Fast geometric approximation techniques and geometric embedding problems

  • Authors:

  • M. W. Bern;H. J. Karloff;P. Raghavan;B. Schieber

  • Affiliations:

  • Xerox Palo Alto Research Center, Palo Alto, CA;tDepartmerd of Computer Science, University of Chicago, Chicago, IL;IBM Research Division, T.J. Watson Research Center, Yorktown Heights, NY;IBM Research Division, T.J. Watson Research Center, Yorktown Heights, NY

  • Venue:
  • SCG '89 Proceedings of the fifth annual symposium on Computational geometry
  • Year:
  • 1989

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Abstract

Given an undirected n-vertex graph G and a set of n points in Rd, we wish to embed the vertices of G onto the points so as to minimize the total embedded edge length. Important special cases of this geometric embedding problem are those in which G is a binary tree, a cycle, or a star. We give fast approximation algorithms for embedding these graphs on the line and in the plane in several metrics. Our principal techniques are: a notion of “approximate geometric sorting” that can be computed in linear time, and fast approximation schemes for the minimum spanning tree problem in the plane. We expect that these approximation techniques can be applied to many geometric problems besides the embedding problem. We give the example of approximating the convex hull of a set of points in the plane.