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Documents authored by Nusser, André

Document
DOI: 10.4230/LIPIcs.SEA.2025.10
Algorithm Engineering of SSSP with Negative Edge Weights

Authors: Alejandro Cassis, Andreas Karrenbauer, André Nusser, and Paolo Luigi Rinaldi

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

Abstract
Computing shortest paths is one of the most fundamental algorithmic graph problems. It is known since decades that this problem can be solved in near-linear time if all weights are nonnegative. A recent break-through by [Aaron Bernstein et al., 2022] presented a randomized near-linear time algorithm for this problem. A subsequent improvement in [Karl Bringmann et al., 2023] significantly reduced the number of logarithmic factors and thereby also simplified the algorithm. It is surprising and exciting that both of these algorithms are combinatorial and do not contain any fundamental obstacles for being practical. We launch the, to the best of our knowledge, first extensive investigation towards a practical implementation of [Karl Bringmann et al., 2023]. To this end, we give an accessible overview of the algorithm and discuss what adaptions are necessary to obtain a fast algorithm in practice. We manifest these adaptions in an efficient implementation. We test our implementation on a benchmark data set that is adapted to be more difficult for our implementation in order to allow for a fair comparison. As in [Karl Bringmann et al., 2023] as well as in our implementation there are multiple parameters to tune, we empirically evaluate their effect and thereby determine the best choices. Our implementation is then extensively compared to one of the state-of-the-art algorithms for this problem [Andrew V. Goldberg and Tomasz Radzik, 1993]. On the hardest instance type, we are faster by up to almost two orders of magnitude.
Cite as

Alejandro Cassis, Andreas Karrenbauer, André Nusser, and Paolo Luigi Rinaldi. Algorithm Engineering of SSSP with Negative Edge Weights. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cassis_et_al:LIPIcs.SEA.2025.10,
  author =	{Cassis, Alejandro and Karrenbauer, Andreas and Nusser, Andr\'{e} and Rinaldi, Paolo Luigi},
  title =	{{Algorithm Engineering of SSSP with Negative Edge Weights}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.10},
  URN =		{urn:nbn:de:0030-drops-232486},
  doi =		{10.4230/LIPIcs.SEA.2025.10},
  annote =	{Keywords: Single Source Shortest Paths, Negative Weights, Near-Linear Time}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.36
Faster Fréchet Distance Under Transformations

Authors: Kevin Buchin, Maike Buchin, Zijin Huang, André Nusser, and Sampson Wong

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the problem of computing the Fréchet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves π and σ of total complexity n and a threshold δ ≥ 0, we present an 𝒪̃(n^{7 + 1/3}) time algorithm to determine whether there exists a translation t ∈ ℝ² such that the Fréchet distance between π and σ + t is at most δ. This improves on the previous best result, which is an 𝒪(n⁸) time algorithm. We then generalize this result to any class of rationally parameterized transformations, which includes translation, rotation, scaling, and arbitrary affine transformations. For a class T of rationally parametrized transformations with k degrees of freedom, we show that one can determine whether there is a transformation τ ∈ T such that the Fréchet distance between π and τ(σ) is at most δ in 𝒪̃(n^{3k+4/3}) time.
Cite as

Kevin Buchin, Maike Buchin, Zijin Huang, André Nusser, and Sampson Wong. Faster Fréchet Distance Under Transformations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{buchin_et_al:LIPIcs.ICALP.2025.36,
  author =	{Buchin, Kevin and Buchin, Maike and Huang, Zijin and Nusser, Andr\'{e} and Wong, Sampson},
  title =	{{Faster Fr\'{e}chet Distance Under Transformations}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{36:1--36:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.36},
  URN =		{urn:nbn:de:0030-drops-234137},
  doi =		{10.4230/LIPIcs.ICALP.2025.36},
  annote =	{Keywords: Fr\'{e}chet distance, curve similarity, shape matching}
}
Artifact
Software
DOI: 10.4230/artifacts.23288
nonObtuseTri

Authors: Mikkel Abrahamsen, Florestan Brunck, Jacobus Conradi, Benedikt Kolbe, and André Nusser

