6 Search Results for "Haverkort, Herman"


Document
Engineering A* Search for the Flip Distance of Plane Triangulations

Authors: Philip Mayer and Petra Mutzel

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)


Abstract
The flip distance for two triangulations of a point set is defined as the smallest number of edge flips needed to transform one triangulation into another, where an edge flip is the act of replacing an edge of a triangulation by a different edge such that the result remains a triangulation. We adapt and engineer a sophisticated A* search algorithm acting on the so-called flip graph. In particular, we prove that previously proposed lower bounds for the flip distance form consistent heuristics for A* and show that they can be computed efficiently using dynamic algorithms. As an alternative approach, we present an integer linear program (ILP) for the flip distance problem. We experimentally evaluate our approaches on a new real-world benchmark data set based on an application in geodesy, namely sea surface reconstruction. Our evaluation reveals that A* search consistently outperforms our ILP formulation as well as a naive baseline, which is bidirectional breadth-first search. In particular, the runtime of our approach improves upon the baseline by more than two orders of magnitude. Furthermore, our A* search successfully solves most of the considered sea surface instances with up to 41 points. This is a substantial improvement compared to the baseline, which struggles with subsets of the real-world data of size 25. Lastly, to allow the consideration of global sea level data, we developed a decomposition-based heuristic for the flip distance. In our experiments it yields optimal flip distance values for most of the considered sea level data and it can be applied to large data sets due to its fast runtime.

Cite as

Philip Mayer and Petra Mutzel. Engineering A* Search for the Flip Distance of Plane Triangulations. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{mayer_et_al:LIPIcs.SEA.2024.23,
  author =	{Mayer, Philip and Mutzel, Petra},
  title =	{{Engineering A* Search for the Flip Distance of Plane Triangulations}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{23:1--23:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.23},
  URN =		{urn:nbn:de:0030-drops-203887},
  doi =		{10.4230/LIPIcs.SEA.2024.23},
  annote =	{Keywords: Computational Geometry, Triangulations, Flip Distance, A-star Search, Integer Linear Programming}
}
Document
Minimum-Error Triangulations for Sea Surface Reconstruction

Authors: Anna Arutyunova, Anne Driemel, Jan-Henrik Haunert, Herman Haverkort, Jürgen Kusche, Elmar Langetepe, Philip Mayer, Petra Mutzel, and Heiko Röglin

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


Abstract
We apply state-of-the-art computational geometry methods to the problem of reconstructing a time-varying sea surface from tide gauge records. Our work builds on a recent article by Nitzke et al. (Computers & Geosciences, 157:104920, 2021) who have suggested to learn a triangulation D of a given set of tide gauge stations. The objective is to minimize the misfit of the piecewise linear surface induced by D to a reference surface that has been acquired with satellite altimetry. The authors restricted their search to k-order Delaunay (k-OD) triangulations and used an integer linear program in order to solve the resulting optimization problem. In geometric terms, the input to our problem consists of two sets of points in ℝ² with elevations: a set 𝒮 that is to be triangulated, and a set ℛ of reference points. Intuitively, we define the error of a triangulation as the average vertical distance of a point in ℛ to the triangulated surface that is obtained by interpolating elevations of 𝒮 linearly in each triangle. Our goal is to find the triangulation of 𝒮 that has minimum error with respect to ℛ. In our work, we prove that the minimum-error triangulation problem is NP-hard and cannot be approximated within any multiplicative factor in polynomial time unless P = NP. At the same time we show that the problem instances that occur in our application (considering sea level data from several hundreds of tide gauge stations worldwide) can be solved relatively fast using dynamic programming when restricted to k-OD triangulations for k ≤ 7. In particular, instances for which the number of connected components of the so-called k-OD fixed-edge graph is small can be solved within few seconds.

Cite as

Anna Arutyunova, Anne Driemel, Jan-Henrik Haunert, Herman Haverkort, Jürgen Kusche, Elmar Langetepe, Philip Mayer, Petra Mutzel, and Heiko Röglin. Minimum-Error Triangulations for Sea Surface Reconstruction. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{arutyunova_et_al:LIPIcs.SoCG.2022.7,
  author =	{Arutyunova, Anna and Driemel, Anne and Haunert, Jan-Henrik and Haverkort, Herman and Kusche, J\"{u}rgen and Langetepe, Elmar and Mayer, Philip and Mutzel, Petra and R\"{o}glin, Heiko},
  title =	{{Minimum-Error Triangulations for Sea Surface Reconstruction}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{7:1--7:18},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.7},
  URN =		{urn:nbn:de:0030-drops-160155},
  doi =		{10.4230/LIPIcs.SoCG.2022.7},
  annote =	{Keywords: Minimum-Error Triangulation, k-Order Delaunay Triangulations, Data dependent Triangulations, Sea Surface Reconstruction, fixed-Edge Graph}
}
Document
Bicriteria Aggregation of Polygons via Graph Cuts

Authors: Peter Rottmann, Anne Driemel, Herman Haverkort, Heiko Röglin, and Jan-Henrik Haunert

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


Abstract
We present a new method for the task of detecting groups of polygons in a given geographic data set and computing a representative polygon for each group. This task is relevant in map generalization where the aim is to derive a less detailed map from a given map. Following a classical approach, we define the output polygons by merging the input polygons with a set of triangles that we select from a constrained Delaunay triangulation of the input polygons' exterior. The innovation of our method is to compute the selection of triangles by solving a bicriteria optimization problem. While on the one hand we aim at minimizing the total area of the outputs polygons, we aim on the other hand at minimizing their total perimeter. We combine these two objectives in a weighted sum and study two computational problems that naturally arise. In the first problem, the parameter that balances the two objectives is fixed and the aim is to compute a single optimal solution. In the second problem, the aim is to compute a set containing an optimal solution for every possible value of the parameter. We present efficient algorithms for these problems based on computing a minimum cut in an appropriately defined graph. Moreover, we show how the result set of the second problem can be approximated with few solutions. In an experimental evaluation, we finally show that the method is able to derive settlement areas from building footprints that are similar to reference solutions.

