Local expansion of vertex-Transitive graphs and random generation in finite groups

Abstract

Heuristic algorithms manipulating finite groups often work under the assumption that certain operations lead to "random" elements of the group. While polynomial time methods to construct uniform random elements of permutation groups have been known for over two decades, no such methods have been known for more general cases such as matrix groups over finite fields. We present a Monte Carlo algorithm which constructs an efficient nearly uniform random generator for finite groups G in a very general setting. The algorithm presumes a priori knowledge of an upper bound n on log IGI. The random generator is constructed and works in time, polynomial in this upper bound n. The process admits high degree of parallelization: After a preprocessing of length O(n log n) with 0(n4) processors, the construction of each random element costs O(log n) time with O(n) processors. We use the computational model of "black box groups": group elements are encoded as (O, I)-strings of uniform length; and an oracle performs group operations at unit cost. The group G is given by a list of generators. The random generator will produce each group element with probability (1/lGl)(l + c) where c can be prescribed to be an arbitrary exponentially small function of n. The resr.dt is surprising because there does not seem to be any hope to estimate the order of a matrix group in polynomial time. A number of previous results have indicated close connection between nearly uniform random generation and approximate counting. The proof involves elementary combinatorial considerations for finite groups as well aa linear algebra and probabilistic techniques to analyse random walks over vertex-Transitive graphs, i.e. graphs with all vertices "alike" (equivalent under the action of the automorphism group). The key tool is a local expansion lemma for groups, which generalizes to vertex-Transitive graphs. As a by-product, we obtain fairly general results on random walks on vertex-Transitive graphs which may be of interest in their own right.