510 likes | 1.12k Views
Agenda. MotivationDefinitionsProblem DefinitionSolved Problems with ResultsFuture Work. Motivation. NASA EPSCoR grantBegan working with Chatham and Skaggs in NovemberDoyle added DLX (Dancing Links) at beginning of semesterNew to me (and the rest of the team, I think)A lot more work!. Category of Problems.
E N D
1. Solving N+k Queens Using Dancing Links Matthew Wolff
CS 499c
May 3, 2006
2. Agenda Motivation
Definitions
Problem Definition
Solved Problems with Results
Future Work
3. 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!
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. 8 Queens Example http://www.jsomers.com/nqueen_demo/nqueens.html
6. 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.
7. Fundamental Solutions for 8 Queens
8. 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
9. Recursion in Art
10. Recursion in Computer Science // precondition: n >= 0// postcondition: n! is returnedfactorial (int n) { if (n == 1) or (n == 0) return 1; else return (n*factorial(n-1));}
11. 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)); }}
12. Backtracking Example Output: A – B – D – E – H – I – C – F - G
13. 4 Queens Backtracking Example Solved by iterating over all solutions, using backtracking
14. N Queens Extend to N board
Similar to 8 Queens
Use a more general board of size NxN
Same algorithm as 8 Queens
15. 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
16. 8x8 Board, 1 Pawn
17. 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
18. Chatham, Fricke, Skaggs Proved N+k queens can be placed on an NxN board with k pawns.
19. 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
20. 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
21. N+1 Results
22. Python versus C++
23. 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
24. Pawn Placements for Parallel N+1 Queens Solution
25. N+1 Queens, Parallel vs. Sequential C++
26. 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
27. 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
28. “The Universe” Multi-Dimensional structure composed of circular, doubly linked-lists
Each row and column is a circular, doubly linked-list
29. Visualization of “The Universe”
30. 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.
31. 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
32. 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
33. Mapping the Chess Board
34. The Amazing TechniColor Chess Board
35. 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
36. 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}
37. 1x1 Universe: Before
38. 1x1 Universe: After
39. 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 ?
As it turns out, the way the universe is built is the only needed modification to go from N Queens to N+k Queens
40. 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?
41. 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
42. Example of Rank “Scan”
43. And now for the moment you’ve all been waiting for! DRUM ROLL!….
GRAPHS AND STUFF!
44. N+1 QueensVarying Language, Algorithm
45. N+1 Queens Parallel Backtracking vs. DLX
46. N+1 QueensSequential DLX vs. Parallel DLX
47. Interesting Tidbit:Sequential DLX vs. Parallel C++
48. N+k Results Still running in Lappin 241L…
Maybe next week ?
49. 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.
50. 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.
51. Questions? Thank you!
Dr. Chatham
Dr. Doyle
Mr. Skaggs
52. References