STORY%MIT-MC@sri-unix.UUCP (04/18/84)
From: Kenneth Byrd Story <STORY @ MIT-MC> [Forwarded from the MIT bboard by Laws@SRI-AI.] DATE: Thursday, April 19, 1984 TIME: Lecture, 4:00pm PLACE: NE43-512a ``Generalized `15-puzzles' and the Diameter of Permutation Groups'' Dan Kornhauser MIT Sam Lloyd's famous ``15-puzzle'' involves 15 numbered unit squares free to move in a 4x4 area with one unit square blank. The problem is to decide whether a given rearrangement of the squares is possible, and to find the shortest sequence of moves to obtain the rearrangement when it is possible. A natural generalization of this puzzle involves a graph with @i(n) vertices, and @i(k<n) tokens numbered @i(1,...,k) on distinct vertices. A legal move consists of sliding a token from its vertex to an adjacent unoccupied vertex. Wilson (1974) obtained a criterion for solvability for biconnected graphs and @i(k=n-1). No polynomial upper bound on number of moves was given. We present a quadratic time algorithm for deciding solvability of the general graph problem. It is also shown that @i[O(n@+{3})] move solutions always exist and can be efficiently planned. Further, @i[O(n@+{3})] is shown to be a matching lower bound for some graph puzzles. We consider related puzzles of the Rubik's cube type, in the context of the general permutation group diameter question. This is joint work with Gary Miller, MIT, and Paul Spirakis, NYU HOST: Professor Silvio Micali