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

I. Topological Interpretation& Generalization

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

II. Geometric Interpretation& Formalization

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

+ = ?

Crossover Design Example

Non-labelled graph neighbourhood

MOVE: Insert/remove an edge

Fixed number of nodes

0

1

2

1

2

3

+

Offspring

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.

IV. Current & Future Work

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)

Questions?