Computing planar Voronoi diagrams in double precision: a further example of degree-driven algorithm design

Geometric algorithms use numerical computations to perform geometric tests, so correct algorithms may produce erroneous results if insufficient arithmetic precision is available. Liotta, Preparata, and Tamassia, in 1999, suggested that algorithm design, which traditionally considers running time and memory space, could also consider precision as a resource. They demonstrated that the Voronoi diagram of n sites on a U × U grid could be rounded to answer nearest neighbor queries on the same grid using only double precision. They still had to compute the Voronoi diagram before rounding, which requires the quadruple-precision InCircle test. We develop a "degree-2 Voronoi diagram" that can be computed using only double precision by a randomized incremental construction in O(n log n log U) expected time and O(n) expected space. Our diagram also answers nearest neighbor queries, even though it doesn't even use sufficient precision to determine a Delaunay triangulation.