Voronoi Diagrams 9 . 1 Post Office Problem

Suppose there are n post offices p 1 ,. .. p n in a city. Someone who is located at a position q within the city would like to know which post office is closest to him. Modeling the city as a planar region, we think of p 1 ,. .. p n and q as points in the plane. Denote the set of post offices by P = {p 1 ,. .. p n }. While the locations of post offices are known and do not change so frequently, we do not know in advance for which—possibly many—query locations the closest post office is to be found. Therefore, our long term goal is to come up with a data structure on top of P that allows to answer any possible query efficiently. The basic idea is to apply a so-called locus approach: we partition the query space into regions on which is the answer is the same. In our case, this amounts to partition the plane into regions such that for all points within a region the same point from P is closest (among all points from P). As a warmup, consider the problem for two post offices p i , p j ∈ P. For which query locations is the answer p i rather than p j ? This region is bounded by the bisector of p i and p j , that is, the set of points which have the same distance to both points.