The number of moves required to solve any state of Rubik's cube has been a matter of long-standing conjecture for over 25 years -since Rubik's cube appeared. This number is sometimes called "God's number". An upper bound of 29 (in the face-turn metric) was produced in the early 1990's, followed by an upper bound of 27 in 2006.An improved upper bound of 26 is produced using 8000 CPU hours. One key to this result is a new, fast multiplication in the mathematical group of Rubik's cube. Another key is efficient outof-core (disk-based) parallel computation using terabytes of disk storage. One can use the precomputed data structures to produce such solutions for a specific Rubik's cube position in a fraction of a second. Work in progress will use the new "brute-forcing" technique to further reduce the bound.

This paper presents some new algorithms to efficiently mine max frequent generalized itemsets (g-itemsets) and essential generalized association rules (g-rules). These are compact and general representations for all frequent patterns and all strong association rules in the generalized environment. Our results fill an important gap among algorithms for frequent patterns and association rules by combining two concepts. First, generalized itemsets employ a taxonomy of items, rather than a flat list of items. This produces more natural frequent itemsets and associations such as (meat, milk) instead of (beef, milk), (chicken, milk), etc. Second, compact representations of frequent itemsets and strong rules, whose result size is exponentially smaller, can solve a standard dilemma in mining patterns: with small threshold values for support and confidence, the user is overwhelmed by the extraordinary number of identified patterns and associations; but with large threshold values, some interesting patterns and associations fail to be identified.Our algorithms can also expand those max frequent g-itemsets and essential g-rules into the much larger set of ordinary frequent g-itemsets and strong g-rules. While that expansion is not recommended in most practical cases, we do so in order to present a comparison with existing algorithms that only handle ordinary frequent g-itemsets. In this case, the new algorithm is shown to be thousands, and in some cases millions, of the time faster than previous algorithms. Further, the new algorithm succeeds in analyzing deeper taxonomies, with the depths of seven or more. Experimental results for previous algorithms limited themselves to taxonomies with depth at most three or four.In each of the two problems, a straightforward lattice-based approach is briefly discussed and then a classificationbased algorithm is developed. In particular, the two classification-based algorithms are MFGI class for mining max frequent g-itemsets and EGR class for mining essential g-rules. The classification-based algorithms are featured with conceptual classification trees and dynamic generation and pruning algorithms.

Binary Decision Diagrams (BDDs) are widely used in formal verification. They are also widely known for consuming large amounts of memory. For larger problems, a BDD computation will often start thrashing due to lack of memory within minutes. This work uses the parallel disks of a cluster or a SAN (storage area network) as an extension of RAM, in order to efficiently compute with BDDs that are orders of magnitude larger than what is available on a typical computer. The use of parallel disks overcomes the bandwidth problem of single disk methods, since the bandwidth of 50 disks is similar to the bandwidth of a single RAM subsystem. In order to overcome the latency issues of disk, the Roomy library is used for the sake of its latency-tolerant data structures. A breadth-first algorithm is implemented. A further advantage of the algorithm is that RAM usage can be very modest, since its largest use is as buffers for open files. The success of the method is demonstrated by solving the 16-queens problem, and by solving a more unusual problem -counting the number of tie games in a three-dimensional 4×4×4 tic-tac-toe board.

Substituting disk for RAM, disk-based computation is a way to increase working memory and achieve results that are not otherwise economical.