Finding constrained and weighted Voronoi diagrams in the plane

Abstract

The Voronoi diagram of a set of weighted points (sites) whose visibilities are constrained by a set of line segments (obstacles) on the plane is studied. The diagram is called constrained and weighted Voronoi diagram. When all the sites are of the same weight, it becomes the constrained Voronoi diagram in which the endpoints of the obstacles need not be sites. An Ω(m 2n 2) lower bound on the combinatorial complexity of both constrained Voronoi diagram and constrained and weighted Voronoi diagram is established, where n is the number of sites and m is the number of obstacles. For constrained Voronoi diagram, an O( m 2 n 2+ n 4) time and space algorithm is presented. The algorithm is optimal when m ≥ cn, for any positive constant c. For constrained and weighted Voronoi diagram, an O( m 2 n 2 + n 42 α( n) ) time and O( m 2 n 2 + n 4) space algorithm (where α( n) is the functional inverse of the Ackermann's function) is presented. The algorithm is near-optimal when m ≥ cn, for any positive constant c.