Weak ε-nets and interval chains

Abstract

We construct weak ε-nets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1/r-nets of size O(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in ℝ d we construct weak 1/r-nets of size r · 2 poly(α(r)) , where the degree of the polynomial in the exponent depends (quadratically) on d. Our constructions result from a reduction to a new problem, which we call stabbing interval chains with j-tuples. Given the range of integers N = [1, n], an interval chain of length k is a sequence of k consecutive, disjoint, nonempty intervals contained in N. A j-tuple $\bar{P}$ = (p1,…,pj) is said to stab an interval chain C = I 1 …I k if each p i falls on a different interval of C. The problem is to construct a small-size family Z of j-tuples that stabs all k-interval chains in N. Let z (j) k (n) denote the minimum size of such a family Z. We derive almost-tight upper and lower bounds for z (j) k (n) for every fixed j; our bounds involve functions α m (n) of the inverse Ackermann hierarchy. Specifically, we show that for j = 3 we have z (3) k (n) = Θ(nα $\lfloor$k/2$\rfloor$ (n)) for all k ≥ 6. For each j≥4, we construct a pair of functions Pʹ j (m), Qʹ j (m), almost equal asymptotically, such that z (j) Pʹ j(m)(n) = O(nα m (n)) and z (j) Qʹ j(m)(n) = Ω(nα m (n)).