Random walks and forbidden minors III: poly(d epsilon(-1))-time partition oracles for minor-free graph classes

Consider the family of bounded degree graphs in any minor-closed family (such as planar graphs). Let d be the degree bound and n be the number of vertices of such a graph. Graphs in these classes have hyperfinite decompositions, where, one removes a small fraction of edges of the graph controlled by a proximity parameter to get connected components of size independent of n. An important tool for sublinear algorithms and property testing for such classes is the partition oracle, introduced by the seminal work of Hassidim-Kelner-Nguyen-Onak (FOCS 2009). A partition oracle is a local procedure that gives consistent access to a hyperfinite decomposition, without any preprocessing. Given a query vertex v, the partition oracle outputs the component containing v in time independent of n. All the answers are consistent with a single hyperfinite decomposition.