Abstract
Cite as

Mikkel Abrahamsen, Florestan Brunck, Jacobus Conradi, Benedikt Kolbe, André Nusser. nonObtuseTri (Software, Source Code). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@misc{dagstuhl-artifact-23288,
   title = {{nonObtuseTri}}, 
   author = {Abrahamsen, Mikkel and Brunck, Florestan and Conradi, Jacobus and Kolbe, Benedikt and Nusser, Andr\'{e}},
   note = {Software, version v3.0., swhId: \href{https://archive.softwareheritage.org/swh:1:dir:b6301970ab626c084f309f9b5e6b91e9acf24949;origin=https://github.com/JacobusTheSecond/nonObtuseTri;visit=swh:1:snp:4b7f079e3cd897cb0826392670078921ce262ba2;anchor=swh:1:rev:bd1842a4272b4e01ad623cf6bb02c7617c3da98a}{\texttt{swh:1:dir:b6301970ab626c084f309f9b5e6b91e9acf24949}} (visited on 2025-06-20)},
   url = {https://github.com/JacobusTheSecond/nonObtuseTri},
   doi = {10.4230/artifacts.23288},
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.22
Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D

Authors: Lotte Blank, Jacobus Conradi, Anne Driemel, Benedikt Kolbe, André Nusser, and Marena Richter

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The Fréchet distance is a computational mainstay for comparing polygonal curves. The Fréchet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in space. It is defined as the minimum Fréchet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fréchet distance under some transformations have been studied, with more work concentrating on the discrete Fréchet distance, leaving a significant gap between the discrete and continuous versions of the Fréchet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fréchet distance under translation on 1-dimensional curves of complexity n with a running time of 𝒪(n^{8/3} log³ n). To achieve this, we develop a novel framework for the problem for 1-dimensional curves, which also applies to other scenarios and leads to our second contribution. We present an algorithm with the same running time of 𝒪(n^{8/3} log³ n) for the Fréchet distance under scaling for 1-dimensional curves. For both algorithms we match the running times of the discrete case and improve the previously best known bounds of 𝒪̃(n⁴). Our algorithms rely on technical insights but are conceptually simple, essentially reducing the continuous problem to the discrete case across different length scales.
Cite as

Lotte Blank, Jacobus Conradi, Anne Driemel, Benedikt Kolbe, André Nusser, and Marena Richter. Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blank_et_al:LIPIcs.SoCG.2025.22,
  author =	{Blank, Lotte and Conradi, Jacobus and Driemel, Anne and Kolbe, Benedikt and Nusser, Andr\'{e} and Richter, Marena},
  title =	{{Transforming Dogs on the Line: On the Fr\'{e}chet Distance Under Translation or Scaling in 1D}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{22:1--22:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.22},
  URN =		{urn:nbn:de:0030-drops-231746},
  doi =		{10.4230/LIPIcs.SoCG.2025.22},
  annote =	{Keywords: Fr\'{e}chet distance under translation, Fr\'{e}chet distance under scaling, time series, shape matching}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.25
Approximating Klee’s Measure Problem and a Lower Bound for Union Volume Estimation

Authors: Karl Bringmann, Kasper Green Larsen, André Nusser, Eva Rotenberg, and Yanheng Wang

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Union volume estimation is a classical algorithmic problem. Given a family of objects O₁,…,O_n ⊂ ℝ^d, we want to approximate the volume of their union. In the special case where all objects are boxes (also called hyperrectangles) this is known as Klee’s measure problem. The state-of-the-art (1+ε)-approximation algorithm [Karp, Luby, Madras '89] for union volume estimation as well as Klee’s measure problem in constant dimension d uses a total of O(n/ε²) queries of three types: (i) determine the volume of O_i; (ii) sample a point uniformly at random from O_i; and (iii) ask whether a given point is contained in O_i. First, we show that if an algorithm learns about the objects only through these types of queries, then Ω(n/ε²) queries are necessary. In this sense, the complexity of [Karp, Luby, Madras '89] is optimal. Our lower bound holds even if the objects are equiponderous axis-aligned polygons in ℝ², if the containment query allows arbitrary (not necessarily sampled) points, and if the algorithm can spend arbitrary time and space examining the query responses. Second, we provide a more efficient approximation algorithm for Klee’s measure problem, which improves the running time from O(n/ε²) to O((n+1/ε²) ⋅ log^{O(d)} (n)). We circumvent our lower bound by exploiting the geometry of boxes in various ways: (1) We sort the boxes into classes of similar shapes after inspecting their corner coordinates. (2) With orthogonal range searching, we show how to sample points from the union of boxes in each class, and how to merge samples from different classes. (3) We bound the amount of wasted work by arguing that most pairs of classes have a small intersection.
Cite as

