A Generic Framework for Local Search : Application to the Sudoku Problem

1. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
A Generic Framework for Local Search :
Application to the Sudoku Problem
ICCS 2006, Reading, UK
Tony LAMBERT
LINA, Université de Nantes, France
LERIA,Université d’Angers, France
Universidad Santa María, Valparaíso, Chile
joint work with E. Monfroy and F. Saubion
Wed, 31 May, 2006
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2. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Outline
6 9 2 1 4
3 6 9 8
8 2
• Local search for solving CSP 1 9 7 2
• A framework for local search techniques 9 5
8 4 6 7
• Generic Iteration Algorithm with reduction
9 5
functions 2 3 5 1
• Some results for the sudoku problem 1 5 6 3 7
• Conclusion 1 2 3 4 5 6 7 8 9
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3. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Constraint satisfaction Problem (CSP)
C2
Y
C1 U
Variables
X
C4 C5
Constraints
Z T
C3
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4. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Local search (1)
Definition :
• Explore a D1 × D2 × · · · × Dn search space
• Move from neighbor to neighbor thanks to an evaluation
function
• Intensification
• Diversification
Properties :
• focus on some “promising” parts of the search space
• does not answer to unsat. problems
• no guaranteed
• “fast” to find a “good” solution
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5. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Local search (2)
• Search space : set of possible configurations
• Tools : neighborhood and evaluation function
Neighborhood
Set of
of s
possible
configurations
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6. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Local search (3)
• Search space : set of possible configurations
• Tools : neighborhood and evaluation function
Neighborhood
Set of
of s
possible
configurations
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7. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Local search (4)
• Search space : set of possible configurations
• Tools : neighborhood and evaluation function
Neighborhood
Set of
of s
possible
configurations
LS(p) = p’
p=( s 1 , s 2 , ..., sn)
s3 p’ = ( s 1 , s 2 , ..., s n , s n+1 )
s4
s6
s1 s2
s5
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8. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
A framework for local search techniques
• Idea :
• Fine grain control
• More strategies
• Technique :
• Decomposing local search resolution into basic functions
• Adapting chaotic iterations
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9. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Our Purpose :
• Use of an existing theoretical model for CSP solving
• Definition of the solving process
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10. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Abstract Model K.R. Apt [CP99]
Abstraction
Reduction functions
Constraint
Application
of functions
CSP
Fixed point
Partial Ordering
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11. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
A Generic Algorithm
F = { set reduction functions}
X = initial CSP
G=F X=
While G = ∅
choose g ∈ G
G = G − {g}
G = G ∪ update(G, g, X )
X = g(X )
EndWhile
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12. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
The theoretical model for CSP solving
Partial ordering :
p
1
Function to pass from one LS
state to another
p
2
State of LS
p
3
p
4
p
5
Terminates with a fixed point : local minimum, maximum number of iteration
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13. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Ls Ordering
Characteristics of a LS path
• notion of solution
• maximum length
Operational (computational) point of view : path = samples
A sample :
• Depends on a CSP
• Corresponds to a local search state
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14. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Neighborhood and Move functions
Neighborhood :
• FullNeighbor : V = {s ∈ D|s ∈ V }
• TabuNeighbor : V = {s ∈ D| ∃k , n − l ≤ k ≤ n, sk = s}
• DescentNeighbor : p = (s1 , . . . , sn ) and V = s ⊂ D s.t.
∃s ∈ V s.t eval(s ) < eval(sn )
Move :
• BestMove : p = p ⊕ s and eval(s ) = mins ∈V eval(s )
• ImproveMove : p = p ⊕ sn and p = p ⊕ s s.t.
eval(s ) < eval(sn )
• RandomMove : p = p ⊕ s and s ∈ V
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15. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Instantiate the GI algorithm
We can precise here the input set of function F .
Random walk :
Tabu search :
• FullNeighbor
• TabuNeighbor
• BestNeighBor
• BestNeighBor
• RandomNeighbor
Random walk + Descent :
TabuSearch + Descent :
• FullNeighbor
• TabuNeighbor
• BestNeighBor
• DescentNeighbor
• RandomNeighbor
• ImproveNeighBor
• DescentNeighbor
• BestNeighBor
• ImproveNeighBor
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16. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Sudoku problem
6 9 2 1 4
3 6 9 8
8 2
1 9 7 2
• n2 × n2 problem 9 5
n4
8 4 6 7
• variables 9 5
• AllDiff constraints 2 3 5 1
1 5 6 3 7
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17. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Sudoku problem
6 9 2 1 4
3 2 1 6 9 8
8 2 1
1 9 7 2 3
9 5
8 4 6 7 9
8 3 9 5 6 2
2 3 5 1
1 5 6 3 7
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18. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Sudoku problem
6 9 5 7 2 8 3 1 4
3 2 1 6 4 9 7 5 8
7 4 8 2 9 1
4 1 9 5 7 2 3
9 5
5 8 4 6 7 9
8 3 9 5 6 2
2 3 9 5 1
1 5 4 6 9 3 7
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19. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Sudoku problem
6 9 5 7 2 8 3 1 4
3 2 1 6 4 9 7 5 8
7 4 8 5 2 9 1
4 1 6 9 5 7 8 2 3
9 6 4 5
5 8 4 6 7 9
8 3 9 5 6 2
2 6 7 3 9 5 4 8 1
1 5 4 6 9 3 7
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20. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Sudoku problem
6 9 5 7 2 8 3 1 4
3 2 1 6 4 9 7 5 8
7 4 8 5 1 3 2 9 1
4 1 6 9 5 7 8 2 3
9 7 3 2 8 1 6 4 5
5 8 2 4 3 6 1 7 9
8 3 9 1 7 4 5 6 2
2 6 7 3 9 5 4 8 1
1 5 4 8 6 2 9 3 7
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21. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
TabuSearch RandomWalk
n2 x n2 16x16 25x25 36x36 16x16 25x25 36x36
cpu time 3,14 115,08 3289,8 3,92 105,22 2495
deviations 1,28 52,3 1347,4 1,47 49,3 1099
mvts 405 3240 22333 443 2318 13975
Descent + TabuSearch Descent +RandomWalk
n2 x n2 16x16 25x25 36x36 16x16 25x25 36x36
cpu time 2,34 111,81 2948 2,41 82,94 2455
deviations 1,42 55,04 1476 1,11 36,99 1092
mvts Avg 534 3666 20878 544 2581 14908
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22. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
Conclusion and Future works
• A generic model local search methods
• Design of strategies
• Dynamic strategies learning
• Providing more tools in a generic environment
• A generic implementation
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23. OUTLINE LOCAL SEARCH FOR SOLVING CSP A FRAMEWORK FOR LS THE SUDOKU RESULTS CONCLUSION
A Generic Framework for Local Search :
Application to the Sudoku Problem
ICCS 2006, Reading, UK
Tony LAMBERT
LINA, Université de Nantes, France
LERIA,Université d’Angers, France
Universidad Santa María, Valparaíso, Chile
joint work with E. Monfroy and F. Saubion
Wed, 31 May, 2006
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