Approximating the minimum weight spanning tree of a set of points in the Hausdorff metric

  • Authors:

  • Victor Alvarez;Raimund Seidel

  • Affiliations:

  • Fachrichtung Informatik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany;Fachrichtung Informatik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2010

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Abstract

We study the problem of approximating MST(P), the minimum weight spanning tree of a set P of n points in [0,1]^d, by a spanning tree of some subset Q@?P. We show that if the weight of MST(P) is to be approximated, then in general Q must be large. If the shape of MST(P) is to be approximated, then this is always possible with a small Q. More specifically, for any 00 arbitrarily small, for d3, see [Pankaj K. Agarwal, Herbert Edelsbrunner, Otfried Schwarzkopf, Emo Welzl, Euclidean minimum spanning trees and bi-chromatic closest pairs, Discrete Comput. Geom. 6 (5) (1991) 407-422]. Also @t"3","1(n) and @t"3","~(n) is known to be O(nlogn), see [Drago Krznaric, Christos Levcopoulos, Bengt J. Nilsson, Minimum spanning trees in d dimensions, Nordic J. of Computing 6 (4) (1999) 446-461].