Animal testing A configuration of unit cubes in three dimensions with integer coordinates is called an animal if the boundary of their union is homeomorphic to a sphere. Shermer discovered several animals from which no single cube may be removed such that the resulting configurations are also animals 14. Here we obtain a dual result: we give an example of an animal to which no cube may be added within its minimal bounding box such that the resulting configuration is also an animal. We also present two O(n)time algorithms for determining whether a given configuration of n unit cubes is an animal.

Monotone probability distributions over the Boolean cube can be learned with sublinear samples A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we give (to our knowledge) the first algorithm that learns an approximation of the distribution in statistical distance using a number of samples that is sublinear in the domain.  :68,do this, we develop a structural lemma describing monotone probability distributions. The structural lemma has further implications to the sample complexity of basic testing tasks for analyzing monotone probability distributions over the Boolean cube: We use it to give nontrivial upper bounds on the tasks of estimating the distance of a monotone distribution to uniform and of estimating the support size of a monotone distribution. In the setting of monotone probability distributions over the Boolean cube, our algorithms are the first to have sample complexity lower than known lower bounds for the same testing tasks on arbitrary (not necessarily monotone) probability distributions.  further :171,consequence of our learning algorithm is an improved sample complexity for the task of testing whether a distribution on the Boolean cube is monotone.