Computing a (1+ε)-Approximate Geometric Minimum-Diameter Spanning Tree

Authors:
Michael J. Spriggs;J. Mark Keil;Sergei Bespamyatnikh;Michael Segal;Jack Snoeyink
Affiliations:
School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada;Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5A9, Canada;Department of Computer Science, Duke University, Box 90129, Durham, NC 27708, USA;Communication Systems Engineering Department, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel;Department of Computer Science, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3175, USA
Venue:
Algorithmica
Year:
2004

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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 algorithm. 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).