An Expected Linear-Time Algorithm for the Farthest-Segment Voronoi Diagram

Abstract: We present an expected linear-time algorithm to construct the farthest-segment Voronoi diagram, given the sequence of its faces at infinity. This sequence forms a Davenport-Schinzel sequence of order 3 and it can be computed in O(n log n) time, where n is the number of input segments. The farthest-segment Voronoi diagram is a tree, with disconnected Voronoi regions, of total complexity Theta(n) in the worst case. Disconnected regions pose a major difficulty in deriving linear-time construction algorithms for such tree-like Voronoi diagrams. In this paper we present a new approach towards this direction.