Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations

Abstract Voronoi diagrams are defined in terms of a given system of planar bisecting curves satisfying some simple combinatorial properties. They offer a unifying framework for a wide range of concrete Voronoi instances on generalized sites and metrics. In this paper, we formulate higher-order abstract color Voronoi diagrams of a set $S$ of $n$ colored abstract sites, simultaneously considering all concrete instances under their umbrella. We prove that the number of vertices in the order-$k$ abstract color Voronoi diagram is at most $4k(n-k)-2n$, and present an iterative construction algorithm. The bound directly applies to a family of $m$ disjoint simple polygons of total complexity $n$. For simple polygons the bound can further improve to $O(\min\{k(n-k),(m-k)^2n\})$. A critical ingredient of our proof is a combinatorial analysis on circular sequences of color permutations derived from the unbounded edges of these diagrams, which is interesting in its own right.