Appendix B Korf’s Optimal Solution to Rubik’s Cube Puzzle and Pattern Databases In this appendix we look at admissible heuristic functions for the Rubik’s cube puzzle and the 15 and 24 tile puzzle based on pattern databases. A pattern database is a heuristic function stored as a lookup table which stores the lengths of optimal solutions for instances of subproblems. The results are from papers by Korf. B.1 Rubik Cube If we consider Rubik’s cube, it can be categorised using a state of 20 elements. This consists of representations of the orientations of the sub–cubes which move. There are 8 corners and 12 edges. Each corner has 3 orientations and each edge 2 orientations. Thus a state is a 20 element vector. Typical a human solving the puzzle takes about 50 – 100 moves. This is done by using learnt algorithms for manipulating pieces leaving others static (e. g. turning an edge cube over while leaving others fixed). There are 8! permutations of the 8 corners and 12! permutations of the 12 edges. Each corner can have 3 orientations and each edge 2, so altogether the total number of states is 8! × 38 × 12! × 212 . But these permutations are split into 12 mutual classes, so that one can only transform between states in the same class. So from one state there are 8! × 38 × 12! × 212 /12 states reachable. This is 4.3 × 1019 . Compared with the 15 puzzle – 1013 states. (16!/2). Take as basic moves 90◦ clockwise, 90◦ counter–clockwise and 180◦ in each of the six planes of the Rubik cube. We consider the centre of the cube as fixed in space and so don’t rotate the three central planes – this determines the colours of the faces for the goal state. The search tree then has 18 (6) way branching initially. Then 15 way branching since rotating the same face consecutively results in a state which appears elsewhere in the tree. However, twisting opposite faces commutes so we can reduce the branching further. It can be shown that the average branching factor is 13.3. xi Depth Nodes 1 18 2 243 3 3,240 4 43,254 5 577,368 6 7,706,988 7 102,876,480 8 1,373,243,544 9 18,330,699,168 10 244,686,773,808 11 3,266,193,870,720 12 43,598,688,377,184 13 581,975,750,199,168 14 7,768,485,393,179,328 15 103,697,388,221,736,960 16 1,384,201,395,738,071,424 17 18,476,969,736,848,122,368 18 246,639,261,965,462,754,048 Table B.1: Nodes in Search Tree From table B.1 we see that there must be some problems requiring at least 18 moves (could be much bigger), since number of reachable states using 17 moves is still less than the total number of states possible. Korf used variant of manhattan distance – distance is min number of moves to correct position and orientation. To be admissible we divide by 8 as 1 basic move moves 4 other corners and 4 edges. He actually used IDA* – the variant of A* called iterative deepening A*. It is also possible to take a simpler heuristic – just use consider the edge cubes only and divided by 4. In general, the basic Manhattan distance gives too low a value and so in general there is too much computation using this directly even if the values are precomputed. Computing IDA* tree to a depth 14 would take 3 days , to 18 250 years. (on a Sparc Ultra). A modified admissible heuristic was evaluated as follows: h(n) = max{hcorners (n), he1 (n), he2 (n)} where hcorners computes the minimum number of moves to ensure the all 8 corners are in the correct position Figure B.1. This is precomputed using a breadth-first search from the goal state. The last cube is determined by the others, and so there are 8!37 possibilities. Each value can be stored using 4 bits since the maximum value is 11. 42M is needed for this table. xii Figure B.1: Corner Database (a) Edge Set1 (b) Edge Set2 Figure B.2: Two Edge Sets The 12 edges are split into two groups of 6 Figure B.2. he1 then evaluates the minimum numbers of moves to position the first set of edges cubes and he2 for the second set. The number of possible combinations is 12!/6! · 26 . The maximum number of moves to orient 6 edges is 10 and so 4-bits can be used per position. Hence 20M is required for each table. So altogether 82M is required for all tables. The stored table is also called a pattern database. It took about 1 hour to compute the table. This table needs to be stored in RAM for the IDA* algorithm, but this is now possible with a reasonable workstation. Korf performed experiments on states produce by making 100 random basic moves. The table B.2 shows the statistics for 10 such cases. All had solutions within 18 moves and 1 only 16 moves. It is thought that 20 moves is sufficient for all start states, but this has not been proved. So by careful choice of a heuristic it is possible to tackle these large combinatorial problems where brute search would clearly be useless. xiii No. Depth Nodes Generated 1 16 3,720,885,493 2 17 11,485,155,726 3 17 64,837,508,623 4 17 126,005,368,381 5 18 262,228,269,081 6 18 344,770,394,346 7 18 502,417,601,953 8 18 562,494,969,937 9 18 626,785,460,346 10 18 1,021,814,815,051 Table B.2: Table of Results B.2 15 and 24 Tile Puzzle Pattern databases for the 15 puzzle were first introduced by Culberson and Schaeffer. One example of splitting the 15 puzzle into subproblems is indicated in Figure B.3. Figure B.3: 15 Puzzle Pattern Database The 8 blue (light grey) tiles are in one set and the 7 red (dark grey) in the other. The two databases are computed (offline) using a backward breadth first search from the goal state. The non-pattern tiles are not distinguished but their moves are counted. The heuristic for the 15 puzzle is then taken as the maximum of these two heuristics. So in the above example, 31 moves can solve the red tiles and 22 the blue. So the heuristic value for the configuration on the left is taken as 31. This is then clearly an admissible heuristic since both sets of tiles have to be put into their final positions. Other divisions of the tiles is possible as alternative for the pattern databases (they could even overlap). Figure B.4 shows another possible split. xiv Figure B.4: Alternative 15 Puzzle Pattern Database Korf changed the way he counted the moves so that the moves of non-pattern tiles are ignored. As long as the subsets don’t overlap, then instead of maximising the two heuristic values they can be added as any one tile will only belong to one set. Taking the configuration in Figure B.5, 20 moves are needed to position the red tiles and 25 moves for the blue. Hence the heuristic value for the configuration will be taken as 20+25 = 45 which will be a lower bound on the minimal number of moves to solve the puzzle and so is admissible. The 7-tile database has about 58 million entries and the 8-tiles database about 519 million entries. Figure B.5: Configuration with Goal xv Korf used an additive pattern database for the 24 puzzle based on the subsets indicated in Figure B.6. All subsets have six tiles and the pattern databases contain about 128 million entries. IDA* can solve optimally random 24-puzzle configurations using this heuristic (sum of the four heuristics on the subsets) in about 100 moves. The running time in his experiments were 18 seconds to 10 days with an average of about 1.5 days. Figure B.6: 24 Puzzle Subsets xvi