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Systematic Search Guided by Local Search with Conflict-based Heuristic in

N-queen problem

Hyoung rae KimDebasis Mitra Ph.D

Florida Institute of TechnologyDepartment of Computer Science

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1. Introduction2. Proposed method3. Implementation design4. Experiments and analysis5. Related work6. Conclusion7. Future works8. References

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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.

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Resource allocation in scheduling

[2]1

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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

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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: QiQj• 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.

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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.

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An example of a search space

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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

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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.

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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.

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Forward checking algorithm

(12)

(4) (4)

(1)

(12)

(5)

(2)

(0)Total comparison: 40

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Forward checking algorithmFC1 (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);

}

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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’;

}

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Min-conflict algorithm

Checking (1)

Checking (3) C(3)

Checking (2)

Checking (1)

Checking (5)

Total comparison: 71

Checking (1)

Ordering (16)

Ordering (10)

Ordering (8)

Ordering (7)

Ordering (7)

Initial status3 12 3

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Min-conflict algorithmInformed_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);

}

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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

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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

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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

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Explanation of hybrid method

• Forward checking after Min-conflict.• K=0 means pure FC, K=n means pure MC.

[8-queens problem]

Solve this portion by MC

Solve this portion by FC

K* Vary this K value

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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].

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Hybrid algorithm

K=2

MC

FC2

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Hybrid algorithmSEARCH (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

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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].

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Compare the label count

k AverageLabel

Std of Label Std. of Label /Sqrt(n)

0 411,608 - - 2 436,652 1,020,620 58,926 4 280,998 738,118 42,615 6 173,603 336,408 19,423 8 104,788 150,904 8,712 10 21,581 20,679 1,194 12 4,753 4,394 254 14 1,300 1,109 64 16 839 836 48 18 1,025 1,025 59 20 1,844 1,800 104 22 2,357 2,749 159 24 2,936 9,096 525 1

Plot label count in a graph

0

100000

200000

300000

400000

500000

600000

1 2 3 4 5 6 7 8 9 10 11 12 13

Pure Forward checking

Pure Min Conflict

k

Complexity

k

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Compare the total count

k AverageConstraint

Std. of Constraints Std. of Constraints /Sqrt(n)

0 9,032,529 - - 2 8,912,273 21,793,943 1,258,274 4 5,810,238 15,920,543 919,173 6 3,608,147 7,195,248 415,418 8 2,124,796 3,110,599 179,590

10 413,155 404,830 23,373 12 154,604 144,286 8,330 14 155,602 140,061 8,086 16 230,620 230,762 13,323 18 379,014 379,427 21,906 20 619,071 600,727 34,683 22 581,590 649,051 37,473 24 643,543 1,916,064 110,624 1

Plot total count in a graph

0

2000000

4000000

6000000

8000000

10000000

12000000

1 2 3 4 5 6 7 8 9 10 11 12 13

Complexity

k

Pure Forward checking

Pure Min Conflict

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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. = 1.32

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

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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].

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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

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7. Future works

• Vary N to bigger number.• For other problems other than N-queens.• Theoretical studies for the result.

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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.

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