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 Queen attacks other at the same row, column

or diagonal line No 2 queens attack each other

A Solution for 6-Queen

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

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

QS1

Pseudo-code

QS1

Gradient-Based Heuristic

QS2

Data Structure Queen placement same as QS1 An array attack is maintained

Store the row indexes of queens that are under attack

QS2

Pseudo-code

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

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

QS4

Algorithm same as QS1 Initialization same as QS4 The fastest algorithm

3,000,000 queens in less than one minute

Results – Statistics of QS1Collisions for random permutation

Max no. and min no. of queens on the most populated diagonal in a random permutation

Permutation Statistics

Swap Statistics

Results – Statistics of QS2

Permutation Statistics

Swap Statistics

Results – Statistics of QS3

Swap Statistics

Number of conflict-free queens during initialization

Results – Time Complexity

Results – Time Complexity

Results – Time Complexity

Results – Time Complexity

Results – Time Complexity

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.