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 degree of transitivity of permutation groups: A short proof
Journal of Combinatorial Theory Series A
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
Randomized algorithms
On the diameter of the symmetric group: polynomial bounds
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
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
On the diameter of Eulerian orientations of graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Hi-index | 0.00 |
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 Sn This is equivalent to polynomial-time (O(nc)) mixing of the (lazy) random walk on Sn where one step is multiplication by a generator or its inverse.We prove that the conjecture is true for almost all pairs of generators. Specifically, our bound is Õ(n7). For almost all pairs of generators, words of this length representing any given permutation can be constructed in Las Vegas polynomial time. The best previous bound on the word length for a random pair of generators was nInn(1/2+o(1)) (Babai-Hetyei, 1992).We build on recent major progress by Babai-Beals-Seress (SODA, 2004), confirming the conjecture under the assumption that at least one of the generators has degree n.The main technical contribution of the present paper is the following near-independence result for permutations. The first cycle of a permutation is the trajectory of the first element of the permutation domain. For a random permutation, the distribution of the length of the first cycle is uniform. We show that if τ ∈ Sn is a given permutation of degree ≥ n3/4 and σ ∈ Sn is chosen at random, then the distributions of the length of the first cycle of σ and the length of the first cycle in στ are nearly independent. The ability of an essentially arbitrarily fixed permutation (τ) to "scramble" another permutation in this technical sense may be of independent interest and suggests new directions in the statistical theory of permutations pioneered by Goncharov and Erdős-Turán.