Computing a (1+ε)-Approximate Geometric Minimum-Diameter Spanning Tree
First Online: 05 December 2003 Received: 23 January 2002 Revised: 26 July 2003 DOI:
Cite this article as: Spriggs, M., Keil, J., Bespamyatnikh, S. et al. Algorithmica (2004) 38: 577. doi:10.1007/s00453-003-1056-z Abstract
Given a set
P of points in the plane,
a geometric minimum-diameter spanning tree (GMDST) of P is a
spanning tree of P such that the longest path through the
tree is minimized.
For several years, the best upper bound on the time to
compute a GMDST was cubic with respect to the
number of points in the input set.
Recently, Timothy Chan introduced a subcubic time
In this paper we present an algorithm
that generates a tree whose diameter is no more than (1 + ε)
times that of a GMDST, for any ε > 0.
Our algorithm reduces the problem to several grid-aligned versions of the
problem and runs within time $O(ε -3+ n)
and space O( n).
Minimum diameter spanning tree Approximation algorithm Geometric graph References
Callahan, P., Kosaraju, R. 1995 A decomposition of multidimensional point sets with applications to k-nearest
neighbors and n-body potential fields Journal of the ACM 42 67 90 MATH CrossRef MathSciNet
Camerini, P.M., Galbiati, G., Maffioli, F. 1980 Complexity
of spanning tree problems, I European Journal of Operational Research 5 346 352 MATH CrossRef MathSciNet
Semi-online maintenance of geometric optima and measures,
Proceedings of the 13th ACM–SIAM Symposium
on Discrete Algorithms (SODA), 2002, pp. 474–483.
Cong, J., He, L., Koh, C., Madden, P. 1996 Performance
optimization of VLSI interconnection layout Integration: the VLSI
Journal 21 1 94 MATH CrossRef
D. Eppstein, Spanning trees and spanners, in Handbook of
Computational Geometry (J.R. Sack and J. Urrutia,
eds.), North-Holland, Amsterdam, 2000, pp. 425–462.
J. Gudmundsson, H. Haverkort, S.-M. Park, C.-S. Shin, and A. Wolff,
Facility location and the geometric minimum-diameter spanning tree,
in Proceedings of the 5th International Workshop on Approximation Algorithms
for Combinatorial Optimization (APPROX), 2002, pp. 146–160.
Hassin, R., Tamir, A. 1995 On the minimum diameter
spanning tree problem Information Processing Letters 53 109 111 MATH CrossRef MathSciNet
Ho, J., Lee, D., Chang, C., Wong, C. 1991 Minimum
diameter spanning trees and related problems SIAM Journal on
Computing 20 987 997 MATH CrossRef MathSciNet
W. D. Jones, Euclidean Communication Spanning Trees,
M.Sc. Thesis, University of Saskatchewan, 1994.
Kariv, O., Hakimi, S. L. 1979 An algorithmic approach to
network location problems, I: the p-centers SIAM Journal on Applied
Mathematics 37 513 537 MATH CrossRef MathSciNet