Geometric Crossover for the Permutation Representation

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GSICE 2005
Geometric Crossover for the Permutation
Representation
Alberto Moraglio Riccardo Poli amoragn,rpoli_at_e
ssex.ac.uk
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Contents

1. Abstract Geometric Operators
2. Geometric Crossover for Permutations
3. Geometric Crossover for TSP
4. Conclusions

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I. Abstract Geometric Operators
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What is crossover?
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Mutation Nearness

• Mutation is naturally interpreted in terms of
  nearness offspring are near the parent
• Example Binary StringP 0 1 0 1 1 1O 0 1 0
  1 0 1
• NEARNESShd(P,O)1

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Crossover Betweenness

• Crossover is naturally interpreted in terms of
  betweenness offspring are between parents
• Example Binary StringP1 0 1 00 1 0P2 1 1
  01 0 1O 0 1 0 1 0 1hd(P1,P2)4hd(P1,O)3
  hd(O,P2)1
• BETWEENNES P1---O-P2

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Geometric Crossover

• DEFINITION geometric crossover is any
  recombination operator for which there is at
  least a (metric) distance such as all offspring
  are between parents
• Definition properties
• - is representation-independent
• clear-cuts crossover from non-crossover
• generalises many pre-existing crossovers

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Geometric Crossovers across Representations

• Many pre-existing recombination operators are
  geometric under suitable distance
• BINARY one-point, two-points, uniform crossovers
• REAL VECTORS line, arithmetic, discrete
  (non-geometric extended line)
• PERMUTATIONS PMX, Edge Recombination, Cycle
  Crossover, Merge Crossover (non-geometric order
  crossover)
• SYNTACTIC TREES homologous one-point uniform
  crossovers (non-geometric subtree swap
  crossover)

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Geometric Operators Formalization
BALL All points within distance r from x
SEGMENT All points between x and y
UNIFORM ?-MUTATION offspring z are taken
uniformly within the ball of radius ? from the
parent x
UNIFORM CROSSOVER offspring z are taken
uniformly within the segment between parents x
and y
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Advantages of Geometric Operators

• REPRESENTATION UNIFICATION many pre-existing
  operators are geometric
• SIMPLIFIED ANALISYS natural interpretation of
  crossover within the classic notion of
  neighbourhood landscape
• GENERAL THEORY formal definition dynamical
  equations ? representation-independent
  evolutionary dynamics
• CROSSOVER DESIGN formal definition specific
  distance ? specific crossover

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II. Geometric Crossover Design for Permutations
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Distance Representation

• IN PRINCIPLE abstract genetic operators are
  well-defined for any distance without any
  reference to solution representation
• IMPLEMENTATION REQUIREMENT however a distance
  must be rooted in the solution representation to
  make the crossover implementation possible
  (practical)
• EDIT DISTANCES firmly rooted in the solution
  representation and guiding crossover
  implementation

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One Representation, Many Crossovers

• Binary Strings are associated with Hamming
  Distance (HD)
• Uniform Geometric Crossover under HD corresponds
  to uniform crossover for binary strings
• Permutation representation can be naturally
  associated with many distances
• Since for each distance, there is one crossover
  there are many different uniform geometric
  crossovers for permutation representation

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Edit Distances for Permutations

• Reversal (A B C D E F) ? (A E D C B F)
• Insert (A B C D E F) ? (A C D E B F)
• Swap (A B C D E F) ? (A D C B E F)
• Adj.Swap (A B C D E F) ? (A C B D E F)

Edit Distance minimum number of edit moves to
transform one permutation into the other
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PermutationEdit Move Neighbourhood Structure
Shortest path distance edit distance
Line segment in the neighbourhood structure
all shortest paths connecting two nodes
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MAGIC OF EDIT DISTANCES Neighbourhood/syntax
DUALITY

• NEIGHBOURHOOD Picking offspring on shortest path
  connecting two nodes
• SYNTAX picking offspring on minimal sorting
  trajectory between parent permutations using the
  edit move as sort move (minimal sorting by x)

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Many sorting algorithms do minimal sorting by X
Ordinary Sorting Algorithm Minimal Sorting by X
Bubble Sort Adj. Swap
Insertion Sort Insert
Selection Sort Swap
Quick Sort No Fix Move!
Geometric Crossovers Sorting Crossovers!
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III. Geometric Crossover Design for TSP
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Distance Problem Knowledge

• IN PRINCIPLE abstract genetic operators are
  well-defined for any distance without any
  reference to the problem at hand
• PROBLEM KNOWLEDGE REQUIREMENT however, a
  problem-independent distance does not put any
  problem knowledge in the search. A good distance
  embeds problem knowledge.
• HEURISTICS Good neighbourhood, Good crossover
  pick the edit distance whose edit move induces a
  neighbourhood structure that is known to be good
  for the problem

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Geometric Crossover for TSP

• A known good neighbourhood structure for TSP is
  2opt structure space of circular permutations
  endowed with reversal edit distance
• Geometric crossover for TSP picking offspring
  on the minimal sorting trajectories by sorting
  one parent circular permutation toward the other
  parent by reversals (sorting circular
  permutations by reversals)

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Approximated Geometric Crossover

• BAD NEWS sorting circular permutations by
  reversals is NP-Hard!
• GOOD NEWS there are approximation algorithms
  that sort within a bounded error to optimality
  (used in genetics)
• A 2-approximation algorithm sorts by reversals
  using sorting trajectories that are at most twice
  the length of the minimal sorting trajectories
• Approximation algorithms can be used to build
  approximated geometric crossovers for TSP

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Results for TSPLIB (typical)
Big Population No mutation Until Convergence
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Good results lot of room for improvement

• SBRX better than ERX for bigger instances
• good empirical results based only on theoretical
  considerations
• Possible improvements
• Fine parameter tuning
• Better approximation algorithm
• Geometric uniform crossover
• Circular permutations instead of linear
  permutations

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IV. Conclusions
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Summary

• Geometric Interpretation Formalization of
  Genetic Operators
• Mutation ? Nearness ? Ball
• Crossover ?Betweenness ? Line Segment
• Crossover Design for Permutations
• Implementation requirement distance based on
  syntax
• One representation, many distances ? many
  crossovers
• Edit distances for permutations geometric
  crossovers sorting algorithms!
• Crossover Design for TSP
• Problem knowledge requirement distance makes
  landscape smooth
• Edit distance for TSP reversal distance (2-opt)
• Sorting circular permutations by reversals
  (NP-Hard)
• 2-approximation algorithm for approximated
  geometric crossover
• Good empirical results based only on theory!

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Thank you for your attention Questions?