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Solving N+k 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 N+1 Queens N+1 attacking queens on an NxN chess board 1 Pawn used to block two or more attacking queens N+k Queens N+k 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 understandrecursion understandrecursionunderstandrecursion understandrecursion “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 >= 0 // postcondition: n! is returned factorial (int n) { if (n == 1) or (n == 0) return 1; else return (n*factorial(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)); depthfirst(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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N+1 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: N+k Queens Why? Instead of focusing on specific solutions (N+1, N+2,...), we will be able to solve any general statement (N+k) 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 N+k queens can be placed on an NxN board with k pawns.

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What did I do? Translate Chatham’s Python Code (for N+1) into a sequential C++ program Modify sequential C++ code to run in Parallel with MPI Design and implement the N+k Queens solution (Iterative) k * (Recursive) N = No. Dancing Links

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N+1 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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N+1 Results NSolutions Fundamental Solutions 6216 7320 816128 952396 102862288 11140311152 12821465172 1354756437848 143898333118664 15292375723387448

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Python versus C++

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N+1 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 N+1 Queens Solution

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N+1 Queens, Parallel vs. Sequential C++

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N+K – what to do? N+k 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? “That’s 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 (R w, F x, A y, or B z ) 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 {R w, F x, A y, or B z } 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(k+1) 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? N+k Queens… hmm What needs to be modified? Do I have to start from scratch?!?! …. Nope Nope As it turns out, the way the universe is built is the only needed modification to go from N Queens to N+k Queens

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Modifying for N+k 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 Wolff’s 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 you’ve all been waiting for! DRUM ROLL! …. GRAPHS AND STUFF!

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N+1 Queens Varying Language, Algorithm

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N+1 Queens Parallel Backtracking vs. DLX

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N+1 Queens Sequential DLX vs. Parallel DLX

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Interesting Tidbit: Sequential DLX vs. Parallel C++

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N+k Results Still running in Lappin 241L… Maybe next week 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 N+1 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

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Solving N+k Queens Using Dancing Links Matthew A. Wolff Morehead State University May 19, 2006.

Solving N+k Queens Using Dancing Links Matthew A. Wolff Morehead State University May 19, 2006.

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