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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Any surface that is intrinsically polyhedral can be represented by a collection of simple polygons (fragments), glued along pairs of equally long oriented edges, where each fragment is endowed with the geodesic metric arising from its Euclidean metric. We refer to such a representation as a portalgon, and we call two portalgons equivalent if the surfaces they represent are isometric.
We analyze the complexity of shortest paths. We call a fragment happy if any shortest path on the portalgon visits it at most a constant number of times. A portalgon is happy if all of its fragments are happy. We present an efficient algorithm to compute shortest paths on happy portalgons.
The number of times that a shortest path visits a fragment is unbounded in general. We contrast this by showing that the intrinsic Delaunay triangulation of any polyhedral surface corresponds to a happy portalgon. Since computing the intrinsic Delaunay triangulation may be inefficient, we provide an efficient algorithm to compute happy portalgons for a restricted class of portalgons.

Maarten Löffler, Tim Ophelders, Rodrigo I. Silveira, and Frank Staals. Shortest Paths in Portalgons. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{loffler_et_al:LIPIcs.SoCG.2023.48, author = {L\"{o}ffler, Maarten and Ophelders, Tim and Silveira, Rodrigo I. and Staals, Frank}, title = {{Shortest Paths in Portalgons}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {48:1--48:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.48}, URN = {urn:nbn:de:0030-drops-178980}, doi = {10.4230/LIPIcs.SoCG.2023.48}, annote = {Keywords: Polyhedral surfaces, shortest paths, geodesic distance, Delaunay triangulation} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We study the problem of constructing a data structure that can store a two-dimensional polygonal curve P, such that for any query segment ab one can efficiently compute the Fréchet distance between P and ab. First we present a data structure of size O(n log n) that can compute the Fréchet distance between P and a horizontal query segment ab in O(log n) time, where n is the number of vertices of P. In comparison to prior work, this significantly reduces the required space. We extend the type of queries allowed, as we allow a query to be a horizontal segment ab together with two points s, t ∈ P (not necessarily vertices), and ask for the Fréchet distance between ab and the curve of P in between s and t. Using O(nlog²n) storage, such queries take O(log³ n) time, simplifying and significantly improving previous results. We then generalize our results to query segments of arbitrary orientation. We present an O(nk^{3+ε}+n²) size data structure, where k ∈ [1,n] is a parameter the user can choose, and ε > 0 is an arbitrarily small constant, such that given any segment ab and two points s, t ∈ P we can compute the Fréchet distance between ab and the curve of P in between s and t in O((n/k)log²n+log⁴ n) time. This is the first result that allows efficient exact Fréchet distance queries for arbitrarily oriented segments.
We also present two applications of our data structure. First, we show that our data structure allows us to compute a local δ-simplification (with respect to the Fréchet distance) of a polygonal curve in O(n^{5/2+ε}) time, improving a previous O(n³) time algorithm. Second, we show that we can efficiently find a translation of an arbitrary query segment ab that minimizes the Fréchet distance with respect to a subcurve of P.

Maike Buchin, Ivor van der Hoog, Tim Ophelders, Lena Schlipf, Rodrigo I. Silveira, and Frank Staals. Efficient Fréchet Distance Queries for Segments. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{buchin_et_al:LIPIcs.ESA.2022.29, author = {Buchin, Maike and van der Hoog, Ivor and Ophelders, Tim and Schlipf, Lena and Silveira, Rodrigo I. and Staals, Frank}, title = {{Efficient Fr\'{e}chet Distance Queries for Segments}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {29:1--29:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.29}, URN = {urn:nbn:de:0030-drops-169671}, doi = {10.4230/LIPIcs.ESA.2022.29}, annote = {Keywords: Computational Geometry, Data Structures, Fr\'{e}chet distance} }

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**Published in:** LIPIcs, Volume 208, 11th International Conference on Geographic Information Science (GIScience 2021) - Part II

An important task in terrain analysis is computing viewsheds. A viewshed is the union of all the parts of the terrain that are visible from a given viewpoint or set of viewpoints. The complexity of a viewshed can vary significantly depending on the terrain topography and the viewpoint position.
In this work we study a new topographic attribute, the prickliness, that measures the number of local maxima in a terrain from all possible angles of view. We show that the prickliness effectively captures the potential of terrains to have high complexity viewsheds. We present near-optimal algorithms to compute it for TIN terrains, and efficient approximate algorithms for raster DEMs. We validate the usefulness of the prickliness attribute with experiments in a large set of real terrains.

