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DOI: 10.4230/LIPIcs.ESA.2023.51

A Sweep-Plane Algorithm for Calculating the Isolation of Mountains

Authors: Daniel Funke, Nicolai Hüning, and Peter Sanders

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract

One established metric to classify the significance of a mountain peak is its isolation. It specifies the distance between a peak and the closest point of higher elevation. Peaks with high isolation dominate their surroundings and provide a nice view from the top. With the availability of worldwide Digital Elevation Models (DEMs), the isolation of all mountain peaks can be computed automatically. Previous algorithms run in worst case time that is quadratic in the input size. We present a novel sweep-plane algorithm that runs in time 𝒪(nlog n+pT_NN) where n is the input size, p the number of considered peaks and T_NN the time for a 2D nearest-neighbor query in an appropriate geometric search tree. We refine this to a two-level approach that has high locality and good parallel scalability. Our implementation reduces the time for calculating the isolation of every peak on Earth from hours to minutes while improving precision.

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Daniel Funke, Nicolai Hüning, and Peter Sanders. A Sweep-Plane Algorithm for Calculating the Isolation of Mountains. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{funke_et_al:LIPIcs.ESA.2023.51,
  author =	{Funke, Daniel and H\"{u}ning, Nicolai and Sanders, Peter},
  title =	{{A Sweep-Plane Algorithm for Calculating the Isolation of Mountains}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{51:1--51:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.51},
  URN =		{urn:nbn:de:0030-drops-187040},
  doi =		{10.4230/LIPIcs.ESA.2023.51},
  annote =	{Keywords: computational geometry, Geo-information systems, sweepline algorithms}
}

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DOI: 10.4230/LIPIcs.SEA.2023.20

Efficient Yao Graph Construction

Authors: Daniel Funke and Peter Sanders

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract

Yao graphs are geometric spanners that connect each point of a given point set to its nearest neighbor in each of k cones drawn around it. Yao graphs were introduced to construct minimum spanning trees in d dimensional spaces. Moreover, they are used for instance in topology control in wireless networks. An optimal 𝒪(n log n)-time algorithm to construct Yao graphs for a given point set has been proposed in the literature but - to the best of our knowledge - never been implemented. Instead, algorithms with a quadratic complexity are used in popular packages to construct these graphs. In this paper we present the first implementation of the optimal Yao graph algorithm. We engineer the data structures required to achieve the 𝒪(n log n) time bound and detail algorithmic adaptations necessary to take the original algorithm from theory to practice. We propose a priority queue data structure that separates static and dynamic events and might be of independent interest for other sweepline algorithms. Additionally, we propose a new Yao graph algorithm based on a uniform grid data structure that performs well for medium-sized inputs. We evaluate our implementations on a wide variety of synthetic and real-world datasets and show that our implementation outperforms current publicly available implementations by at least an order of magnitude.

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Daniel Funke and Peter Sanders. Efficient Yao Graph Construction. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{funke_et_al:LIPIcs.SEA.2023.20,
  author =	{Funke, Daniel and Sanders, Peter},
  title =	{{Efficient Yao Graph Construction}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-279-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{265},
  editor =	{Georgiadis, Loukas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.20},
  URN =		{urn:nbn:de:0030-drops-183706},
  doi =		{10.4230/LIPIcs.SEA.2023.20},
  annote =	{Keywords: computational geometry, geometric spanners, Yao graphs, sweepline algorithms, optimal algorithms}
}

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A Sweep-Plane Algorithm for Calculating the Isolation of Mountains

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Efficient Yao Graph Construction

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