Fine-Grained Complexity Theory: Conditional Lower Bounds for Computational Geometry

TL;DR

This paper introduces fine-grained complexity theory in computational geometry, focusing on conditional lower bounds for problems like nearest neighbor search based on the Orthogonal Vectors Hypothesis.

Contribution

It provides the first detailed exploration of conditional lower bounds in computational geometry, especially for polynomial-time problems.

Findings

Conditional lower bounds for nearest neighbor search under Euclidean distance

Conditional lower bounds for Fréchet distance computations

Application of the Orthogonal Vectors Hypothesis to geometric problems

Abstract

Fine-grained complexity theory is the area of theoretical computer science that proves conditional lower bounds based on the Strong Exponential Time Hypothesis and similar conjectures. This area has been thriving in the last decade, leading to conditionally best-possible algorithms for a wide variety of problems on graphs, strings, numbers etc. This article is an introduction to fine-grained lower bounds in computational geometry, with a focus on lower bounds for polynomial-time problems based on the Orthogonal Vectors Hypothesis. Specifically, we discuss conditional lower bounds for nearest neighbor search under the Euclidean distance and Fr\'echet distance.