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Long Plane Trees

Authors Sergio Cabello , Michael Hoffmann , Katharina Klost, Wolfgang Mulzer , Josef Tkadlec



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Author Details

Sergio Cabello
  • Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
  • Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Switzerland
Katharina Klost
  • Institut für Informatik, Freie Universität Berlin, Germany
Wolfgang Mulzer
  • Institut für Informatik, Freie Universität Berlin, Germany
Josef Tkadlec
  • Department of Mathematics, Harvard University, Cambridge, MA, USA

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Sergio Cabello, Michael Hoffmann, Katharina Klost, Wolfgang Mulzer, and Josef Tkadlec. Long Plane Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 23:1-23:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.23

Abstract

In the longest plane spanning tree problem, we are given a finite planar point set 𝒫, and our task is to find a plane (i.e., noncrossing) spanning tree T_OPT for 𝒫 with maximum total Euclidean edge length |T_OPT|. Despite more than two decades of research, it remains open if this problem is NP-hard. Thus, previous efforts have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates |T_OPT|. The approximate trees in these algorithms all have small unweighted diameter, typically three or four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms. We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree T_ALG with diameter at most four and |T_ALG| ≥ 0.546 ⋅ |T_OPT|. This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter d ≥ 3, we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most d (compared to a longest plane tree without constraints).

Subject Classification

ACM Subject Classification
  • Theory of computation → Routing and network design problems
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Computational geometry
  • Mathematics of computing → Trees
Keywords
  • geometric network design
  • spanning trees
  • plane straight-line graphs
  • approximation algorithms

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