Approximating a Voronoi Cell

Given a set of points in IR , called sites, we consider the problem of approximating the Voronoi cell of a site by a convex polyhedron with a small number of facets or, equivalently, of finding a small set of approximate Voronoi neighbors of . More precisely, we define an -approximate Voronoi neighborhood of , denoted , to be a subset of satisfying the following property: is an -approximate nearest neighbor for any point inside the convex polyhedron defined by the bisectors between and the sites in . We show that there exists a set of -approximate Voronoi neighbors with cardinality for ! and cardinality " # $ $ % '&)(#*,+.-0/2143 # $ $ for any fixed 6587 . We also provide a worst-case lower bound of 9: ; % '&0( *,+.on the number of approximate Voronoi neighbors. Thus, our bound is tight in the plane and within a factor of /2143 # $ $ from optimal in dimension <5=7 . Finally, based on our existence proofs, we design efficient algorithms for computing approximate Voronoi neighborhoods. Figure 1: The Voronoi cell of is shaded. Its approximation is the outer polygon; the approximate Voronoi neighborhood is represented by the white dots. Note that the approximation is better when we are close to .