Topological Interpretation of Crossover

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GECCO 2004
Topological Interpretation of Crossover
Alberto Moraglio Riccardo Poli amoragn,rpoli_at_e
ssex.ac.uk
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Contents

1. Topological Interpretation Generalization of
   Crossover Mutation
2. Geometric Interpretation Formalization
3. Implications
4. Current Future Work

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I. Topological Interpretation Generalization
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What is crossover?
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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

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Direct Neighbour Mutation
Representation Binary String Move Bit
Flip Neighbourhood Hamming Representation
Move Neighbourhood
100
101
000
001
111
110
?
010
011
Mutation Offspring in the direct
neighbourhood What is crossover?
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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?

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Shortest Path Crossover
Parent1 011101 Parent2 010111 Children
0111
Children are on shortest paths More than one
shortest path in general
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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

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II. Geometric Interpretation Formalization
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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

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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.

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Squared balls Chunky segments
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Uniform Mutation Uniform Crossover

• Uniform topological crossover
• Uniform topological e-mutation

Genetic operators have a geometric nature
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Representation independentand rigorous
definition ofcrossover and mutation in the
neighbourhood seen as a geometric space
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III. Implications- Crossover Principled
Design- Simplification Clarification-
Unification General Theory
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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

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Crossover Design Example

?
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Non-labelled graph neighbourhood
MOVE Insert/remove an edge Fixed number of
nodes
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Offspring
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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))

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Clarification Equivalences Theorem

• Topological Crossover Topological Mutation
  Isomorphism
• One Distance, One Mutation, One Crossover
• One Representation, Various Edit Distances, One
  Crossover for each Distance

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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.

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IV. Current Future Work
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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)

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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)
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Questions?