Systematic Search Guided by Local Search with Conflict-based Heuristic in N-queen problem

1. Systematic Search Guided by Local Search with Conflict-based Heuristic in N-queen problem Hyoung rae Kim Debasis Mitra Ph.D Florida Institute of Technology Department of Computer Science

2. Contents • Introduction • Proposed method • Implementation design • Experiments and analysis • Related work • Conclusion • Future works • References > > > > > 1

3. 1. Introduction • Constraint Satisfaction Problem(CSP) does very important role in Artificial Intelligence (AI). CSP appears in many areas, for instance vision, resource allocation in scheduling and temporal reasoning [2]. • What is a constraint satisfaction problem • A CSP is a problem composed of a finite set of variables, each of which is associated with a finite domain, and a set of constraints. • The task is to assign a value to each variable satisfying all the constraints. 1

4. Resource allocation in scheduling [2] 1

5. N-queens problem • Place eight queens on an 8 × 8 chessboard satisfying the constraint that no two queens should be on the same row, column or diagonal. [4 × 4 queens problem] 1

6. N-queens problem • Problem formalization • The set of variables: Z = {Q1, Q2, …, Q8} • Domain: DQ1 = DQ2 = … = DQ8 = {1,2,3,4,5,6,7,8} • Constraint (1): i,j: QiQj • Constraint (2): i,j, if Qi=a and Qj=b, then i-j  a-b, and i-j  b-a. • The variable is considered row number. • The domain of each variable is set of column numbers. 1

7. Problem reduction and search • There are two approaches to solve CSP • Problem reduction • Pruning off search spaces that contain no solution • Reducing the size of domains of the variables* Tightening constraints potentially reduce the search space at a later stage of the search • Pruning off branches in the search space • It can be performed at any stage of the search • Search • Find solution in the search space, all or one of the solutions. • One often has to find a balance between the efforts made and the potential gains in problem reduction. 1

8. An example of a search space 1

9. Search strategies • Systematic algorithm • Starts from an empty variable assignment that is extended until obtaining a complete assignment that satisfies all the constraints in the problem. • Look-back enhancements (backward checking, back jumping, etc.) • Look-ahead enhancements (forward checking, etc.) • Local search algorithm • Perform an incomplete exploration of the search space by repairing infeasible complete assignments (min-conflict, GSAT, tabu search). • Hybrid approach • Performing a local search before or after a systematic search. • Performing a systematic search improved with a local search at some point of the search. • Performing an overall local search, and using systematic search either to select a candidate neighbor or to prune the search space [1]. 1

10. The contributions of this work • Explain the relationship between Local search algorithm (MC) and Systematic algorithm (FC). • Trying to find faster searching algorithm by combining them. 1

11. 2. Proposed method • We improve the speed by hybrid of Forward Checking and Min-Conflict: Forward checking after Min-conflict. • We examine the complexity and accuracy as gradually varying the coverage of Min-conflict. 1

12. Forward checking algorithm (12) (12) (4) (4) (5) (1) (2) Total comparison: 40 (0) 1

13. Forward checking algorithm FC1 (UNLABELLED, COMPOUND_LABEL, D,C){ if (UNLABELLED={}) {return UNLABELLED;}; Pick one variable x from UNLABELLED; { pick one value v from Dx; Delete v from Dx; D’=Update1(UNLABELLED-{X}, D, C, <x,v>); Result = FC1 (UNLABELLED-{X}, COMPOUND_LABEL+{<x,v>}, D’, C); if (Result != Nil) {return Result;}; } until (Dx={}); return (NIL); } S

14. Update1(W,D,C,Lable) { D’=D; for each variable y in W { for each value v in D’y { if (<y,v> is incompatible with Label with respect to the constraints in C) D’y=D’y-{v}; } } return D’; } S

15. 3 2 3 1 Min-conflict algorithm Initial status Checking (1) Ordering (16) Checking (1) Checking (1) Ordering (10) Ordering (7) Checking (3) C(3) Checking (5) Ordering (8) Total comparison: 71 Checking (2) 2 Ordering (7)

16. Min-conflict algorithm Informed_Backtrack(Z,D,C) { LEFT = {}; for each variable x in Z { pick a random value from Dx; add <x,v> to LEFT; } InfBack(LEFT, {}, D, C); } S

17. InfBack(LEFT, DONE, D,C) { if (LEFT+DONE is compatible with constraints) {return LEFT+DONE;}; x = any variable such that label <x,v> is in LEFT; Queue = Order_values(x, Dx, Labels_left, Labels_done, C); while (Queue != {}){ w = first element in Queue; Delete w from Queue; DONE = DONE + {<x,w>}; Result = InfBack(LEFT-{<x,v>}, DONE, D, C): if (Result != Nil) {return Result;}; } return Nil; } S

18. Order_values(x, Dx, LEFT, DONE, C) { List = {}; for each v in Dx { if (<x,v> is compatible with all the labels in DONE) { Count [v] = 0; for each <y,w> in LEFT { if NOT satisfies ((<x,v><y,w>), Cx,y) Count[v]=count[v]+1; } List = List + {v}; } } Queue = the values in List ordered in ascending order of Count[v]; return Queue; } S

