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

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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).

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Authors and Affiliations

  1. School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada

    Michael J. Spriggs

  2. Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5A9, Canada

    J. Mark Keil

  3. Department of Computer Science, Duke University, Box 90129, Durham, NC 27708, USA

    Sergei Bespamyatnikh

  4. Communication Systems Engineering Department, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

    Michael Segal

  5. Department of Computer Science, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3175, USA

    Jack Snoeyink

Authors
  1. Michael J. Spriggs
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  2. J. Mark Keil
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  3. Sergei Bespamyatnikh
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  4. Michael Segal
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  5. Jack Snoeyink
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Corresponding authors

Correspondence to Michael J. Spriggs, J. Mark Keil, Sergei Bespamyatnikh, Michael Segal or Jack Snoeyink.

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Spriggs, M., Keil, J., Bespamyatnikh, S. et al. Computing a (1+ε)-Approximate Geometric Minimum-Diameter Spanning Tree. Algorithmica 38, 577–589 (2004). https://doi.org/10.1007/s00453-003-1056-z

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  • DOI: https://doi.org/10.1007/s00453-003-1056-z

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