A trace bound for the hereditary discrepancy

Abstract

Let A be the incidence matrix of a set system with m sets and n points, m ≤ n, and let t = tr M, where M = AT A. Finally, let σ = tr M2 be the sum of squares of the elements of M. We prove that the hereditary discrepancy of the set system is at least 1/4 cnσ/t2 √t/n, with c = 1/324. This general trace bound allows us to resolve discrepancy-type questions for which spectral methods had previously failed. Also, by using this result in conjunction with the spectral lemma for linear circuits, we derive new complexity bounds for range searching. • We show that the (red-blue) discrepancy of the set system formed by n points and n lines in the plane is Ω(n1/6) in the worst case and always1 Õ(n1/6). • We give a simple explicit construction of n points and n halfplanes with hereditary discrepancy Ω̃(n1/4). • We show that in any dimension d = Ω(log n/log log n), there is a set system of n points and n axis-parallel boxes in Rd with discrepancy nΩ(1/log log n). • Applying these discrepancy results together with a new variation of the spectral lemma, we derive a lower bound of Ω(n log n) on the arithmetic complexity of off-line range searching for points and lines (for nonmonotone circuits). We also prove a lower bound of Ω(n log n/log log n) on the complexity of orthogonal range searching in any dimension Ω(log n/log log n).