Rook Polynomial Relaxation Labeling Ofir Cohen Shay Horonchik.
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Transcript of Rook Polynomial Relaxation Labeling Ofir Cohen Shay Horonchik.
Rook PolynomialRelaxation Labeling
Ofir Cohen Shay Horonchik
Problem Domain
Rooks can only move horizontally or vertically.
Place n Rooks on a n*n chess board with holes, where no piece can challenge other rooks.
This is an NP Complete problem
Problem Domain (cont.)
Rook Polynomial can be reduced to: Resource distribution under constraints
Known Solutions Algorithms using back tracking Include / exclude mechanism
Rook Polynomial via Relaxation Labeling
Set of Objects: We declared each cell (except holes) as an object.
Set of Labels: We declared two labels: {Empty, Rook}
Initial Confidence: Rook => 1 / Maximum between empty cells in row
and clumn Empty => 1 - Empty
bb nB ,...,
1
m,...,2,1
im
i
i
p
p
1
0
1
0
0
Rook Polynomial via Relaxation Labeling
Compatibility -
Example:
Rook Polynomial via Relaxation Labeling
Results: Very long running time it doesn’t converge to the correct solution The algorithm doesn’t try to achieve maximum rook
number on board Successful runs. (only on small boards)
Rook Polynomial via Relaxation Labeling (phase b)
We perform the following changes: Initial confidence
Randomize rooks on several cells on the board Support function
Zeroing cells where found rooks in both row and column Increasing cells value where found an empty
row/column
Implementation
Input: Number Of Columns Number Of Rows Number Of Cells With Holes
Problems And Conclusion
Relaxation algorithm purpose don’t match the problem specification . Relaxation labeling purpose is to match objects and
labels The rook polynomial problem purpose is to find
maximal “Rook labels”
Any Questions ?
Thank You