
100 101
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
Uniform Mutation & Uniform Crossover
Uniform topological crossover:
Uniform topological ε-mutation:
|],[|
]),[(}2,1|Pr{),|(
yx
yxzyPxPzUXyxzfUX
],[}0),|(|{)],(Im[ yxyxzfSzyxUX UX
|),(|
)),((}|Pr{)|(
xB
xBzxPzUMxzfUM
),(}0)|(|{)](Im[ xBxzfSzxUM M
Genetic operators have a geometric nature
Representation independentand rigorous definition of
crossover and mutation in the neighbourhood seen as a
geometric space
III. Implications:- Crossover Principled Design- Simplification & Clarification- Unification & General Theory
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
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))
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
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.
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)
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)