Density-Based clustering in mapReduce with guarantees on parallel time, space, and solution quality

A well-known clustering problem called Density-Based Spatial Clustering of Applications with Noise (DBSCAN) involves computing the solutions of at least one disk range query per input point, computing the connected components of a graph, and bichromatic fixed-radius nearest neighbor. MapReduce class is a model where a sublinear number of machines, each with sublinear memory, run for a polylogarithmic number of parallel rounds.  Most of these problems either require quadratic time in the sequential model or are hard to compute in a constant number of rounds in MapReduce. In the Euclidean plane, DBSCAN algorithms with near-linear time and a randomized parallel algorithm with a polylogarithmic number of rounds exist.  We solve DBSCAN in the Euclidean plane in a constant number of rounds in MapReduce, assuming the minimum number of points in range queries is constant and each connected component fits inside the memory of a single machine and has a constant diameter.