Product Geometric Crossover for the Sudoku Puzzle

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IEEE CEC 2006
Product Geometric Crossover for the Sudoku Puzzle
Alberto Moraglio, Julian Togelius Simon
Lucas
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Contents

• Geometric Crossover and Product Geometric
  Crossover
• Design of Geometric Crossover for the Sudoku
  Puzzle
• Experimental Results and Conclusions

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I. Geometric Crossover
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Geometric Crossover

• Line segment
• A binary operator GX is a geometric crossover if
  all offspring are in a segment between its
  parents.
• Geometric crossover is dependent on the metric

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Geometric Crossover

• The traditional n-point crossover is geometric
  under the Hamming distance.

H(A,X) H(X,B) H(A,B)
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Many Recombinations are Geometric

• Traditional Crossover extended to multary strings
• Recombinations for real vectors
• PMX, Cycle Crossovers for permutations
• Homologous Crossover for GP trees
• Ask me for more examples over a coffee!

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Being geometric crossover is important because.

• We know how the search space is searched by
  geometric crossover for any representation
  convex search
• We know a rule-of-thumb on what type of
  landscapes geometric crossover will perform well
  smooth landscape
• This is just a beginning of general theory, in
  the future we will know more!

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Product Geometric Crossover

• GX1AxA?A geometric under d1
• GX2BxB? B geometric under d2
• A product crossover of GX1 and GX2 is an operator
  defined on the cartesian product of their domains
  PGX(A,B)x(A,B)?(A,B) that applies GX1on the
  first projection and GX2 on the second
  projection. GX1 and GX2 do not need to be
  independent.
• Theorem PGX is a geometric crossover under the
  distance d d1d2

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Properties of Product Geometric Crossover

• It is a simple and general method to build more
  complex geometric crossovers from simple
  geometric crossovers
• Multi-crossover same representation, same
  crossover n times
• Hybrid crossover same representation, different
  crossover for each projection
• Hybrid representation different representation
  (and crossover) for each projection
• No independence required base crossovers do not
  need to be independent

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II. Geometric Design for Sudoku
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The Sudoku Game
Fill in the grid so that every row,every column,
and every 3x3 boxcontains the digits 1 through 9
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Constraints

• It is a constraint-satisfaction problem with 4
  types of constraints
• Fixed Elements
• Rows are permutations
• Columns are permutations
• Boxes are permutations

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Computational Complexity

• The general Sudoku puzzle is based on a
  (n2)x(n2) grid
• The problem is NP-Complete
• Relaxation (3 constraints)
• Latin square completion (123) NP-Hard
• Sudoku puzzle generator (234) Polynomial?
• Initialisation problem (124 or 134) NP-Hard?
• Relaxation (2 constraints) Polynomial!

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Geometric Design

• Look at the problem and build a nice fitness
  landscape ( fitness function distance)
• the smaller search space the better
• the smoother landscape the better
• Pick genetic operators that match the landscape
  mutation and crossover should be geometric under
  the distance chosen

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Soft Hard Constraints

• Hard constraints all feasible solutions must
  respect them. Search operators take feasible
  solutions and produce feasible solutions
• Soft constraints level of fulfillment is the
  fitness of a solution
• More than one combination of soft and hard
  constraints available!

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Restricted Hamming space

• Hard constraint fixed positions
• Soft constraints permutations on rows, columns
  and boxes
• Distance Hamming distance between grids
• Feasible Mutation change any non-fixed element
• Feasible Crossover traditional crossover over
  the vector obtained by joining the rows of the
  grid

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Row-swap space

• Hard constraints fixed positions and
  permutations on rows
• Soft constraints permutations on columns and
  boxes
• Distance sum of swap distances between paired
  rows (row-swap distance)
• Feasible mutation swap two non-fixed elements in
  a row

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Geometric crossovers for row-swap space

• Row-wise PMX and row-wise cycle crossover
• Feasibility
• Row permutation simple PMX and cycle crossovers
  recombine permutations and produce permutations
• Fixed elements they both preserve fixed
  positions in the parents
• Geometricity
• Known simple PMX and cycle crossovers are
  geometric under swap distance
• For the product geometric theorem row-wise PMX
  and row-wise cycle crossovers are geometric under
  row-swap distance

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Fitness Function

• Fitness level of fulfilment of soft constraints
• Fitness to maximize
• Sum of unique elements in each row, plus,
• Sum of unique elements in each column, plus,
• Sum of unique elements in each box
• For a 9x9 grid the fitness corresponding to a
  fully correct grid is 243

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Smooth Fitness Landscapes

• Restricted Hamming Space
• a single element change affects the current
  fitness of -1, 0 or 1 for its row, for its
  column and for its box. Absolute maximum total
  change in fitness for a single change is 3
• Row-swap space
• A single swap in a row affects the current
  fitness of 0 for its row, between -2 and 2 for
  the columns touched, and the same for the boxes
  touched. The absolute maximum total change in
  fitness for a single swap in a row is 4
• Maximum delta fitness
• Max fitness for 9x9 grid 243
• Min fitness for 9x9 grid 27
• Max delta fitness in the landscape 243 27216
• Index of smoothness
• Change in fitness at distance one divided maximum
  change in fitness
• 0 perfectly smooth landscape, 1 max and min
  fitness are neighbours
• Index for Restricted Hamming Space 3/216
• Index for Row-swap Space 4/216
• Both Fitness Landscapes are very smooth!

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Prediction!

• Both fitness landscapes are very smooth so
  geometric crossovers and mutations associated
  with both spaces should work well
• Advantages of the row-swap search space
• it is much smaller because it restricts the
  search to feasible rows
• The restriction includes the optimum grid and
  prunes only grids with lower fitness
• Bet Row-swap operators will win!

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III. Experimental Results
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Hamming space crossovers with uniform swap
mutation
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Row-swap space crossovers with row-swap mutation
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Hill Climbers
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Results summary

• Crossovers based on row-swap space better than
  those based on hamming space
• Crossover (with mutation) better than hill
  climbers
• Many more experiments in the paper!
• Future work smartsquare crossover

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Conclusions

• Extended the geometric crossover with the notion
  of Product Geometric Crossover
• Product geometric crossover for Sudoku
• Designed geometric crossovers to deal naturally
  with constraints
• New geometric crossovers for the entire grid by
  using simple geometric crossover for each rows
• The associated distance has allowed us to analyse
  the crossover fitness landscape and predict that
  the crossovers will perform well
• Extensive experimental results confirm that the
  crossovers designed perform well