Solving N k Queens Using Dancing Links

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Solving Nk Queens Using Dancing Links

• Matthew A. Wolff
• Morehead State University
• May 19, 2006

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Agenda

• Motivation
• Terms
• Problem Definition
• Timing Results for various N1 Programs
• Future Work

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Motivation

• NASA EPSCoR grant
• (Subcontract WKURF 516140-06-15)
• Began working with Chatham and Skaggs in November
• Doyle added DLX (Dancing Links) at beginning of
  Spring '06 semester
• Senior Project

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Category of Problems

• 8 Queens
• 8 attacking queens on an 8x8 chess board
• N Queens
• N attacking queens on an NxN chess board
• N1 Queens
• N1 attacking queens on an NxN chess board
• 1 Pawn used to block two or more attacking queens
• Nk Queens
• Nk attacking queens on an NxN chess board
• k Pawns used to block numerous attacking queens

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Recursion

• "To understand recursion, one must first
  understand recursion" -- Tina Mancuso
• A function is recursive if it can be called
  while active (on the stack).
• i.e. It calls itself

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Recursion in Art
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Recursion in Computer Science

• // precondition n gt 0// postcondition n! is
  returnedfactorial (int n) if (n 1) or (n
  0) return 1 else return
  (nfactorial(n-1))

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Backtracking

• An example of backtracking is used in a
  depth-first search in a binary tree
• Let t be a binary tree
• depthfirst(t) if (t is not empty) access
  root item of t depthfirst(left(t)) depthfi
  rst(right(t))

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Backtracking Example

• Output A B D E H I C F - G

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Main Focus Nk Queens

• Why?
• Instead of focusing on specific solutions (N1,
  N2, ...), we will be able to solve any general
  statement (Nk) of the Queens Problem.
• Implementing a solution is rigorous and utilizes
  many important techniques in computer science
  such as parallel algorithm development,
  recursion, and backtracking

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Chatham, Fricke, Skaggs

• Proved Nk queens can be placed on an NxN board
  with k pawns.

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NK what to do?

• Nk presents a very large problem
• 1 Pawn meant an extra for loop around everything
• k Pawns would imply k for loops around everything
• Dynamic for loops?
• Search for a better way
• Dancing Links

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Why Dancing Links?

• Structure Algorithm
• Comprehendible (Open for Debate)
• Increased performance
• DLX is supposedly quicker than a standard
  backtracking algorithm
• Made popular by Knuth via his circa 2000 article

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The Universe

• Multi-Dimensional structure composed of circular,
  doubly linked-lists
• Each row and column is a circular, doubly
  linked-list

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Visualization of The Universe
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The Header node

• The root node of the entire structure
• Members
• Left pointer
• Right pointer
• Name (H)
• Size Number of Column Headers in its row.

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Column Headers

• Column Headers are nodes linked horizontally with
  the Header node
• Members
• Left pointer
• Right pointer
• Up pointer
• Down pointer
• Name (Rw, Fx, Ay, or Bz)
• Size the number of Column Objects linked
  vertically in their column

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Column Objects

• Grouped in two ways
• All nodes in the same column are members of the
  same Rank, File, or Diagonal on the chess board
• Linked horizontally in sets of 4
• Rw, Fx, Ay, or Bz
• Each set represents a space on the chess board
• Same members as Column Headers, but with an
  additional top pointer which points directly to
  the Column Header

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Mapping the Chess Board
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The Amazing TechniColor Chess Board
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The Dance Steps

• The entire algorithm is based off of two simple
  ideas
• Cover remove an item
• Node.right.left Node.left
• Node.left.right Node.right
• Uncover insert the item back
• Node.right.left Node
• Node.left.right Node

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Dance Steps, cont.

• void search(k) if (header.right header)
  finished else c choose_column() cover(c)
  r c.down while (r ! c) j
  r.right while (j ! r) cover(j.top) j
  j.right place next queen search(k1)
  c r.top j r.left while (j !
  r) uncover(j.top) j j.left
  completed search(k) uncover(c) finished

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1x1 Universe Before
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1x1 Universe After
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Modifying for Nk Queens

• 1 Pawn will cut its row, column, and diagonal
  into 2 separate pieces
• Just add these 4 new Column Headers to the
  universe, along with their respective Column
  Objects
• k Pawns will cut their rows, columns, and
  diagonals into. ? separate pieces.
• Still need to add these extra Column Headers, but
  how many are there and how many Column Objects
  are in each?

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It Slices, It Dices

• Find ALL valid Pawn Placements
• (N-2)2 choose k lots of combinations
• Then build 4 NxN arrays
• One for each Rank, File, and Diagonal
• Scan through arrays
• For Ranks scan horizontally (Files vertically,
  Diagonals diagonally)
• Reach the end or a Pawn, increment 1

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Example of Rank Scan
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N1 QueensVarying Language, Algorithm
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N1 Queens Parallel Backtracking vs. DLX
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N1 QueensSequential DLX vs. Parallel DLX
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Interesting TidbitSequential DLX vs. Parallel
C
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Questions?

• Thank you!
• Dr. Chatham
• Dr. Doyle
• Mr. Skaggs

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References