Bowties and Hourglasses: Intersections of Double-Wedges (or Stabbing and Avoiding Line Segments)

TL;DR

This paper investigates the intersection properties of arrangements of double-wedges, including bowties and hourglasses, revealing complex intersection structures and providing algorithms for their computation.

Contribution

It generalizes previous work by considering both bowties and hourglasses, analyzing their intersection complexity, and developing optimal algorithms for intersection computation.

Findings

The intersection of n double-wedges can have Ω(n^2) interior-disjoint regions.

Provides algorithms with worst-case optimal running time for computing intersections.

Establishes a connection between intersection points and the 3SUM problem complexity.

Abstract

We study the common intersection of arrangements of double-wedges. We consider arrangements where double-wedges may be either bowties (which do not contain a vertical line) or hourglasses (which contain a vertical line), in contrast to earlier studies that focused on arrangements of only bowties. This generalization changes the setting drastically, in particular, with respect to all arguments involving the point-line duality. Namely, a point in the intersection of all double-wedges is equivalent to a line that stabs a set of segments $\mathcal{S}$ (corresponding to the bowties) while it avoids a different set of segments $\mathcal{A}$ (corresponding to the complement of the hourglasses). We show that in this general setting, the intersection of $n$ double-wedges may consist of $\Omega(n^2)$ interior-disjoint regions. Further, we discuss Gallai-type results for arrangements of segments and anti-segments, and we provide algorithms for computing the intersection of such arrangements with worst-case optimal running time. Finally, we also prove that we can find a single intersection point in almost optimal running time, assuming that 3SUM admits no truly subquadratic-time algorithm.