40
Topological Crossover for the Permutation Representation Alberto Moraglio & Riccardo Poli {amoragn,rpoli}@essex.ac.uk GECCO 2005

Topological Crossover for the Permutation Representation

  • Upload
    cain

  • View
    57

  • Download
    1

Embed Size (px)

DESCRIPTION

GECCO 2005. Topological Crossover for the Permutation Representation. Alberto Moraglio & Riccardo Poli {amoragn,rpoli}@essex.ac.uk. Sorry… Name Change!. Topological Crossover Abstract Geometric Crossover. Contents. Abstract Geometric Operators Geometric Crossover for Permutations - PowerPoint PPT Presentation

Citation preview

Page 1: Topological Crossover for the Permutation Representation

Topological Crossover for the Permutation Representation

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

GECCO 2005

Page 2: Topological Crossover for the Permutation Representation

Topological Crossover

Abstract Geometric Crossover

Sorry… Name Change!

Page 3: Topological Crossover for the Permutation Representation

ContentsI. Abstract Geometric Operators

II. Geometric Crossover for Permutations

III. Geometric Crossover for TSP

IV. Conclusions

Page 4: Topological Crossover for the Permutation Representation

I. Abstract Geometric Operators

Page 5: Topological Crossover for the Permutation Representation

What is crossover?

CrossoverIs there anycommon

aspect ?

Is it possible to give arepresentation-

independent definitionof crossover and

mutation?

100000011101000

100111100011100

100110011101000

100001100011100

Binary StringsPermutations

Real Vectors

Syntactic Trees

Page 6: Topological Crossover for the Permutation Representation

Shortest Path Crossover011001

010001 011101 011011

010101 011111

010011

010111

D0 : P1

D2 : P2

D1

Parent1: 011101

Parent2: 010111

Children: 01*1*1

Crossover in the Neighbourhood: offspring between parents

Mask-based crossover: children are on shortest paths

Hamming Neighbourhood Structure

Page 7: Topological Crossover for the Permutation Representation

From graphs to geometry• Neighbourhood Structure=Metric Space • 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

Page 8: Topological Crossover for the Permutation Representation

Balls & SegmentsIn 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

Page 9: Topological Crossover for the Permutation Representation

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

Page 10: Topological Crossover for the Permutation Representation

Uniform Mutation & Uniform Crossover

Uniform topological crossover:

Uniform topological ε-mutation:

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

yxyxzyPxPzUXyxzfUX

],[}0),|(|{)],(Im[ yxyxzfSzyxUX UX

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

xBxBzxPzUMxzfUM

),(}0)|(|{)](Im[ xBxzfSzxUM M

Genetic operators have a geometric nature

Page 11: Topological Crossover for the Permutation Representation

Representation-independentand rigorous definition of

crossover and mutation in the neighbourhood seen as a

geometric space

Page 12: Topological Crossover for the Permutation Representation

So what? Claims at Gecco 2004

(i) EAs Unification: most pre-existing genetic operators for main representations are geometric

(ii) Simplification & Clarification: crossover as function of classical neighbourhood structure simplifies the established notion of crossover landscape (hyper-neighbourhood) as function of crossover

(iii) General theory: formal representation-independent definitions allow for a general theory

(iv) Crossover principled design: specifying the formal definition of crossover for a specific representation and distance one gets automatically a specific crossover

Page 13: Topological Crossover for the Permutation Representation

II. Geometric Crossover

for Permutations

Page 14: Topological Crossover for the Permutation Representation

Many Distances Dilemma

Page 15: Topological Crossover for the Permutation Representation

Many Distances Dilemma

WHAT IS A GOOD DISTANCE? WHAT IS THE RIGTH CROSSOVER?

Representation Binary Strings Permutations

Distance One distance =Hamming distance Many distances

Geometric Crossover Mask-based crossover Many types of crossover

Geometric Uniform Crossover Uniform crossover Many uniform crossovers

Page 16: Topological Crossover for the Permutation Representation

What is a good distance?– IN PRINCIPLE: abstract genetic operators are well-

defined for any distance. However:– IMPLEMENTATION: a distance not rooted in the

solution syntax does not tell how to implement crossover– PROBLEM KNOWLEDGE: a problem-independent

distance does not put any problem knowledge in the search

– A GOOD DISTANCE: – (i) suggests how to implement crossover– (ii) embeds problem knowledge in the algorithm

Page 17: Topological Crossover for the Permutation Representation

Crossover Implementation & Edit

Distances

Page 18: Topological Crossover for the Permutation Representation

Mutations/Edit moves 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

Page 19: Topological Crossover for the Permutation Representation

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

Page 20: Topological Crossover for the Permutation Representation

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)

Page 21: Topological Crossover for the Permutation Representation

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!

