Linear-Time Algorithms for the Farthest-Segment Voronoi Diagram and Related Tree Structures

We present linear-time algorithms to construct tree-structured Voronoi diagrams, after the sequence of their regions at infinity or along a given boundary is known. We focus on Voronoi diagrams of line segments, including the farthest-segment Voronoi diagram, the order-(k+1) subdivision within a given order-k Voronoi region, and deleting a segment from a nearest-neighbor diagram. Although tree-structured, these diagrams illustrate properties surprisingly different from their counterparts for points. The sequence of their faces along the relevant boundary forms a Davenport-Schinzel sequence of order 3. Once this sequence is known, we show how to compute the corresponding Voronoi diagram in linear time, expected or deterministic, augmenting the existing linear frameworks for points in convex position, with the ability to handle non-uniqueness of Voronoi faces. Our techniques contribute towards the question of linear-time construction algorithms for tree-like Voronoi diagrams.