A Lower Bound for Shallow Partitions

TL;DR

This paper establishes a lower bound on the crossing number of shallow partitions in planar point sets, resolving a long-standing open problem in computational geometry.

Contribution

It provides the first non-trivial lower bound for the crossing number of k-partitions, answering a 20-year-old open question by Matousek.

Findings

Lower bound of Omega(log(n/k)/loglog(n/k)) for crossing number

Answers a 20-year-old open problem in the field

Advances understanding of shallow partitions in computational geometry

Abstract

Let P be a planar n-point set. A k-partition of P is a subdivision of P into n/k parts of roughly equal size and a sequence of triangles such that each part is contained in a triangle. A line is k-shallow if it has at most k points of P below it. The crossing number of a k-partition is the maximum number of triangles in the partition that any k-shallow line intersects. We give a lower bound of Omega(log (n/k)/loglog(n/k)) for this crossing number, answering a 20-year old question of Matousek.