A Lower Bound for Shallow Partitions
Computational Geometry
2012-02-03 v2 Discrete Mathematics
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.
Cite
@article{arxiv.1201.2267,
title = {A Lower Bound for Shallow Partitions},
author = {Wolfgang Mulzer and Daniel Werner},
journal= {arXiv preprint arXiv:1201.2267},
year = {2012}
}
Comments
6 pages, 1 figure. The result was also obtained independently by Peyman Afshani