RNC algorithms for the uniform generation of combinatorial structures

We describe several RNC algorithms for generating graphs and subgraphs uniformly at random. For example, unla-belled undirected graphs are generated in O(lg³ n lg lg n) time using O (ϵn^1.5/lg³n lg lg n) processors if their number is known in advance and in O(lg n) time using O(ϵn²/lg n) processors otherwise. In both cases the error probability is the inverse of a polynomial in ϵ. Thus ϵ may be chosen to trade-off processors for error probability. Also, for an arbitrary graph, we describe RNC algorithms for the uniform generation of its subgraphs that are either non-simple paths or spanning trees. The work measure for the subgraph algorithms is essentially determined by the transitive closure bottleneck. As for sequential algorithms, the general notion of constructing generators from counters also applies to parallel algorithms although this approach is not employed by all the algorithms of this paper.