On the Average Complexity of the $k$-Level

TL;DR

This paper establishes new upper bounds on the expected complexity of the k-level in arrangements of great circles and spheres, improving understanding of geometric arrangements in computational geometry.

Contribution

It provides the first expected complexity bounds for the k-level in random great-circle and great-sphere arrangements, extending previous worst-case bounds.

Findings

Expected complexity of the k-level in great-circle arrangements is O((k+1)^2).

Expected complexity in arrangements of great (d-1)-spheres is Θ((k+1)^{d-1}).

Results generalize to arrangements on the sphere with random points.

Abstract

Let ${\cal L}$ be an arrangement of $n$ lines in the Euclidean plane. The \emph{$k$-level} of ${\cal L}$ consists of all vertices $v$ of the arrangement which have exactly $k$ lines of ${\cal L}$ passing below $v$. The complexity (the maximum size) of the $k$-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of $O(n\cdot (k+1)^{1/3})$. Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the $k$-level denotes the vertices at distance at most $k$ to a marked cell, the \emph{south pole}. We prove an upper bound of $O((k+1)^2)$ on the expected complexity of the $k$-level in great-circle arrangements if the south pole is chosen uniformly at random among all cells. We also consider arrangements of great $(d-1)$-spheres on the sphere $\mathbb{S}^d$ which are orthogonal to a set of random points on $\mathbb{S}^d$. In this model, we prove that the expected complexity of the $k$-level is of order $\Theta((k+1)^{d-1})$.