Permutations of bounded degree generate groups of polynomial diameter
Information Processing Letters
The complexity of finding minimum-length generator sequences
Theoretical Computer Science
Computing short generator sequences
Information and Computation
On the diameter of Cayley graphs of the symmetric group
Journal of Combinatorial Theory Series A
The probability of generating the symmetric group
Journal of Combinatorial Theory Series A
On the diameter of permutation groups
European Journal of Combinatorics
Computation with permutation groups
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
Finding optimal solutions to Rubik's cube using pattern databases
AAAI'97/IAAI'97 Proceedings of the fourteenth national conference on artificial intelligence and ninth conference on Innovative applications of artificial intelligence
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On the diameter of Eulerian orientations of graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
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We address the long-standing conjecture that all permutations have polynomially bounded word length in terms of any set of generators of the symmetric group. The best available bound on the maximum required word length is exponential in n log n. Polynomial bounds on the word length have previously been established for very special classes of generating sets only.In this paper we give a polynomial bound on the word length under the sole condition that one of the generators fix at least 67% of the domain. Words of the length claimed can be found in Las Vegas polynomial time.The proof involves a Markov chain mixing estimate which permits us, apparently for the first time, to break the "element order bottleneck."As a corollary, we obtain the following average-case result: for a 1 -- δ fraction of the pairs of generators for the symmetric group, the word length is polynomially bounded. It is known that for almost all pairs of generators, the word length is less than exp(√n ln n(1 + o(1))).