Geometric Crossover for the Permutation Representation

Alberto Moraglio & Riccardo Poli{amoragn,rpoli}@essex.ac.uk

GSICE 2005

ContentsI. Abstract Geometric Operators

II. Geometric Crossover for Permutations

III. Geometric Crossover for TSP

IV. Conclusions

I. Abstract Geometric Operators

What is crossover?

CrossoverIs there anycommon

aspect ?

Is it possible to give arepresentation-

independent definitionof crossover and mutation?

100000011101000

100111100011100

100110011101000

100001100011100

Binary Strings

Permutations

Real Vectors

Syntactic Trees

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

• NEARNESS:hd(P,O)=1

Crossover & Betweenness• Crossover is naturally interpreted in terms of

betweenness: offspring are between parents• Example: Binary StringP1 = 0 1 0|0 1 0P2 = 1 1 0|1 0 1O = 0 1 0 1 0 1hd(P1,P2)=4hd(P1,O)=3 hd(O,P2)=1

• BETWEENNES: P1---O-P2

Geometric CrossoverDEFINITION: 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

Geometric Crossovers across Representations

Many pre-existing recombination operators are geometric under suitable distance:

BINARY: one-point, two-points, uniform crossoversREAL 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)

Geometric Operators Formalization

|),(|)),((}|Pr{)|(

xBxBzxPzUMxzfUM

}),(|{);( ryxdSyrxB

)},(),(),(|{];[ yxdyzdzxdSzyx

BALL: All points within distance r from x

SEGMENT: All points between x and y

|],[|]),[(}2,1|Pr{),|(

yxyxzyPxPzUXyxzfUX

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

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

II. Geometric Crossover

Design for Permutations

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

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

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

Permutation+Edit Move = Neighbourhood Structure

Shortest path distance = edit distance

abc

bac acb

bca cab

cba

B(abc; 1)Adjacent swap space

abc

bac acb

bca cab

cba

[abc; bca]1 geodesic

Adjacent swap space

B(abc; 1)Swap space & Reversal space

abc

bac acb

bca cab

cba

abc

bac acb

bca cab

cba

[abc; bca]3 geodesics

Swap space & Reversal space

B(abc; 1)Insertion space

[abc; bca]1 geodesic

Insertion space

abc

bac acb

bca cab

cba

abc

bac acb

bca cab

cba

Line segment in the neighbourhood structure = all shortest paths connecting two nodes

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)

Many sorting algorithms do minimal sorting by X

Ordinary Sorting Algorithm

Minimal Sorting by X

Bubble Sort Adj. SwapInsertion Sort Insert

Selection Sort SwapQuick Sort No Fix Move!

Geometric Crossovers = Sorting Crossovers!

III. Geometric Crossover

Design for TSP

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

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)

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

Results for TSPLIB (typical)

0

50000

100000

150000

200000

250000

300000

350000

1 22 43 64 85 106 127 148 169 190 211 232

PMXERXSBRX

Big Population – No mutation – Until Convergence

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

IV. Conclusions

SummaryGeometric 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!

Thank you for your attention… Questions?