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Solving N+k 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 N+1 Programs Future Work

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Motivation NASA EPSCoR grant (Subcontract # WKURF ) 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 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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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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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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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? 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 (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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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(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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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 (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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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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Questions? Thank you! Dr. Chatham Dr. Doyle Mr. Skaggs

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References

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