Notes in Computational Geometry Voronoi Diagrams

If the points in S are uniformly distributed, a reasonable approach would be to use a grid-based structure. Thus, with a √ n× √ n grid, we have, on an average, 1 point per square, hence, a constant query time complexity on expectation. This is similar to the grid-based method used for the closest pair problem. For a general distribution of points, in 1 dimension, an effective approach would be to just sort all the points and performa binary search to answer a query. Here, the query time is O(log n). The space complexity would be O(n) and preprocessing time O(n log n). Moving to 2 dimensions, we start with a case where there are only two points in S. The division of the plane in to the regions of influence of each point is obtained by the perpendicular bisector of the line joining the two points. Thus, once we have the division of the plane in to the two regions of influence, we can determine which region a point lies in by answering a point location query, as was discussed in the previous topic. We can easily extend this idea to 3 points by constructing the perpendicular bisectors of each pair of points. We notice that the 3 bisectors intersect at a single point (since, the 3 points form a triangle) which is the circum-centre of the 3 points. These two cases are depicted in Fig. 1.