Page 22: Topological Crossover for the Permutation Representation

Geometric Crossovers = Sorting Crossovers!

• Sorting Crossover by X:– sorting one parent permutation toward the other

using X sort move– stop the sorting at random and return the

partially sorted permutation as offspring• Bubble Sort Crossover = Geometric

Crossover under adj. swap edit distance

Page 23: Topological Crossover for the Permutation Representation

EmbeddingProblem Knowledge

Page 24: Topological Crossover for the Permutation Representation

Edit Distances & Problem Knowledge

How can we pick an edit distance that embeds problem knowledge?

• Minimal fitness change: pick the edit distance whose edit move corresponds to a minimal fitness change

• Good mutation, Good crossover: pick the edit distance whose edit move corresponds to a good mutation for the problem at hand

• Good neighbourhood, Good crossover: pick the edit distance whose edit move induces a neighbourhood structure that is known to be good for the problem

Page 25: Topological Crossover for the Permutation Representation

N-queens - mutations

0

5

10

15

20

25

30

35

40

45

1 40 79 118 157 196 235 274 313 352 391 430 469

swpadj_swpins

Page 26: Topological Crossover for the Permutation Representation

N-queens - crossovers

0

5

10

15

20

25

30

35

40

45

1 37 73 109 145 181 217 253 289 325 361 397 433 469

pmxss1xssuxbs1xbsuxis1xisux

Page 27: Topological Crossover for the Permutation Representation

Crossover Rank vs. Mutation Rank

1. Selection Sort Uniform 1. Swap2. PMX -3. Selection Sort 1-point 1. Swap4. Insertion Sort Uniform 2. Insertion5. Insertion Sort 1-point 2. Insertion

6. Bubble Sort Uniform 3. Adj. Swap

7. Bubble Sort 1-point 3. Adj. Swap

Good mutation, good crossover heuristic holds!

Uniform crossovers are better than 1-point crossovers

Page 28: Topological Crossover for the Permutation Representation

III. Geometric Crossover

for TSP

Page 29: Topological Crossover for the Permutation Representation

Geometric Crossover for TSP

• A 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)

Page 30: Topological Crossover for the Permutation Representation
Page 31: Topological Crossover for the Permutation Representation

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

Page 32: Topological Crossover for the Permutation Representation

Experiments - ParametersTest-bed• TSPLIB: eil51, gr96, eil101, lin105, d198, kroA200, lin318, pcb442Crossovers• PMX: partially matched crossover• ERX: edge recombination• SBRX: sorting by reversal crossover (limitations: no circular permutation,

uniform on one fixed geodesic, 2-approxiamtion) Parameter Setting• BIG POPULATION: Population Size = Instance Size * 20• Until Population Convergence• No Mutation• Runs=30 (average of bests in population)• No Fine Tuning. The settings have been chosen to allow the best crossover to

reach a near optimal solution before convergence.

Page 33: Topological Crossover for the Permutation Representation

Results for eil51 (small)

0

200

400

600

800

1000

1200

1400

1600

1 13 25 37 49 61 73 85 97 109 121 133

PMXERXSBRX

Page 34: Topological Crossover for the Permutation Representation

Results for lin105 (medium)

0

20000

40000

60000

80000

100000

120000

1 28 55 82 109 136 163 190 217 244 271 298 325

PMXERXSBRX

Page 35: Topological Crossover for the Permutation Representation

Results for kroA200 (medium-big)

0

50000

100000

150000

200000

250000

300000

350000

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

PMXERXSBRX

Page 36: Topological Crossover for the Permutation Representation

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– Non-deterministic approx algorithm (uniform

crossover)– Circular Permutations instead of Linear Permutations

Page 37: Topological Crossover for the Permutation Representation

IV. Conclusions

Page 38: Topological Crossover for the Permutation Representation

ConclusionsPermutations & Many Distances

– Many types of geometric crossovers!– What is a good distance?

Implementation & Edit Distance: – Edit Distances are good– For permutations: geometric crossovers = sorting algorithms!

Problem Knowledge and Edit Move: – Good mutation, good crossover heuristics– For permutations: good mutation, good crossover holds for the N-

queen problem using sorting crossovers Geometric Crossover for TSP

– Sorting circular permutation by reversals (NP-Hard)– 2-approximation algorithm for approximated geometric crossover– Good empirical results based only on theory!

Page 39: Topological Crossover for the Permutation Representation

Thank you for your attention… Questions?

Page 40: Topological Crossover for the Permutation Representation

N-queens - parametersProblem size 100

Population size 5000

Mutation probability 0.1 (0)

Crossover probability (0) 1

Generation 500

Selection tournament size 5

Statistics Average 30 runs