47 Epsilon - Approximations & Epsilon - Nets

The use of random samples to approximate properties of geometric configurations has been an influential idea for both combinatorial and algorithmic purposes. This chapter considers two related notions— -approximations and -nets—that capture the most important quantitative properties that one would expect from a random sample with respect to an underlying geometric configuration. An example problem: given a set P of points in the plane and a parameter > 0, is it possible to choose a set N of O( 1 ) points of P such that N contains at least one point from each disk containing |P | points of P? More generally, what is the smallest non-empty set A ⊆ P that can be chosen such that for any disk D in the plane, the proportion of points of P contained in D is within to the proportion of points of A contained in D? In both these cases, a random sample provides an answer ‘in expectation;’ establishing worst-case guarantees is the topic of this chapter.