Abstract Voronoi Diagrams with Disconnected Regions

Abstract Voronoi diagrams [15,16] are based on bisecting curves enjoying simple combinatorial properties, rather than on the geometric notions of sites and distance. They serve as a unifying concept. Once the bisector system of any concrete type of Voronoi diagram is shown to fulfill the AVD axioms, structural results and efficient algorithms become available without further effort; for example, the first optimal algorithms for constructing nearest Voronoi diagrams of disjoint convex objects, or of line segments under the Hausdorff metric, have been obtained this way [18]. One of these axioms stated that all Voronoi regions must be pathwise connected, a property quite useful in divide&conquer and randomized incremental construction algorithms. Yet, there are concrete Voronoi diagrams where this axiom fails to hold.

In this paper we consider, for the first time, abstract Voronoi diagrams with disconnected regions. By combining the randomized incremental construction technique [18] with trapezoidal decomposition [21] we obtain an algorithm that runs in expected time

$O(s^2 n \sum_{j=3}^n {{m_j}/{j}})$

, where

s

is the maximum number of faces a Voronoi region in a subdiagram of three sites can have, and

m

j

denotes the average number of faces per region in any subdiagram of

j

sites. In the connected case, where

s

 = 1 = 

m

j

, this results in the known optimal bound

$O( n \ \sum_{j=3}^n 1/j) \ = \ O(n \log n)$

.