19. Comparison between MC and FC • Forward checking (FC) • Advantage: Completeness – it always find a solution if one exists. One of the best Systematic algorithm. • Disadvantage: FC is typically cursed with early mistakes in the search, a wrong variable value can cause a whole sub-tree to be explored with no success. • Min-conflict (MC) • Advantage: Do not suffer from the early-mistake problem. It may be far more efficient than systematic ones to find a first solution. • Disadvantage: Not complete. It can be undone, without having anything to prove. 1

20. Explanation of hybrid method • Forward checking after Min-conflict. • K=0 means pure FC, K=n means pure MC. Solve this portion by MC K * Vary this K value Solve this portion by FC 1 [8-queens problem]

21. 3. Implementation design • Input variable: N-queens problem • Output variable: • Counted number of visited label. • Counted number of executed constraints. • MC-FC algorithm runs MC and then FC with the results from MC. • We use standard MC algorithm [2]. • We use standard FC algorithm [2]. 1

22. Hybrid algorithm K=2 MC FC 2

23. Hybrid algorithm SEARCH (n) { for each k=0 to n 1. MC_FC (k, Success, Count_Label, Count_Constraint); 2. print (k, Success, Count_Label, Count_Constraint); } MC_FC (k, Success, Count_Label, Count_Constraint) { Repeat until it gets a result or reach to the max iteration 1. Initialize cZ, cD, CC 2. COMPOUND_LABEL = MC(k, cZ, cD, cC, Count_Label, Count_Constraint); 3. If COMPOUND_LABEL is valid Result=FC(k,COMPOUND_LABEL, cZ,cD,cC,Count_Label,Count_Constraint) 4. If Result is valid Success = True return; } S

24. 4. Experiment and analysis • We use 24-queens problem. • We ran the algorithm 300 times on a Sun Ultra 60. • The max iteration number was 1000 (if FC part does not have solutions, it randomly re-execute MC part). • We recorded every k value from 0 through n with an interval of 2. • An output parameter ‘Label count’ is the number of label that the algorithm visited. • The other parameter ‘Total count’ is the number of how many times the constraint is checked. ‘Total count’ subsumes the ‘Label count’. • We analyze the ‘Label count’ and ‘Total count’. • We use this formula to compare the quality of data points, which is often referred to as standard error of the mean: S.D. of Total count / Sqrt(n) [3]. 1

25. Compare the label count 1

26. Complexity Pure Forward checking Plot label count in a graph Pure Min Conflict 1 k k

27. Compare the total count 1

28. Complexity Pure Forward checking Plot total count in a graph Pure Min Conflict 1 k

29. Explanation of the results • The reason of gradual shrinking of the width. • 4-queens problem has two solutions with following conditions. • When K=1, • There are two solution marks (called A), it takes 4 steps to know the results. • There are two un-solution marks (called B), it takes 6 steps to know the results. • Starts with solution mark A (50%): 4 • Starts with un-solution mark B (50%): 10 • When K=2, • There are two solution marks (called A), it takes 2 steps to know the results. • There are four un-solution marks, 2 has 1step (called B), 2 has 2 steps (called C). • Starts with solution mark (34%): 2 • Starts with un-solution mark, B -> A (16.5%) : 3 • Starts with un-solution mark, B -> C -> A (16.5%): 5 • Starts with un-solution mark, C -> A (16.5%): 4 • Starts with un-solution makr, C-> B -> A (16.5%): 5 • This tells when K=2 the S.D is much smaller. • Case 1=12={4,10,4,10,…}, S.D. = 3.1; Case 2=12={2,2,3,5,4,5,…}, S.D. = 1.3 2

30. As for the reduction of the complexity We are trying to find the explanation. 1

31. 5. Related work • A research tried to show that the look-back and look-ahead enhancements of backtracking-based algorithms can be exploited for local search algorithms, and can greatly improve their behavior too. They propose a generic search technique over CSP which is called decision-repair, which show great performance [1]. S

32. 6. Conclusion • We performed a hybrid search: Performing a local search (MC) before a systematic search (FC). • The purpose of our research is to understand the relationship between MC and FC and to improve the speed of searching algorithm. • The algorithm shows the best performance when K value is in the middle. • We need theoretical explanation for this results. • Even without the theoretical explanation, the Hybrid algorithm is better than pure MC and FC. 1

33. 7. Future works • Vary N to bigger number. • For other problems other than N-queens. • Theoretical studies for the result. 1

34. 8. References • [1] N. Jussien, O. Lhomme, Local Search with Constraint Propagation and Conflict-absed Heuristics, Artificial Intelligence 139 (2002) 21-45. • [2] E. Tsang, “Foundations of Constraint Satisfaction”, University of Essex Colchester Essex, UK., (1995). • [3] John Mandel, The statistical analysis of experimental data, Dover, (1964) 63. 1