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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

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).

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)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2022.23, author = {Cabello, Sergio and Hoffmann, Michael and Klost, Katharina and Mulzer, Wolfgang and Tkadlec, Josef}, title = {{Long Plane Trees}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {23:1--23:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.23}, URN = {urn:nbn:de:0030-drops-160311}, doi = {10.4230/LIPIcs.SoCG.2022.23}, annote = {Keywords: geometric network design, spanning trees, plane straight-line graphs, approximation algorithms} }

Document

**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Let S ⊆ ℝ² be a set of n planar sites, such that each s ∈ S has an associated radius r_s > 0. Let 𝒟(S) be the disk intersection graph for S. It has vertex set S and an edge between two distinct sites s, t ∈ S if and only if the disks with centers s, t and radii r_s, r_t intersect. Our goal is to design data structures that maintain the connectivity structure of 𝒟(S) as sites are inserted and/or deleted.
First, we consider unit disk graphs, i.e., r_s = 1, for all s ∈ S. We describe a data structure that has O(log² n) amortized update and O(log n/log log n) amortized query time. Second, we look at disk graphs with bounded radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ r_s ≤ Ψ, for a Ψ ≥ 1 known in advance. In the fully dynamic case, we achieve amortized update time O(Ψ λ₆(log n) log⁷ n) and query time O(log n/log log n), where λ_s(n) is the maximum length of a Davenport-Schinzel sequence of order s on n symbols. In the incremental case, where only insertions are allowed, we get logarithmic dependency on Ψ, with O(α(n)) query time and O(logΨ λ₆(log n) log⁷ n) update time. For the decremental setting, where only deletions are allowed, we first develop an efficient disk revealing structure: given two sets R and B of disks, we can delete disks from R, and upon each deletion, we receive a list of all disks in B that no longer intersect the union of R. Using this, we get decremental data structures with amortized query time O(log n/log log n) that support m deletions in O((nlog⁵ n + m log⁷ n) λ₆(log n) + nlog Ψ log⁴n) overall time for bounded radius ratio Ψ and O((nlog⁶ n + m log⁸n) λ₆(log n)) for arbitrary radii.

Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Dynamic Connectivity in Disk Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 49:1-49:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kaplan_et_al:LIPIcs.SoCG.2022.49, author = {Kaplan, Haim and Kauer, Alexander and Klost, Katharina and Knorr, Kristin and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul}, title = {{Dynamic Connectivity in Disk Graphs}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {49:1--49:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.49}, URN = {urn:nbn:de:0030-drops-160572}, doi = {10.4230/LIPIcs.SoCG.2022.49}, annote = {Keywords: Disk Graphs, Connectivity, Lower Envelopes} }

Document

**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Let S subset R^2 be a set of n sites, where each s in S has an associated radius r_s > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t in S if and only if |st| <= r_s + r_t, i.e., if the disks with centers s and t and respective radii r_s and r_t intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| <= r_s, i.e., if t lies in the disk with center s and radius r_s.
We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest.

Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Triangles and Girth in Disk Graphs and Transmission Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kaplan_et_al:LIPIcs.ESA.2019.64, author = {Kaplan, Haim and Klost, Katharina and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul and Sharir, Micha}, title = {{Triangles and Girth in Disk Graphs and Transmission Graphs}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {64:1--64:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.64}, URN = {urn:nbn:de:0030-drops-111859}, doi = {10.4230/LIPIcs.ESA.2019.64}, annote = {Keywords: disk graph, transmission graph, triangle, girth} }

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