Bucketing and filtering in computational geometry
Author:Jyrki Katajainen
Publication:Ph.D. Thesis, Department of Computer Science, University of Turku (1987), vi+120 pp.
Abstract:This thesis summarizes the work on computational geometry which I have done during the last five years at the University of Turku. The computational complexity of three geometrical proximity problems is examined: How to construct a minimum spanning tree, the relative neighbourhood graph, and the Delaunay diagram (a Delaunay triangulation) for a finite set of points in a two-dimensional coordinate space with the Lp metric as a distance measure.

   The main contributions of this work are:

  • A simple expected-time optimal algorithm for relative neighbourhood graphs. The algorithm works for any Lp metric, 1 ≤ p ≤ ∞. Similar ideas can be used to obtain a simple algorithm for minimum spanning trees, as well as for Delaunay diagrams (triangulations).
  • A simple expected-time and worst-case time optimal algorithm for Delaunay diagrams (triangulations). The algorithm works for any Lp metric, 1 < p < ∞.
  • A worst-case optimal algorithm for relative neighbourhood graphs in L1 and L metrics.

   To some extent, the minimum spanning tree and relative neighbourhood graph problems are also considered in a multidimensional space. Two methodologies, bucketing and filtering, are extensively used in developing the algorithms. Most of the algorithms proposed are implemented, and compared experimentally with existing algorithms.

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  author = {Jyrki Katajainen},
  title = {Bucketing and filtering in computational geometry},
  type = {Ph.D. Thesis},
  institution = {Department of Computer Science, University of Turku},
  year = {1987},
  pagetotal = {vi+120},
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