Karl Bringmann, Kasper Green Larsen, André Nusser, Eva Rotenberg, and Yanheng Wang. Approximating Klee’s Measure Problem and a Lower Bound for Union Volume Estimation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2025.25,
  author =	{Bringmann, Karl and Larsen, Kasper Green and Nusser, Andr\'{e} and Rotenberg, Eva and Wang, Yanheng},
  title =	{{Approximating Klee’s Measure Problem and a Lower Bound for Union Volume Estimation}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{25:1--25:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.25},
  URN =		{urn:nbn:de:0030-drops-231778},
  doi =		{10.4230/LIPIcs.SoCG.2025.25},
  annote =	{Keywords: approximation, volume of union, union of objects, query complexity}
}
Document
CG Challenge
DOI: 10.4230/LIPIcs.SoCG.2025.79
Computing Non-Obtuse Triangulations with Few Steiner Points (CG Challenge)

Authors: Mikkel Abrahamsen, Florestan Brunck, Jacobus Conradi, Benedikt Kolbe, and André Nusser

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We present the winning implementation of the Seventh Computational Geometry Challenge (CG:SHOP 2025). The task in this challenge was to find non-obtuse triangulations for given planar regions, respecting a given set of constraints consisting of extra vertices and edges that must be part of the triangulation. The goal was to minimize the number of introduced Steiner points. Our approach is to maintain a constrained Delaunay triangulation, for which we repeatedly remove, relocate, or add Steiner points. We use local search to choose the action that improves the triangulation the most, until the resulting triangulation is non-obtuse.
Cite as

Mikkel Abrahamsen, Florestan Brunck, Jacobus Conradi, Benedikt Kolbe, and André Nusser. Computing Non-Obtuse Triangulations with Few Steiner Points (CG Challenge). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 79:1-79:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2025.79,
  author =	{Abrahamsen, Mikkel and Brunck, Florestan and Conradi, Jacobus and Kolbe, Benedikt and Nusser, Andr\'{e}},
  title =	{{Computing Non-Obtuse Triangulations with Few Steiner Points}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{79:1--79:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.79},
  URN =		{urn:nbn:de:0030-drops-232311},
  doi =		{10.4230/LIPIcs.SoCG.2025.79},
  annote =	{Keywords: non-obtuse triangulation, local search, competition}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.63
Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs

Authors: Ivor van der Hoog, André Nusser, Eva Rotenberg, and Frank Staals

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
A classical problem in computational geometry and graph algorithms is: given a dynamic set 𝒮 of geometric shapes in the plane, efficiently maintain the connectivity of the intersection graph of 𝒮. Previous papers studied the setting where, before the updates, the data structure receives some parameter P. Then, updates could insert and delete disks as long as at all times the disks have a diameter that lies in a fixed range [1/P, 1]. As a consequence of that prerequisite, the aspect ratio ψ (i.e. the ratio between the largest and smallest diameter) of the disks would at all times satisfy ψ ≤ P. The state-of-the-art for storing disks in a dynamic connectivity data structure is a data structure that uses O(Pn) space and that has amortized O(P log⁴ n) expected amortized update time. Connectivity queries between disks are supported in O(log n / log log n) time. In the dynamic setting, one wishes for a more flexible data structure in which disks of any diameter may arrive and leave, independent of their diameter, changing the aspect ratio freely. Ideally, the aspect ratio should merely be part of the analysis. We restrict our attention to axis-aligned squares, and study fully-dynamic square intersection graph connectivity. Our result is fully-adaptive to the aspect ratio, spending time proportional to the current aspect ratio ψ, as opposed to some previously given maximum P. Our focus on squares allows us to simplify and streamline the connectivity pipeline from previous work. When n is the number of squares and ψ is the aspect ratio after insertion (or before deletion), our data structure answers connectivity queries in O(log n / log log n) time. We can update connectivity information in O(ψ log⁴ n + log⁶ n) amortized time. We also improve space usage from O(P ⋅ n log n) to O(n log³ n log ψ) - while generalizing to a fully-adaptive aspect ratio - which yields a space usage that is near-linear in n for any polynomially bounded ψ.
Cite as

