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

2. Agenda  Motivation  Terms  Problem Definition  Timing Results for various N+1 Programs  Future Work

3. 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

4. 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

5. 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

6. Recursion in Art

7. 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)); }

8. 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)); } }

9. Backtracking Example Output: A – B – D – E – H – I – C – F - G

10. 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

11. Chatham, Fricke, Skaggs  Proved N+k queens can be placed on an NxN board with k pawns.

12. 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

13. 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

14. “The Universe”  Multi-Dimensional structure composed of circular, doubly linked-lists  Each row and column is a circular, doubly linked-list

15. Visualization of “The Universe”

16. 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.

17. 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

18. 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

19. Mapping the Chess Board

20. The Amazing TechniColor Chess Board

21. 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

22. 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}

23. 1x1 Universe: Before

24. 1x1 Universe: After

25. 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?

26. 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

27. Example of Rank “Scan”

28. N+1 Queens Varying Language, Algorithm

29. N+1 Queens Parallel Backtracking vs. DLX

30. N+1 Queens Sequential DLX vs. Parallel DLX

31. Interesting Tidbit: Sequential DLX vs. Parallel C++

32. Questions?  Thank you!  Dr. Chatham  Dr. Doyle  Mr. Skaggs

33. References