Recognizing Voronoi Diagrams with Linear Programming

Abstract

Let P = {P1, …, Pn} be a finite set of distinct points in ℝd and let Ri be the set of points whose distance from Pi is less than or equal to the distance from every other point in P. The collection of regions R1, …, Rn is called the Voronoi diagram generated by P. Our main result is a polynomial time algorithm for recognizing whether or not a given tessellation of ℝd into polyhedra is a Voronoi diagram. This is accomplished by describing a linear program that has a solution if and only if the given tessellation is a Voronoi diagram. As a consequence, for each Ri of a Voronoi diagram, the set of points in Ri contained in some generating set P is either a singleton or the interior of a polyhedron. We also give a polynomial time algorithm for describing this set for each Ri. Finally, this leads to a second algorithm for recognizing Voronoi diagrams; this algorithm also relies on linear programming but, for fixed dimension d, it is strongly polynomial and has linear time complexity. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.