Ivor van der Hoog, André Nusser, Eva Rotenberg, and Frank Staals. Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 63:1-63:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vanderhoog_et_al:LIPIcs.MFCS.2024.63,
  author =	{van der Hoog, Ivor and Nusser, Andr\'{e} and Rotenberg, Eva and Staals, Frank},
  title =	{{Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{63:1--63:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.63},
  URN =		{urn:nbn:de:0030-drops-206197},
  doi =		{10.4230/LIPIcs.MFCS.2024.63},
  annote =	{Keywords: Computational geometry, planar geometry, data structures, geometric intersection graphs, fully-dynamic algorithms}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.2
Clustering with Few Disks to Minimize the Sum of Radii

Authors: Mikkel Abrahamsen, Sarita de Berg, Lucas Meijer, André Nusser, and Leonidas Theocharous

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
Given a set of n points in the Euclidean plane, the k-MinSumRadius problem asks to cover this point set using k disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV '12]; however, the running time of this algorithm is 𝒪(n^881), and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the k-MinSumRadius problem is that of small k. For the 2-MinSumRadius problem, a near-quadratic time algorithm with expected running time 𝒪(n² log² n log² log n) was given over 30 years ago [Eppstein '92]. We present the first improvement of this result, namely, a near-linear time algorithm to compute the 2-MinSumRadius that runs in expected 𝒪(n log² n log² log n) time. We generalize this result to any constant dimension d, for which we give an 𝒪(n^{2-1/(⌈d/2⌉ + 1) + ε}) time algorithm. Additionally, we give a near-quadratic time algorithm for 3-MinSumRadius in the plane that runs in expected 𝒪(n² log² n log² log n) time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution.
Cite as

Mikkel Abrahamsen, Sarita de Berg, Lucas Meijer, André Nusser, and Leonidas Theocharous. Clustering with Few Disks to Minimize the Sum of Radii. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2024.2,
  author =	{Abrahamsen, Mikkel and de Berg, Sarita and Meijer, Lucas and Nusser, Andr\'{e} and Theocharous, Leonidas},
  title =	{{Clustering with Few Disks to Minimize the Sum of Radii}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{2:1--2:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.2},
  URN =		{urn:nbn:de:0030-drops-199472},
  doi =		{10.4230/LIPIcs.SoCG.2024.2},
  annote =	{Keywords: geometric clustering, minimize sum of radii, covering points with disks}
}
Document
CG Challenge
DOI: 10.4230/LIPIcs.SoCG.2023.66
Constructing Concise Convex Covers via Clique Covers (CG Challenge)

Authors: Mikkel Abrahamsen, William Bille Meyling, and André Nusser

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

Abstract
This work describes the winning implementation of the CG:SHOP 2023 Challenge. The topic of the Challenge was the convex cover problem: given a polygon P (with holes), find a minimum-cardinality set of convex polygons whose union equals P. We use a three-step approach: (1) Create a suitable partition of P. (2) Compute a visibility graph of the pieces of the partition. (3) Solve a vertex clique cover problem on the visibility graph, from which we then derive the convex cover. This way we capture the geometric difficulty in the first step and the combinatorial difficulty in the third step.
Cite as

Mikkel Abrahamsen, William Bille Meyling, and André Nusser. Constructing Concise Convex Covers via Clique Covers (CG Challenge). In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 66:1-66:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2023.66,
  author =	{Abrahamsen, Mikkel and Bille Meyling, William and Nusser, Andr\'{e}},
  title =	{{Constructing Concise Convex Covers via Clique Covers}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{66:1--66:9},
  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.2023.66},
  URN =		{urn:nbn:de:0030-drops-179164},
  doi =		{10.4230/LIPIcs.SoCG.2023.66},
  annote =	{Keywords: Convex cover, Polygons with holes, Algorithm engineering, Vertex clique cover}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.20
Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves

