Distribution-free connectivity testing for sparse graphs
We consider distribution-free property-testing of graph connectivity. In this setting of property testing, the distance between functions is measured with respect to a fixed but unknown distribution D on the domain, and the testing algorithm has an oracle access to random sampling from the domain according to this distribution D. This notion of distribution-free testing was previously defined, and testers were shown for very few properties. However, no distribution-free property testing algorithm was known for any graph property. We present the first distribution-free testing algorithms for one of the central properties in this area-graph connectivity (specifically, the problem is mainly interesting in the case of sparse graphs). We introduce three testing models for sparse graphs: A model for bounded-degree graphs, A model for graphs with a bound on the total number of edges (both models were already considered in the context of uniform distribution testing), and A model which is a combination of the two previous testing models; i.e., bounded-degree graphs with a bound on the total number of edges.
connectivity testing property testing distribution-free testing properties of sparse graphs