Polynomial Time Algorithms for the N-Queen Problem

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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 nlt100

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QS1

• Data Structure
• The i-th queen is placed at row i and column
  queeni
• 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 dniqueeni and
  dpi-queeni
• 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
Go through the attacking queen only
Reduce cost of bookeeping
C232 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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Results Time Complexity
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Results Time Complexity
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Results Time Complexity
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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.