Computing the Implicit Voronoi Diagram in Triple Precision

In a paper that considered arithmetic precision as a limited resource in the design and analysis of algorithms, Liotta, Preparata and Tamassia defined an “implicit Voronoi diagram” supporting logarithmictime proximity queries using predicates of twice the precision of the input and query coordinates. They reported, however, that computing this diagram uses five times the input precision. We define a reduced-precision Voronoi diagram that similarly supports proximity queries, and describe a randomized incremental construction using only three times the input precision. The expected construction time is O(n(log n+log μ)), where μ is the length of the longest Voronoi edge; we can construct the implicit Voronoi from the reduced-precision Voronoi in linear time.