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Polynomial Time Algorithms for the N-Queen Problem

Polynomial Time Algorithms for the N-Queen Problem. Rok sosic and Jun Gu. Outline. N-Queen Problem Previous Works Probabilistic Local Search Algorithms QS1, QS2, QS3 and QS4 Results. N-Queen Problem. A classical combinatorial problem n x n chess board n queens on the same board

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Polynomial Time Algorithms for the N-Queen Problem

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

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

  3. 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

  4. A Solution for 6-Queen

  5. 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

  6. 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

  7. QS1 • Pseudo-code

  8. QS1 • Gradient-Based Heuristic

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

  10. QS2 • Pseudo-code

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

  12. 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

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

  14. 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

  15. Results – Statistics of QS2 Permutation Statistics Swap Statistics

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

  17. Results – Time Complexity

  18. Results – Time Complexity

  19. Results – Time Complexity

  20. Results – Time Complexity

  21. Results – Time Complexity

  22. 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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