Sublinear algorithms for testing monotone and unimodal distributions

Abstract

The complexity of testing properties of monotone and unimodal distributions, when given access only to samples of the distribution, is investigated. Two kinds of sublinear-time algorithms - those for testing monotonicity and those that take advantage of monotonicity - are provided. The first algorithm tests if a given distribution on [n] is monotone or far away from any monotone distribution in L1-norm; this algorithm uses Õ(√n) samples and is shown to be nearly optimal. The next algorithm, given a joint distribution on [n] × [n], tests if it is monotone or is far away from any monotone distribution in L1-norm; this algorithm uses Õ(n3/2) samples. The problems of testing if two monotone distributions are close in L1-norm and if two random variables with a monotone joint distribution are close to being independent in L 1-norm are also considered. Algorithms for these problems that use only poly(log n) samples are presented. The closeness and independence testing algorithms for monotone distributions are significantly more efficient than the corresponding algorithms as well as the lower bounds for arbitrary distributions. Some of the abovs results are also extended to unimodal distributions.