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

2026-04-25

AI Summary

This study addresses the intersection problem of two classes of double-wedge arrangements—bowtie and hourglass—by equivalently formulating it as the geometric task of finding a line that simultaneously stabs one set of segments while avoiding another. Leveraging point-line duality, the problem is transformed into a unified dual setting involving segment stabbing and avoidance, enabling a combined analysis through arrangement complexity and conditional lower bounds under the 3SUM conjecture. The main contributions include the first unified treatment of both double-wedge types, a proof that their intersection can yield Ω(n²) pairwise-disjoint regions, a worst-case optimal algorithm for computing the full intersection, and, assuming the 3SUM hypothesis, a near-optimal algorithm for detecting a single intersecting solution.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $Ω(n^2)$ 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.

Problem

• double-wedges
• bowties
• hourglasses
• line stabbing
• segment avoidance

Innovation

• double-wedges
• stabbing and avoiding
• point-line duality
• arrangement intersection
• 3SUM hardness