The Anchored Voronoi Diagram: Static, Dynamic Versions and Applications

For a given set S of n points in the plane and a fixed point o, we introduce the Voronoi diagram of S anchored at o. It will be defined as an abstract Voronoi diagram that uses as bisectors the following curves. For each pair of points p, q in S, the bisecting curve between p and q is the locus of points x in the plane such that the line segment ox is equidistant to both p and q. We show that those bisectors have nice properties and, therefore, this new structure can be computed in O(n log n) time and O(n) space both for nearest-site and furthest-site versions. Also, we prove that the dynamic version of this diagram can be built in O(nλ6s+2(n) log n) time complexity, where s is a constant depending on the function that describes the motion of the points. Finally, we show how to use these structures for solving several locational optimization problems.