On the diameter of permutation groups

We show that any group represented by generators that are cycles of bounded degree has <italic>O(n</italic><supscrpt>2</supscrpt>) diameter, i.e., that the longest product of generators required to reach any permutation in the group is <italic>O(n</italic><supscrpt>2</supscrpt>). We also show how such “short” products can be found in polynomial time. The techniques presented are applicable to generalizations of many permutation-group puzzles such as Alexander's Star and the Hungarian Rings.