Authors: Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, Dániel Marx, and André Nusser

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

Abstract
The Dynamic Time Warping (DTW) distance is a popular measure of similarity for a variety of sequence data. For comparing polygonal curves π, σ in ℝ^d, it provides a robust, outlier-insensitive alternative to the Fréchet distance. However, like the Fréchet distance, the DTW distance is not invariant under translations. Can we efficiently optimize the DTW distance of π and σ under arbitrary translations, to compare the curves' shape irrespective of their absolute location? There are surprisingly few works in this direction, which may be due to its computational intricacy: For the Euclidean norm, this problem contains as a special case the geometric median problem, which provably admits no exact algebraic algorithm (that is, no algorithm using only addition, multiplication, and k-th roots). We thus investigate exact algorithms for non-Euclidean norms as well as approximation algorithms for the Euclidean norm. For the L₁ norm in ℝ^d, we provide an 𝒪(n^{2(d+1)})-time algorithm, i.e., an exact polynomial-time algorithm for constant d. Here and below, n bounds the curves' complexities. For the Euclidean norm in ℝ², we show that a simple problem-specific insight leads to a (1+ε)-approximation in time 𝒪(n³/ε²). We then show how to obtain a subcubic 𝒪̃(n^{2.5}/ε²) time algorithm with significant new ideas; this time comes close to the well-known quadratic time barrier for computing DTW for fixed translations. Technically, the algorithm is obtained by speeding up repeated DTW distance estimations using a dynamic data structure for maintaining shortest paths in weighted planar digraphs. Crucially, we show how to traverse a candidate set of translations using space-filling curves in a way that incurs only few updates to the data structure. We hope that our results will facilitate the use of DTW under translation both in theory and practice, and inspire similar algorithmic approaches for related geometric optimization problems.
Cite as

Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, Dániel Marx, and André Nusser. Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2022.20,
  author =	{Bringmann, Karl and Kisfaludi‑Bak, S\'{a}ndor and K\"{u}nnemann, Marvin and Marx, D\'{a}niel and Nusser, Andr\'{e}},
  title =	{{Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{20:1--20:17},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.20},
  URN =		{urn:nbn:de:0030-drops-160287},
  doi =		{10.4230/LIPIcs.SoCG.2022.20},
  annote =	{Keywords: Dynamic Time Warping, Sequence Similarity Measures}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.21
Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs

Authors: Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, André Nusser, and Zahra Parsaeian

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

Abstract
We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in d-dimensional Euclidean space, such as balls, segments, or hypercubes, and whose edges correspond to pairs of intersecting shapes. The diameter of a graph is the largest distance realized by a pair of vertices in the graph. Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since in the related class of planar graphs the diameter can be computed in 𝒪̃(n^{5/3}) time [Cabello 2019, Gawrychowski et al. 2021]. In this work we (conditionally) rule out sub-quadratic algorithms in several classes of intersection graphs, i.e., algorithms of running time 𝒪(n^{2-δ}) for some δ > 0. In particular, there are no sub-quadratic algorithms already for fat objects in small dimensions: unit balls in ℝ³ or congruent equilateral triangles in ℝ². For unit segments and congruent equilateral triangles, we can even rule out strong sub-quadratic approximations already in ℝ². It seems that the hardness of approximation may also depend on dimensionality: for axis-parallel unit hypercubes in ℝ^{12}, distinguishing between diameter 2 and 3 needs quadratic time (ruling out (3/2-ε)- approximations), whereas for axis-parallel unit squares, we give an algorithm that distinguishes between diameter 2 and 3 in near-linear time. Note that many of our lower bounds match the best known algorithms up to sub-polynomial factors. Ultimately, this fine-grained perspective may enable us to determine for which shapes we can have efficient algorithms and approximation schemes for diameter computation.
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Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, André Nusser, and Zahra Parsaeian. Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2022.21,
  author =	{Bringmann, Karl and Kisfaludi‑Bak, S\'{a}ndor and K\"{u}nnemann, Marvin and Nusser, Andr\'{e} and Parsaeian, Zahra},
  title =	{{Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{21:1--21:16},
  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.2022.21},
  URN =		{urn:nbn:de:0030-drops-160294},
  doi =		{10.4230/LIPIcs.SoCG.2022.21},
  annote =	{Keywords: Hardness in P, Geometric Intersection Graph, Graph Diameter, Orthogonal Vectors, Hyperclique Detection}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.22
Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time

