Let 0 < α≤ 1/2. We say that a finite point set P in ℝ^d is α-split by a hyperplane h if each of the closed half-spaces determined by h, contains at least α|P| of the points of P. We further say P is α-split by a k-dimensional flat τ if P is α-split by any hyperplane through τ. In the standard notation (which coincides with Tukey depth for k= 0), the k-flat τ has depth α with respect to P. We establish interesting Helly-type theorems for splitting families of finite point sets in ℝ^d. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of k-flats of arbitrary dimensionality 0 ≤ k ≤ d-1.
We prove new upper and lower bounds for the Online Orthogonal Vectors Problem (𝖮𝗇𝗅𝗂𝗇𝖾𝖮𝖵_n,d). In this problem, a preprocessing algorithm receives n vectors x_1,…,x_n∈{0,1}^d and constructs a data structure of size S. A query algorithm subsequently receives a query vector q∈{0,1}^d and in time T decides whether q is orthogonal to any of the input vectors x_i. We design a new deterministic data structure for 𝖮𝗇𝗅𝗂𝗇𝖾𝖮𝖵_n,d. In low dimensions (d = c log n), our data structure matches the performance of the best known randomized algorithm due to Chan [SoCG 2017]. Furthermore, in moderate dimensions (d=n^ε), we give the first improvement since Charikar, Indyk and Panigrahy [ICALP 2002]. Along the way, we give the first deterministic refutation of a conjecture on the hardness of 𝖮𝗇𝗅𝗂𝗇𝖾𝖮𝖵 posed by Goldstein, Lewenstein and Porat [ISAAC 2017]. This data structure also extends to a number of problems, including Partial Match, Orthogonal Range Search, and DNF Evaluation. We use a novel structure-versus-randomness decomposition to design our algorithm. Under the Non-Uniform Strong Exponential Time Hypothesis, we also prove arbitrarily large polynomial space lower bounds for any 𝖮𝗇𝗅𝗂𝗇𝖾𝖮𝖵 data structure with sublinear query time even with computationally unbounded preprocessing. These lower bounds extend to several other problems, including Polynomial Evaluation, Partial Match, Orthogonal Range Search, and Approximate Nearest Neighbors. We also prove similar lower bounds for 3-𝖲𝖴𝖬 with preprocessing under the Non-Uniform Hamiltonian Path Conjecture.
