Fast Distributed Algorithms for Testing Graph Properties

We initiate a thorough study of \emph{distributed property testing} -- producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called \emph{dense} testing model we emulate sequential tests for nearly all graph properties having $1$-sided tests, while in the \emph{sparse} and \emph{general} models we obtain faster tests for triangle-freeness and bipartiteness respectively. We then provide a clustering construction with applications. In most cases, aided by parallelism, the distributed algorithms have a much shorter running time as compared to their counterparts from the sequential querying model of traditional property testing. The simplest property testing algorithms allow a relatively smooth transitioning to the distributed model. For the more complex tasks we develop new machinery that is of independent interest. This includes a method for distributed maintenance of multiple random walks, and a comprehensive clustering framework. Our clustering method allows us to construct a forest that has a large portion of the edges of a spanning tree of a connected graph, as well as a variant with weights where the forest is a subgraph of a minimum spanning tree (MST). The forest construction implies an efficient test for connectivity, and the variant producing a subgraph of an MST allows us to construct a \emph{lightweight sparse spanning subgraph} in a number of rounds that overcomes the lower bound for approximating MST by a tree.