Distributed Uniformity Testing

In the uniformity testing problem, we are given access to samples from some unknown distribution μ on a fixed domain \set1,..,n , and our goal is to distinguish the case where μ is the uniform distribution from the case where μ is ε-far from uniform in L_1 distance. Centralized uniformity testing has been extensively studied, and it is known that Θ(√ /ε^2) samples are necessary and sufficient. In this paper we study distributed uniformity testing : in a network of k nodes, each node i has access to s_i samples from the underlying distribution μ. Our goal is to test uniformity, while minimizing the number of samples per node, as well as the running time. We consider several distributed models: the ŁOCAL model, the \CONGEST model, and a 0-round model where nodes cannot communicate with each other at all. We give upper bounds for each model, and a lower bound for the 0-round model. The key to our results is analyzing the centralized uniformity-testing problem in an unusual error regime, for which we give new upper and lower bounds.