1. Finding Optimal Solutions to Rubik's Cube Using Pattern Databases
Richard E. Korf
Computer Science
UCLA
Presented By
Christopher Brown

2. The Cube Erno Rubik of Hungary, 1974. Cubies:
Edges = 12 w/ 2 orientations each
Corners = 8 w/ 3 orientations each
12 different “universes” of cubes

3. The Cube 43,252,003,274,489,856,000 Possible Cubes!! Cubies:
Edges = 12 w/ 2 orientations each
Corners = 8 w/ 3 orientations each
12! * 2^12 * 8! * 3^8 = 519,024,039,293,878,272,000
/ 12 =
43,252,003,274,489,856,000
43 Quintillion

4. How Many Moves?
At Depth 17, the total number of nodes is not even 20 Quintillion..there must be at least some at depth 18.
Superflip

5. Heuristics Expected Values Manhattan Distance Problem?
Expected value is an average – gives a good approximation, since otherwise you could only compare similar heuristics and hard to combine – not often feasible
Calculated through random sampling
Problem w/ Manhattan Distance
1 Move = 8 changes
Divide total by 8 to stay admissible.

6. Heuristics Pattern Database Corners Edges Major Restriction: Memory
Problem w/ Manhattan Distance
1 Move = 8 changes
Divide total by 8 to stay admissible.
Expected value = 5.5
Corners = Expected value 8.764
4 bits / entry
Table: 42 MB
6 Edges = 7.668
Table: 20 MB
7 Edges = ? 244 MB of memory

7. Experiment Max of 3 heuristics admissible 10 Random Cubes
100 move random walks to generate
700,000 nodes / second
Conjecture that the diameter of the space is 20 moves. (superflip)
100 moves is reasonable

8. Experiment Sun Ultra Sparc 1 workstation # nodes stable
116 – 127 for depth 17
complete to depth 16 = 9.5 billion
complete to depth 17 = 122 billion
2 days
complete depth 18 – 4 weeks?

9. Results Performance e – expected value d – depth of search
IDA* should be similar to brute force of d – e
Is this true?
At depth 17
brute force for depth (17 – 9 = 8) -> 1.37 billion nodes
Experimentally: IDA * at depth 17 -> 122 billion nodes
What happened?
Experimentally, the search ignores larger heuristic values, and focuses on smaller ones, so the average of encountered nodes is below the expected value.
But, estimating expected value from branching factor is pessimistic.
Cancel?
n - states
m – # of heuristic values in memory
t – running time o fIDA*
d – depth ≈ logb n
e – ex. val ≈ logb m
t ≈ bd-e ≈ blog n-log m ≈ n / m – estimates well

10. Other Research Fiat (1989) - Bidirectional search
O(bd/2)time, O(bd/4) space
10,000 vs 700,000 nodes/sec
500 bytes vs ½ byte per permutation
Not faster with more memory
t ≈ n/m, then O(n1/2) time takes O(n1/2) space
build time?
One-time cost

11. Finding Optimal Solutions to Rubik's Cube Using Pattern Databases
First Optimal Solver
Expected
Heuristic
Values
Pattern
Databases
Christopher Brown
Korf, Richard E. “Finding Optimal Solutions to Rubik's Cube Using Pattern Databases” Proceedings of the Fourteenth National Conference on Artificial Intelligence (AAAI-97), Providence, RI, July, 1997, pp Reprinted in Games in AI Research, Jaap ven den Herik and Hiroyuki Iida, Universiteit Maastricht, 1999, pp