Polynomial Data Structure Lower Bounds in the Group Model

Abstract

Proving super-logarithmic data structure lower bounds in the static group model has been a fundamental challenge in computational geometry since the early 80's. We prove a polynomial $(n^{\Omega(1)})$ lower bound for an explicit range counting problem of $n^{3}$ convex polygons in $\mathbb{R}^{2}$ (each with $n^{\tilde{O}(1)}$ facets/semialgebraic-complexity), against linear storage arithmetic data structures in the group model. Our construction and analysis are based on a combination of techniques in Diophantine approximation, pseudorandomness, and compressed sensing—in particular, on the existence and partial derandomization of optimal binary compressed sensing matrices in the polynomial sparsity regime $(k=n^{1-\delta})$ . As a byproduct, this establishes a (logarithmic) separation between compressed sensing matrices and the stronger RIP property.