New lower bounds for Hopcroft's problem

Abstract

We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in ℝd, is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time Ω(n log m + n2/3m2/3 +m log n) in two dimensions, or Ω(n log m+n5/6 m1/2 +n1/2m5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log*(n+m)) of the best known upper bound, due to Matoušek. Previously, the best known lower bound, in any dimension, was Ω(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive lower bounds on its size in the worst case. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover. As a related result, using a straightforward adversary argument, we derive a quadratic lower bound on the complexity of Hopcroft's problem in a surprisingly powerful decision tree model of computation.