The Complexity of One or Many Faces in the Overlay of Many Arrangements

TL;DR

This paper extends the Combination Lemma to analyze the complexity of faces in arrangements of geometric objects, providing new bounds and simpler proofs for existing complexity measures in computational geometry.

Contribution

It introduces an extended Combination Lemma that relates face complexity in arrangements to individual arrangements, and applies it to derive new bounds and simplified proofs for geometric arrangement complexities.

Findings

Complexity of a face in arrangements of polygons is Θ(n α(k)).

New proof of the bound O(√m λ_{s+2}(n)) for edges in arrangements of Jordan arcs.

Extended bounds for edges in sparse arrangements involving arrangement complexity w.

Abstract

We present an extension of the Combination Lemma of [GSS89] that expresses the complexity of one or several faces in the overlay of many arrangements, as a function of the number of arrangements, the number of faces, and the complexities of these faces in the separate arrangements. Several applications of the new Combination Lemma are presented: We first show that the complexity of a single face in an arrangement of $k$ simple polygons with a total of $n$ sides is $\Theta(n \alpha(k) )$, where $\alpha(\cdot)$ is the inverse of Ackermann's function. We also give a new and simpler proof of the bound $O \left( \sqrt{m} \lambda_{s+2}( n ) \right)$ on the total number of edges of $m$ faces in an arrangement of $n$ Jordan arcs, each pair of which intersect in at most $s$ points, where $\lambda_{s}(n)$ is the maximum length of a Davenport-Schinzel sequence of order $s$ with $n$ symbols. We extend this result, showing that the total number of edges of $m$ faces in a sparse arrangement of $n$ Jordan arcs is $O \left( (n + \sqrt{m}\sqrt{w}) \frac{\lambda_{s+2}(n)}{n} \right)$, where $w$ is the total complexity of the arrangement. Several other applications and variants of the Combination Lemma are also presented.