Long Plane Trees
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Date
2022
Publication Type
Conference Paper
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yes
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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 TOPT for P with maximum total Euclidean edge length |TOPT|. 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 |TOPT|. 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 TALG with diameter at most four and |TALG| = 0.546 · |TOPT|. 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).
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Publication status
published
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Book title
38th International Symposium on Computational Geometry
Volume
224
Pages / Article No.
23
Publisher
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Event
38th International Symposium on Computational Geometry (SoCG 2022)
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Methods
Software
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Date created
Subject
geometric network design; spanning trees; plane straight-line graphs; approximation algorithms
Organisational unit
03457 - Welzl, Emo (emeritus) / Welzl, Emo (emeritus)