Ankush Acharyya, Ramesh K. Jallu, Maarten Löffler, Gert G.T. Meijer, Maria Saumell, Rodrigo I. Silveira, and Frank Staals. Terrain Prickliness: Theoretical Grounds for High Complexity Viewsheds. In 11th International Conference on Geographic Information Science (GIScience 2021) - Part II. Leibniz International Proceedings in Informatics (LIPIcs), Volume 208, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{acharyya_et_al:LIPIcs.GIScience.2021.II.10, author = {Acharyya, Ankush and Jallu, Ramesh K. and L\"{o}ffler, Maarten and Meijer, Gert G.T. and Saumell, Maria and Silveira, Rodrigo I. and Staals, Frank}, title = {{Terrain Prickliness: Theoretical Grounds for High Complexity Viewsheds}}, booktitle = {11th International Conference on Geographic Information Science (GIScience 2021) - Part II}, pages = {10:1--10:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-208-2}, ISSN = {1868-8969}, year = {2021}, volume = {208}, editor = {Janowicz, Krzysztof and Verstegen, Judith A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GIScience.2021.II.10}, URN = {urn:nbn:de:0030-drops-147697}, doi = {10.4230/LIPIcs.GIScience.2021.II.10}, annote = {Keywords: Digital elevation model, Triangulated irregular network, Viewshed complexity} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We study augmenting a plane Euclidean network with a segment, called shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Questions of this type have received considerable attention recently, mostly for discrete variants of the problem. We study a fully continuous setting, where all points on the network and the inserted segment must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model, together with several results for networks that are paths, restricted to two types of shortcuts: shortcuts with a fixed orientation and simple shortcuts.

Delia Garijo, Alberto Márquez, Natalia Rodríguez, and Rodrigo I. Silveira. Computing Optimal Shortcuts for Networks. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 15:1-15:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{garijo_et_al:LIPIcs.ISAAC.2018.15, author = {Garijo, Delia and M\'{a}rquez, Alberto and Rodr{\'\i}guez, Natalia and Silveira, Rodrigo I.}, title = {{Computing Optimal Shortcuts for Networks}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {15:1--15:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.15}, URN = {urn:nbn:de:0030-drops-99634}, doi = {10.4230/LIPIcs.ISAAC.2018.15}, annote = {Keywords: graph augmentation, shortcut, diameter, geometric graph} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

In memory-constrained algorithms we have read-only access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space bottleneck is a stack into memory-constrained algorithms. Given an algorithm A that runs in O(n) time using a stack of length Theta(n), we can modify it so that it runs in O(n^2/2^s) time using a workspace of O(s) variables (for any s \in o(log n)) or O(n log n/log p)$ time using O(p log n/log p) variables (for any 2 <= p <= n). We also show how the technique can be applied to solve various geometric problems, namely computing the convex hull of a simple polygon, a triangulation of a monotone polygon, the shortest path between two points inside a monotone polygon, 1-dimensional pyramid approximation of a 1-dimensional vector, and the visibility profile of a point inside a simple polygon. Our approach exceeds or matches the best-known results for these problems in constant-workspace models (when they exist), and gives a trade-off between the size of the workspace and running time. To the best of our knowledge, this is the first general framework for obtaining memory-constrained algorithms.

Luis Barba, Matias Korman, Stefan Langerman, Rodrigo I. Silveira, and Kunihiko Sadakane. Space-Time Trade-offs for Stack-Based Algorithms. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 281-292, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{barba_et_al:LIPIcs.STACS.2013.281, author = {Barba, Luis and Korman, Matias and Langerman, Stefan and Silveira, Rodrigo I. and Sadakane, Kunihiko}, title = {{Space-Time Trade-offs for Stack-Based Algorithms}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {281--292}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.281}, URN = {urn:nbn:de:0030-drops-39411}, doi = {10.4230/LIPIcs.STACS.2013.281}, annote = {Keywords: space-time trade-off, constant workspace, stack algorithms} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10491, Representation, Analysis and Visualization of Moving Objects (2011)

In the group discussions we discussed distance measures focussing on real world applications specifically on domain areas where trajectories have been generated by animals (birds, primates...) and humans in urban areas.

Joachim Gudmundsson, Harvey Miller, Rodrigo Silveira, Mathias Versichele, and Stefan van der Spek. 10491 Results of the break-out group: Similarity measures. In Representation, Analysis and Visualization of Moving Objects. Dagstuhl Seminar Proceedings, Volume 10491, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{gudmundsson_et_al:DagSemProc.10491.7, author = {Gudmundsson, Joachim and Miller, Harvey and Silveira, Rodrigo and Versichele, Mathias and van der Spek, Stefan}, title = {{10491 Results of the break-out group: Similarity measures}}, booktitle = {Representation, Analysis and Visualization of Moving Objects}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2011}, volume = {10491}, editor = {J\"{o}rg-R\"{u}diger Sack and Bettina Speckmann and Emiel Van Loon and Robert Weibel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10491.7}, URN = {urn:nbn:de:0030-drops-29893}, doi = {10.4230/DagSemProc.10491.7}, annote = {Keywords: Similarity, Movement Analysis} }