Cite as

Peter Rottmann, Anne Driemel, Herman Haverkort, Heiko Röglin, and Jan-Henrik Haunert. Bicriteria Aggregation of Polygons via Graph Cuts. In 11th International Conference on Geographic Information Science (GIScience 2021) - Part II. Leibniz International Proceedings in Informatics (LIPIcs), Volume 208, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{rottmann_et_al:LIPIcs.GIScience.2021.II.6,
  author =	{Rottmann, Peter and Driemel, Anne and Haverkort, Herman and R\"{o}glin, Heiko and Haunert, Jan-Henrik},
  title =	{{Bicriteria Aggregation of Polygons via Graph Cuts}},
  booktitle =	{11th International Conference on Geographic Information Science (GIScience 2021) - Part II},
  pages =	{6:1--6:16},
  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.6},
  URN =		{urn:nbn:de:0030-drops-147658},
  doi =		{10.4230/LIPIcs.GIScience.2021.II.6},
  annote =	{Keywords: map generalization, aggregation, graph cuts, bicriteria optimization}
}
Document
Media Exposition
Plane-Filling Trails (Media Exposition)

Authors: Herman Haverkort

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)


Abstract
The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to be close to each other in the plane, and vice versa. However, sketches of plane-filling curves do not show this well: they are hard to read on different levels of detail and it is hard to see how far apart points are along the curve. This paper presents a software tool to produce compelling visualisations that may give more insight in the structure of the curves.

Cite as

Herman Haverkort. Plane-Filling Trails (Media Exposition). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 81:1-81:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{haverkort:LIPIcs.SoCG.2020.81,
  author =	{Haverkort, Herman},
  title =	{{Plane-Filling Trails}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{81:1--81:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.81},
  URN =		{urn:nbn:de:0030-drops-122396},
  doi =		{10.4230/LIPIcs.SoCG.2020.81},
  annote =	{Keywords: space-filling curve, plane-filling curve, spatial indexing}
}
Document
Hyperorthogonal Well-Folded Hilbert Curves

Authors: Arie Bos and Herman J. Haverkort

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes - smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor Omega(2^{d/2}) larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.

Cite as

Arie Bos and Herman J. Haverkort. Hyperorthogonal Well-Folded Hilbert Curves. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 812-826, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{bos_et_al:LIPIcs.SOCG.2015.812,
  author =	{Bos, Arie and Haverkort, Herman J.},
  title =	{{Hyperorthogonal Well-Folded Hilbert Curves}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{812--826},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.812},
  URN =		{urn:nbn:de:0030-drops-50962},
  doi =		{10.4230/LIPIcs.SOCG.2015.812},
  annote =	{Keywords: space-filling curve, Hilbert curve, multi-dimensional, range query, R-tree}
}
Document
The Priority R-Tree: A Practically Efficient and Worst-Case-Optimal R-Tree

Authors: Lars Arge, Mark de Berg, Herman J. Haverkort, and Ke Yi

Published in: Dagstuhl Seminar Proceedings, Volume 4301, Cache-Oblivious and Cache-Aware Algorithms (2005)


Abstract
The query efficiency of a data structure that stores a set of objects, can normally be assessed by analysing the number of objects, pointers etc. looked at when answering a query. However, if the data structure is too big to fit in main memory, data may need to be fetched from disk. In that case, the query efficiency is easily dominated by moving the disk head to the correct locations, rather than by reading the data itself. To reduce the number of disk accesses, once can group the data into blocks, and strive to bound the number of different blocks accessed rather than the number of individual data objects read. An R-tree is a general-purpose data structur that stores a hierarchical grouping of geometric objects into blocks. Many heuristics have been designed to determine which objects should be grouped together, but none of these heuristics could give a guarantee on the resulting worst-case query time. We present the Priority R-tree, or PR-tree, which is the first R-tree variant that always answers a window query by accessing $O((N/B)^{1-1/d} + T/B)$ blocks, where $N$ is the number of $d$-dimensional objects stored, $B$ is the number of objects per block, and $T$ is the number of objects whose bounding boxes intersect the query window. This is provably asymptotically optimal. Experiments show that the PR-tree performs similar to the best known heuristics on real-life and relatively nicely distributed data, but outperforms them significantly on more extreme data.

Cite as

Lars Arge, Mark de Berg, Herman J. Haverkort, and Ke Yi. The Priority R-Tree: A Practically Efficient and Worst-Case-Optimal R-Tree. In Cache-Oblivious and Cache-Aware Algorithms. Dagstuhl Seminar Proceedings, Volume 4301, pp. 1-26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)


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@InProceedings{arge_et_al:DagSemProc.04301.3,
  author =	{Arge, Lars and de Berg, Mark and Haverkort, Herman J. and Yi, Ke},
  title =	{{The Priority R-Tree: A Practically Efficient and Worst-Case-Optimal R-Tree}},
  booktitle =	{Cache-Oblivious and Cache-Aware Algorithms},
  pages =	{1--26},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4301},
  editor =	{Lars Arge and Michael A. Bender and Erik Demaine and Charles Leiserson and Kurt Mehlhorn},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04301.3},
  URN =		{urn:nbn:de:0030-drops-1554},
  doi =		{10.4230/DagSemProc.04301.3},
  annote =	{Keywords: R-Trees}
}
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