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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
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Topological Crossover for the Permutation Representation
Alberto Moraglio & Riccardo Poli{amoragn,rpoli}@essex.ac.uk
GECCO 2005
Topological Crossover
Abstract Geometric Crossover
Sorry… Name Change!
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 StringsPermutations
Real Vectors
Syntactic Trees
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
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
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
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{),|(
yxyxzyPxPzUXyxzfUX
],[}0),|(|{)],(Im[ yxyxzfSzyxUX UX
|),(|)),((}|Pr{)|(
xBxBzxPzUMxzfUM
),(}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
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
II. Geometric Crossover
for Permutations
Many Distances Dilemma
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
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
Crossover Implementation & Edit
Distances
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
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
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. Swap
Insertion Sort Insert
Selection Sort Swap
Quick Sort No Fix Move!
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
EmbeddingProblem Knowledge
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
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
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
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
III. Geometric Crossover
for TSP
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)
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
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.
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
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
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
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
IV. Conclusions
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!
Thank you for your attention… Questions?
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