Dancing Links and N k Queens Funded by: Kentucky NSF EPSCoR: UKRF 304688440007419

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Dancing Links and N k QueensFunded by
Kentucky NSF EPSCoR UKRF 3046884400-07-419
John Miller (millerj10_at_nku.edu) Dr. Doug
Chatham Dr. Maureen Doyle Dr. Jeffrey Ward Amber
Rogers Luke Thompson
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Agenda

• N Queens Problem
• Dancing Links
• N k Queens Problem
• Finding Solutions
• Results
• Future Work

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The N Queens Problem

• First proposed by Max Bezzel in 1848 as the 8
  Queens Problem.
• The problem has been investigated by
  mathematicians since its inception, including
  Gauss and Cantor
• After solutions were found in 1850 by Franz Nauk,
  it was expanded to the N Queens Problem

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The N Queens Problem

• How many ways can n Queens on an n x n chessboard
  so that no queen can attack another queen
• Commonly used to introduce recursion

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What are Dancing Links?

• Dancing Links is a data structure that can be
  used to help solve any full cover problem.
• N k Queens
• Sudoku
• Introduced in 1979 by Hitotumatu and Noshita .
• Popularized by Knuth in 2000

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The Dancing Links Universe
Things get complicated very quickly For an n x n
chessboard, the universe will have 6n-2 columns
and 4n26n-1 nodes
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The Dance Steps
solve( int x ) if ( x n) solution found
return choose a row r for placing
queen remove row r for every block b in row
r place queen on block b remove bs
column remove bs diagonals solve(x1) repla
ce bs diagonals replace bs
column replace row r
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Results

• N Queens Standard Backtracking vs. Dancing
• Links (DLX)

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N k Queens Problem

• Consider a chessboard where we have blocked k
  squares. This is equivalent to placing a pawn on
  this square. The pawns are unable to attack by
  default.

This problem was first proposed in 1995 by
Michael Anshel.
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N k Queens Problem

• We have shown that there exist integers n and k
  such that we can safely place nk queens and k
  pawns on an n x n chessboard
• For each positive integer k and N gt max87 k,
  25k
• Also, no more than n k queens can safely be
  placed on an n x n chessboard with k pawns.

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Approach to Solving N k Queens

• How do we take advantage of the universe with the
  pawns.
• Look at all pawn positions and split the
  universe.
• Can we mimic the process of removing and
  replacing portions of the universe?

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Splitting Pawns

• Before finding Solutions, we must split the
  universe based on the positions of the pawns

Given a block b to place a pawn Create a new
header block Move everything below b into the
group Remove b from its group
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Splitting Pawns
Before Split
After Split
Head
Head
New Head
X
X
Y
Pawn
Pawn
Y
Notice Pawn has nothing pointing to it, but It is
still pointing to its former up and down nodes
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Reducing Search Space

• We have shown the following
• In order to place n k queens and k pawns on an
  n x n board
• No two pawns may share an edge
• No pawn may be placed on the outer edge of the
  board
• No pawn may be positioned on a square adjacent to
  a corner square.

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How fast is it on n k Queens?
Results obtained on Dual Core 64-bit Athlon
Processor, running red hat v4
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How Fast is it? (cont.)

• For some small values of n and k, the cost of
  creating the universe outweighs the benefit.
• For large value of and k, the speedup of DLX is
  invaluable.

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Parallel Results

• Approach to implementing with MPI

Current results indicate a 30 speedup on a 16
node cluster We are investigating other
approaches to achieve a More appropriate speedup
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Whats Next?

• Logic Programming
• Learning when we reach a dead end
• 3D Queens
• Can the DLX universe be extended to a 3D full
  cover problem
• Theoretical Chess Pieces
• Extending movement beyond straight lines

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References
1 CHATHAM, R. D., DOYLE, M, FRICKE, G.H.,
SKAGGS, R.D., AND WOLFF, M. Separating queens
on the chessboard 2 KNUTH D.E. Dancing links.
Millennial Perspectives in Computer Science.