A Deterministic Partition Tree and Applications

TL;DR

This paper introduces a deterministic partition tree variant that improves query efficiency for high-dimensional geometric problems, surpassing previous lower bounds without using bit-packing techniques.

Contribution

It presents a deterministic version of Chan's randomized partition tree, enabling faster geometric queries and deterministic solutions for several classical problems.

Findings

Achieves $O(n^{1-1/d} / \\log^{\\Omega(1)} n)$ query time for simplex range counting

Breaks the $\\Omega(n^{1-1/d})$ lower bound in the semigroup setting

Provides deterministic improvements for multiple geometric problems

Abstract

In this paper, we present a deterministic variant of Chan's randomized partition tree [Discret. Comput. Geom., 2012]. This result leads to numerous applications. In particular, for $d$-dimensional simplex range counting (for any constant $d \ge 2$), we construct a data structure using $O(n)$ space and $O(n^{1+\epsilon})$ preprocessing time, such that each query can be answered in $o(n^{1-1/d})$ time (specifically, $O(n^{1-1/d} / \log^{\Omega(1)} n)$ time), thereby breaking an $\Omega(n^{1-1/d})$ lower bound known for the semigroup setting. Notably, our approach does not rely on any bit-packing techniques. We also obtain deterministic improvements for several other classical problems, including simplex range stabbing counting and reporting, segment intersection detection, counting and reporting, ray-shooting among segments, and more. Similar to Chan's original randomized partition tree, we expect that additional applications will emerge in the future, especially in situations where deterministic results are preferred.