Weak ε-nets and interval chains

Abstract

Listen

Translate article

We construct weak e-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 ℝd we construct weak 1/r-nets of size r · 2poly(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 p is said to stab an interval chain C = I1 … Ik if each pi 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 zk(j)(n) denote the minimum size of such a family Z. We derive almost-tight upper and lower bounds for zk(j)(n) for every fixed j; our bounds involve functions αm(n) of the inverse Ackermann hierarchy. Specifically, we show that for j = 3 we have zk(3)(n) = Θ for all k ≥ 6. For each j ≥ 4 we construct a pair of functions P'j(m), Q'j(m), almost equal asymptotically, such that z(j)P'j(m)(n)=O(nαm(n)) and z(j)Q'j(m)(n)=Ω(nαm(n)).