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
I.
Abstract Geometric Operators
II.
Geometric Crossover for Permutations
III.
Geometric Crossover for TSP
IV. Conclusions
I. Abstract Geometric
Operators
What is crossover?
Binary Strings
Permutations
100000011101000
100111100011100
100110011101000
100001100011100
Is there any
common
aspect ?
Crossover
Is it possible to give a
representationindependent definition
of crossover and
mutation?
Real Vectors
Syntactic Trees
Shortest Path Crossover
Hamming Neighbourhood
Structure
011001
D0 : P1
010001
011101
011011
Parent1: 011101
Parent2: 010111
D1
010101
011111
Children: 01*1*1
010011
D2 : P2
010111
Crossover in the Neighbourhood: offspring between parents
Mask-based crossover: children are on shortest paths
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 & Segments
In a metric space (S, d) the closed ball is the set of the form
B ( x; r )  { y  S | d ( x, y )  r}
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
[ x; y ]  {z  S | d ( x, z )  d ( z, y )  d ( x, y )}
where x and y belong to S and are called extremes of the segment and identify the segment.
Squared balls & Chunky segments
Balls
100
000
101
001
111
110
3
3
011
010
3
3
B((3, 3); 1)
Euclidean space
B(000; 1)
Hamming space
B((3, 3); 1)
Manhattan space
Line segments
100
000
101
001
2
1
1
111
110
010
2
011
[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
Uniform Mutation &
Uniform Crossover
Uniform topological crossover:
fUX ( z | x, y )  Pr{UX  z | P1  x, P2  y} 
 ( z  [ x, y ])
| [ x, y ] |
Im[UX ( x, y )]  {z  S | fUX ( z | x, y )  0}  [ x, y ]
Uniform topological ε-mutation:
fUM ( z | x )  Pr{UM  z | P  x} 
 ( z  B( x,  ))
| B( x,  ) |
Im[UM  ( x)]  {z  S | f M ( z | x)  0}  B( x,  )
Genetic operators have a geometric nature
Representation-independent
and 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 (hyperneighbourhood) 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
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?
WHAT IS THE RIGTH CROSSOVER?
What is a good distance?
– IN PRINCIPLE: abstract genetic operators are welldefined 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
abc
abc
abc
bac
acb
bac
cba
cab
bca
bac
acb
bca
cab
acb
cab
bca
cba
cba
B(abc; 1)
Adjacent swap space
B(abc; 1)
Swap space &
Reversal space
B(abc; 1)
Insertion space
abc
abc
abc
bac
acb
bca
cab
bac
bca
cba
acb
bca
cab
cab
cba
cba
[abc; bca]
1 geodesic
Adjacent swap space
bac
acb
[abc; bca]
3 geodesics
Swap space &
Reversal space
[abc; bca]
1 geodesic
Insertion space
Shortest path distance = edit distance
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
Bubble Sort
Minimal
Sorting by X
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
Embedding
Problem 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
45
40
35
30
swp
adj_swp
ins
25
20
15
10
5
0
1
40 79 118 157 196 235 274 313 352 391 430 469
N-queens - crossovers
45
40
35
pmx
ss1x
ssux
bs1x
bsux
is1x
isux
30
25
20
15
10
5
0
1
37 73 109 145 181 217 253 289 325 361 397 433 469
Crossover Rank vs. Mutation Rank
1. Selection Sort Uniform 1. Swap
2. PMX
-
3. Selection Sort 1-point
1. Swap
4. Insertion Sort Uniform
2. Insertion
5. 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 - Parameters
Test-bed
• TSPLIB: eil51, gr96, eil101, lin105, d198, kroA200, lin318, pcb442
Crossovers
• 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)
1600
1400
1200
1000
PMX
ERX
SBRX
800
600
400
200
0
1
13
25
37
49
61
73
85
97 109 121 133
Results for lin105 (medium)
120000
100000
80000
PMX
ERX
SBRX
60000
40000
20000
0
1
28 55 82 109 136 163 190 217 244 271 298 325
Results for kroA200 (medium-big)
350000
300000
250000
PMX
ERX
SBRX
200000
150000
100000
50000
0
1
22 43 64 85 106 127 148 169 190 211 232
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
Conclusions
Permutations & 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 Nqueen 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 - parameters
Problem size
100
Population size
5000
Mutation probability
0.1 (0)
Crossover probability
(0) 1
Generation
500
Selection
tournament size 5
Statistics
Average 30 runs