Solving N k Queens Using Dancing Links

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Solving Nk Queens Using Dancing Links

• Matthew Wolff
• CS 499c
• May 3, 2006

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Agenda

• Motivation
• Definitions
• Problem Definition
• Solved Problems with Results
• Future Work

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Motivation

• NASA EPSCoR grant
• Began working with Chatham and Skaggs in November
• Doyle added DLX (Dancing Links) at beginning of
  semester
• New to me (and the rest of the team, I think)
• A lot more work!

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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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8 Queens Example

• http//www.jsomers.com/nqueen_demo/
  nqueens.html

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Solutions

• Solutions A class of Queen placements such
  that no two Queens can attack each other.
• Fundamental Solutions A class of solutions
  such that all members of the class are simply
  rotations or reflections of one another.
• Given the set of solutions, a set of fundamental
  solutions can be generated. And vice versa
• The fundamental solutions are a subset of all
  solutions.

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Fundamental Solutions for 8 Queens
http//mathworld.wolfram.com/QueensProblem.html
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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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4 Queens Backtracking Example

• Solved by iterating over all solutions, using
  backtracking

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N Queens

• Extend to N board
• Similar to 8 Queens
• Use a more general board of size NxN
• Same algorithm as 8 Queens

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N1 Queens

• What happens when you add a pawn?
• For a large enough board, we can add an extra
  Queen
• Slightly more complex
• Another loop over Pawn placements
• More checking for fundamental solutions

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8x8 Board, 1 Pawn
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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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What did I do?

• Translate Chathams Python Code (for N1) into a
  sequential C program
• Modify sequential C code to run in Parallel
  with MPI
• Design and implement the Nk Queens solution
• (Iterative)k (Recursive)N No.
• Dancing Links

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N1 Sequential Solution

• Optimized to exploit the geometry of the problem
• Pawns may not be placed in first or last column
  or row
• Pawns are only placed on roughly 1/8 of the board
  (in a wedge shape)
• The Need for Speed
• Even with optimizations, program can run for days
  for large N
• Roughly 6x faster than Python

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N1 Results
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Python versus C
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N1 Parallel Solution

• Almost exactly the same as Sequential except
• For-loop over Pawn Placements is distributed
  over p processors
• Evidence suggests that more solutions are found
  when the Pawns are near the center of the chess
  board
• More solutions implies more computations, thus
  more time
• Pawns are specially numbered for more optimization

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Pawn Placements for Parallel N1 Queens Solution
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N1 Queens, Parallel vs. Sequential C
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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? Thats Unpossible Ralph
  Wiggum
• Search for a better way
• Dancing Links

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Why Dancing Links?

• Structure Algorithm
• Comprehendible (Open for Debate)
• Increased performance
• Parallel computing is utilized mainly for
  performance advantages so, why run a sub-par
  algorithm (backtracking) when the goal is to
  achieve the quickest run-time?
• 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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Dance, Dance Revolution

• 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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The Latest Dance Craze

• 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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The Aha! Moment

• N Queens worked, now what?
• Nk Queens hmm
• What needs to be modified?
• Do I have to start from scratch?!?!
• .
• Nope ?
• As it turns out, the way the universe is built is
  the only needed modification to go from N Queens
  to Nk Queens

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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
• Wolffs Theorem
• (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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And now for the moment youve all been waiting
for!

• DRUM ROLL!.
• GRAPHS AND STUFF!

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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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Nk Results

• Still running in Lappin 241L
• Maybe next week ?

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Further Work

• Finish module that will properly count the
  fundamental solutions
• Since run-times will decrease over time (newer
  processors, etc), compare amount of updates to
  the structure to see if Dancing Links is actually
  doing less work, which would explain the decrease
  in run-time.

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Future Work (Project)

• Find a more efficient way to account for k Pawns
  in the universe
• Using Dancing Links itself?
• Find patterns so parallelization can be done
  efficiently, similar to N1 specific parallel
  program
• Find more results for larger values of N and k
• May involve use of Genetic Algorithms
• Domination Problem?
• Fewest number of Queens to cover entire chess
  board.

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Questions?

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

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References