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

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Systematic Search Guided by Local Search with
Conflict-based Heuristic in N-queen problem
Florida Institute of Technology Department of
Computer Science

• Hyoung rae Kim
• Debasis Mitra Ph.D

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Contents

1. Introduction
2. Proposed method
3. Implementation design
4. Experiments and analysis
5. Related work
6. Conclusion
7. Future works
8. 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
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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
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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 Qi?Qj
• Constraint (2) ?i,j, if Qia and Qjb, 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.

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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)
(12)
(4)
(4)
(5)
(1)
(2)
Total comparison 40
(0)
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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
• DUpdate1(UNLABELLED-X, D, C, ltx,vgt)
• Result FC1 (UNLABELLED-X,
  COMPOUND_LABELltx,vgt, D, C)
• if (Result ! Nil) return Result
• until (Dx)
• return (NIL)

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• Update1(W,D,C,Lable)
• DD
• for each variable y in W
• for each value v in Dy
• if (lty,vgt is incompatible with Label with
  respect to the constraints in C)
• DyDy-v
• return D

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Min-conflict algorithm
Checking (1)
Initial status
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)
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Min-conflict algorithm

• Informed_Backtrack(Z,D,C)
• LEFT
• for each variable x in Z
• pick a random value from Dx
• add ltx,vgt to LEFT
• InfBack(LEFT, , D, C)

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• InfBack(LEFT, DONE, D,C)
• if (LEFTDONE is compatible with constraints)
• return LEFTDONE
• x any variable such that label ltx,vgt 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 ltx,wgt
• Result InfBack(LEFT-ltx,vgt, DONE, D, C)
• if (Result ! Nil) return Result
• return Nil

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• Order_values(x, Dx, LEFT, DONE, C)
• List
• for each v in Dx
• if (ltx,vgt is compatible with all the labels in
  DONE)
• Count v 0
• for each lty,wgt in LEFT
• if NOT satisfies ((ltx,vgtlty,wgt), Cx,y)
• Countvcountv1
• List List v
• Queue the values in List ordered in ascending
  order of Countv
• return Queue

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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.

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Explanation of hybrid method

• Forward checking after Min-conflict.
• K0 means pure FC, Kn means pure MC.

Solve this portion by MC
K Vary this K value
Solve this portion by FC
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8-queens problem
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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
K2
MC
FC
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Hybrid algorithm

• SEARCH (n)
• for each k0 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
• ResultFC(k,COMPOUND_LABEL,
• 
  cZ,cD,cC,Count_Label,Count_Constraint)
• 4. If Result is valid
• Success True
• return

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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
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Complexity
Pure Forward checking
Plot label count in a graph
Pure Min Conflict
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k
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Compare the total count
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Complexity
Pure Forward checking
Plot total count in a graph
Pure Min Conflict
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k
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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 K1,
• 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 K2,
• 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 -gt A (16.5) 3
• Starts with un-solution mark, B -gt C -gt A
  (16.5) 5
• Starts with un-solution mark, C -gt A (16.5) 4
• Starts with un-solution makr, C-gt B -gt A (16.5)
  5
• This tells when K2 the S.D is much smaller.
• Case 1124,10,4,10,, S.D. 3.1 Case
  2122,2,3,5,4,5,, S.D. 1.3

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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.

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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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