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Polynomial Time Algorithms for the N-Queen Problem Rok sosic and Jun Gu

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Outline N-Queen Problem Previous Works Probabilistic Local Search Algorithms QS1, QS2, QS3 and QS4 Results

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N-Queen Problem A classical combinatorial problem n x n chess board n queens on the same board Queen attacks other at the same row, column or diagonal line No 2 queens attack each other

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A Solution for 6-Queen

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Previous Works Analytical solution Direct computation, very fast Generate only a very restricted class of solutions Backtracking Search Generate all possible solutions Exponential time complexity Can only solve for n<100

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QS1 Data Structure The i-th queen is placed at row i and column queen[i] 1 queen per row The array queen must contain a permutation of integers {1,…,n} 1 queen per column 2 arrays, dn and dp, of size 2n-1 keep track of number of queen on negative and positive diagonal lines The i-th queen is counted at dn[i+queen[i]] and dp[i-queen[i]] Problem remains Resolve any collision on the diagonal lines

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QS1 Pseudo-code

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QS1 Gradient-Based Heuristic

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QS2 Data Structure Queen placement same as QS1 An array attack is maintained Store the row indexes of queens that are under attack

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QS2 Pseudo-code

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QS2 Reduce cost of bookeeping Go through the attacking queen only C2=32 to maximize the speed for small N, no effect on large N

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QS3 Improvement on QS2 Random permutation generates approximately 0.53n collisions Conflict-free initialization The position of a new queen is randomly generated until a conflict-free place is found After a certain of queens, m, the remaining c queens are placed randomly regardless of conflicts

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QS4 Algorithm same as QS1 Initialization same as QS4 The fastest algorithm 3,000,000 queens in less than one minute

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Results – Statistics of QS1 Collisions for random permutation Max no. and min no. of queens on the most populated diagonal in a random permutation Permutation Statistics Swap Statistics

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Results – Statistics of QS2 Permutation Statistics Swap Statistics

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Results – Statistics of QS3 Swap Statistics Number of conflict-free queens during initialization

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Results – Time Complexity

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References R. Sosic, and J. Gu, “A polynomial time algorithm for the n-queens problem,” SIGART Bulletin, vol.1(3), Oct. 1990, pp.7-11. R. Sosic, and J. Gu, “Fast Search Algorithms for the N-Queens Problem,” IEEE Transactions on Systems, Man, and Cybernetics, vol.21(6), Nov. 1991, pp.1572- 1576. R. Sosic, and J. Gu, “3,000,000 Queens in Less Than One Minute,” SIGART Bulletin, vol.2(2), Apr. 1991, pp.22-24.

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1 Chapter 5 Advanced Search. 2 Chapter 5 Contents l Constraint satisfaction problems l Heuristic repair l The eight queens problem l Combinatorial optimization.

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