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Long Plane Trees

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

TL;DR

This paper studies the problem of finding a longest plane spanning tree on a planar point set $\mathcal{P}$, aiming to maximize the total edge length while preserving planarity. It introduces a polynomial-time approximation based on a diameter-bounded construction that achieves at least $0.5467$ of the optimum $OPT$ (and a $2/3$-approximation to $OPT_{cr}$ for flat point sets), and provides a polynomial-time algorithm for the longest diameter-3 case, along with upper bounds on achievable ratios for larger diameters. The work clarifies how diameter constraints interact with optimality in the longest-plane spanning-tree problem and contrasts plane and crossing-tree benchmarks in the analysis. Overall, it advances understanding of the trade-offs between planarity, diameter, and length in geometric network design.

Abstract

In the longest plane spanning tree problem, we are given a finite planar point set $\mathcal{P}$, and our task is to find a plane (i.e., noncrossing) spanning tree for $\mathcal{P}$ with maximum total Euclidean edge length. Despite more than two decades of research, it remains open whether this problem is NP-hard. Thus, previous efforts have focused on olynomial-time algorithms that produce plane trees whose total edge length approximates $\text{OPT}$, the maximum possible length. 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 with diameter at most four and total edge length at least $0.546 \cdot \text{OPT}$. 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 $d \geq 3$, 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).

Long Plane Trees

TL;DR

This paper studies the problem of finding a longest plane spanning tree on a planar point set , aiming to maximize the total edge length while preserving planarity. It introduces a polynomial-time approximation based on a diameter-bounded construction that achieves at least of the optimum (and a -approximation to for flat point sets), and provides a polynomial-time algorithm for the longest diameter-3 case, along with upper bounds on achievable ratios for larger diameters. The work clarifies how diameter constraints interact with optimality in the longest-plane spanning-tree problem and contrasts plane and crossing-tree benchmarks in the analysis. Overall, it advances understanding of the trade-offs between planarity, diameter, and length in geometric network design.

Abstract

In the longest plane spanning tree problem, we are given a finite planar point set , and our task is to find a plane (i.e., noncrossing) spanning tree for with maximum total Euclidean edge length. Despite more than two decades of research, it remains open whether this problem is NP-hard. Thus, previous efforts have focused on olynomial-time algorithms that produce plane trees whose total edge length approximates , the maximum possible length. 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 with diameter at most four and total edge length at least . 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 , we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most (compared to a longest plane tree without constraints).

Paper Structure

This paper contains 2 sections, 17 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A general approximation algorithm

Key Result

Theorem 1

For any finite point set $\mathcal{P}$ in general position (the set $\mathcal{P}$ contains no three collinear points), we can compute in polynomial time a plane tree of Euclidean length at least $f \cdot \mathrm{OPT}$, where $\mathrm{OPT}$ denotes the length of a longest plane tree on $\mathcal{P}$

Figures (2)

  • Figure 1: A tree $S_a$ and a tree $T_{a,b}$.
  • Figure 6: For $p\in E$, we have $\ell_{T_{\mathrm{OPT}}}(p) \leq 2d$, for $p \in N$, we have $\ell_{T_{\mathrm{OPT}}}(p) \leq \max\{\| pp_a\|,\| pp_u\| \}$, and for $p \in S$, we have $\ell_{T_{\mathrm{OPT}}}(p) \leq \max\{\| pp_a\|,\| pp_u\|\}$.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma 9
  • Theorem 9
  • ...and 7 more