geometric crossover for the permutation representation alberto moraglio & riccardo poli...
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Geometric Crossover for the Permutation Representation
Alberto Moraglio & Riccardo Poli
{amoragn,rpoli}@essex.ac.uk
GSICE 2005
Contents
I. Abstract Geometric Operators
II. Geometric Crossover for Permutations
III. Geometric Crossover for TSP
IV. Conclusions
I. Abstract Geometric Operators
What is crossover?
CrossoverIs there any
common
aspect ?
Is it possible to give a
representation-
independent definition
of 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 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
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)
Geometric Operators Formalization
|),(|
)),((}|Pr{)|(
xB
xBzxPzUMxzfUM
}),(|{);( ryxdSyrxB
)},(),(),(|{];[ yxdyzdzxdSzyx
BALL: All points within distance r from x
SEGMENT: All points between x and y
|],[|
]),[(}2,1|Pr{),|(
yx
yxzyPxPzUXyxzfUX
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. Swap
Insertion Sort Insert
Selection Sort Swap
Quick 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
PMX
ERX
SBRX
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?