Proximity: Fundamental Algorithms

  • Franco P. Preparata
  • Michael Ian Shamos
Part of the Texts and Monographs in Computer Science book series (MCS)


In Chapter 4 we illustrated anO(NlogN)algorithm for finding the two farthest points of a plane set. One may think that finding the two closest points would be a simple extension, but it is not. The two farthest points are necessarily hull vertices and we may exploit convexity to give a fast algorithm; the two closest points do not necessarily bear any relation to the convex hull, so a new technique must be developed, which is the subject of this chapter. We will be concerned with a large class of problems that involve the proximity of points in the plane and our goal will be to deal with all of these seemingly unrelated tasks via a single algorithm, one that discovers, processes, and stores compactly all of the relevant proximity information. To do this, we revive a classical mathematical object, the Voronoi diagram, and turn it into an efficient computational structure that permits vast improvement over the best previously known algorithms. In this chapter, several of the geometric tools we have discussed earlier—such as hull-finding and searching—will be used to attack this large and difficult class—theclosest-pointorproximity problems.


Convex Hull Voronoi Diagram Steiner Tree Delaunay Triangulation Close Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Franco P. Preparata
    • 1
  • Michael Ian Shamos
    • 2
  1. 1.Department of Computer ScienceBrown UniversityProvidenceUSA
  2. 2.Department of Computer ScienceCarnegie–Mellon UniversityPittsburghUSA

Personalised recommendations