Random Graph Generation in Hyperedge Replacement Languages

We present a novel approach for the random generation of graphs in context-free hypergraph languages. It is obtained by adapting both of Mairson's generation algorithms for context-free string grammars to the setting of hyperedge replacement grammars. It provides a concrete instrument for the generation of unbiased graph data where a solid mathematical proof is required for the validation of procedures, filling an important gap in the field of random testing. Our main result is that for non-ambiguous hyperedge replacement grammars, the method is guaranteed to efficiently generate hypergraphs uniformly at random in user-specified domains. It means that testing for the sought properties in the generated graph is no longer required since they are directly inferred by the grammar. The efficiency of the method is ensured by the proofs of polynomial time and space asymptotic behaviors. Our secondary result is that it greatly extends the range of either context-free and non-context-free string languages to sample from with a uniform distribution through the use of string graph grammars. We prove how ambiguous string grammars can be expressed with equivalent non-ambiguous hyperedge replacement grammars, overcoming the current limitation for the achievement of the uniformity of the sampling. Our contribution also proposes several case studies of relevant hyperedge replacement languages proving the existence of a representative grammar for a uniform sampling, or, otherwise, their inherent ambiguity by the analysis of particular structures. These languages form a basis for a proof by reduction for more complex languages, greatly improving the possible search for non-ambiguous solutions.