Worst-case complexity of nearest Voronoi diagrams for arbitrary lines

Determine the exact worst-case combinatorial complexity of the nearest Voronoi diagram of an arbitrary set of n lines in R^3, closing the gap between the known Ω(n^2) lower bound and the O(n^{3+ε}) upper bound for every fixed ε>0.

Background

The paper places its restricted ruling class in the context of the general problem for arbitrary line sites in three-dimensional Euclidean space. For arbitrary lines, the known lower bound is quadratic, while the best general upper bound is near-cubic, leaving the exact asymptotic complexity unresolved. The present paper proves a tight quadratic bound only for lines contained in one ruling of a smooth regulus, and therefore does not resolve the unrestricted problem.

References

The combinatorial complexity of the nearest Voronoi diagram of arbitrary lines in R3 is a long-standing open problem.

— Quadratic Complexity of Voronoi Diagrams in $\mathbb{R}^3$ for Lines in a Single Ruling of a Regulus  (2608.27114 - Park, 27 Aug 2026) in Section 1, Introduction

An important direction for future research is to identify natural classes of n lines in R3 for which the nearest Voronoi diagram has worst-case combinatorial complexity Θ(n2). The present work establishes such a bound when the lines belong to one ruling of a smooth regulus, but it remains to determine which geometric or algebraic properties of a line family force a quadratic upper bound while allowing a quadratic lower bound.

— Quadratic Complexity of Voronoi Diagrams in $\mathbb{R}^3$ for Lines in a Single Ruling of a Regulus  (2608.27114 - Park, 27 Aug 2026) in Section 1, Conclusion