Authors: Kevin Buchin, André Nusser, and Sampson Wong

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

Abstract
Dynamic Time Warping is arguably the most popular similarity measure for time series, where we define a time series to be a one-dimensional polygonal curve. The drawback of Dynamic Time Warping is that it is sensitive to the sampling rate of the time series. The Fréchet distance is an alternative that has gained popularity, however, its drawback is that it is sensitive to outliers. Continuous Dynamic Time Warping (CDTW) is a recently proposed alternative that does not exhibit the aforementioned drawbacks. CDTW combines the continuous nature of the Fréchet distance with the summation of Dynamic Time Warping, resulting in a similarity measure that is robust to sampling rate and to outliers. In a recent experimental work of Brankovic et al., it was demonstrated that clustering under CDTW avoids the unwanted artifacts that appear when clustering under Dynamic Time Warping and under the Fréchet distance. Despite its advantages, the major shortcoming of CDTW is that there is no exact algorithm for computing CDTW, in polynomial time or otherwise. In this work, we present the first exact algorithm for computing CDTW of one-dimensional curves. Our algorithm runs in time 𝒪(n⁵) for a pair of one-dimensional curves, each with complexity at most n. In our algorithm, we propagate continuous functions in the dynamic program for CDTW, where the main difficulty lies in bounding the complexity of the functions. We believe that our result is an important first step towards CDTW becoming a practical similarity measure between curves.
Cite as

Kevin Buchin, André Nusser, and Sampson Wong. Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2022.22,
  author =	{Buchin, Kevin and Nusser, Andr\'{e} and Wong, Sampson},
  title =	{{Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{22:1--22:16},
  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.2022.22},
  URN =		{urn:nbn:de:0030-drops-160307},
  doi =		{10.4230/LIPIcs.SoCG.2022.22},
  annote =	{Keywords: Computational Geometry, Curve Similarity, Fr\'{e}chet distance, Dynamic Time Warping, Continuous Dynamic Time Warping}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.18
Translating Hausdorff Is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation

Authors: Karl Bringmann and André Nusser

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Computing the similarity of two point sets is a ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance, which assigns to each point from one set the closest point in the other set and then evaluates the maximum distance of any assigned pair. A drawback is that this distance measure is not translational invariant, that is, comparing two objects just according to their shape while disregarding their position in space is impossible. Fortunately, there is a canonical translational invariant version, the Hausdorff distance under translation, which minimizes the Hausdorff distance over all translations of one of the point sets. For point sets of size n and m, the Hausdorff distance under translation can be computed in time 𝒪̃(nm) for the L₁ and L_∞ norm [Chew, Kedem SWAT'92] and 𝒪̃(nm (n+m)) for the L₂ norm [Huttenlocher, Kedem, Sharir DCG'93]. As these bounds have not been improved for over 25 years, in this paper we approach the Hausdorff distance under translation from the perspective of fine-grained complexity theory. We show (i) a matching lower bound of (nm)^{1-o(1)} for L₁ and L_∞ assuming the Orthogonal Vectors Hypothesis and (ii) a matching lower bound of n^{2-o(1)} for L₂ in the imbalanced case of m = 𝒪(1) assuming the 3SUM Hypothesis.
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Karl Bringmann and André Nusser. Translating Hausdorff Is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2021.18,
  author =	{Bringmann, Karl and Nusser, Andr\'{e}},
  title =	{{Translating Hausdorff Is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.18},
  URN =		{urn:nbn:de:0030-drops-138177},
  doi =		{10.4230/LIPIcs.SoCG.2021.18},
  annote =	{Keywords: Hausdorff Distance Under Translation, Fine-Grained Complexity Theory, Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.25
When Lipschitz Walks Your Dog: Algorithm Engineering of the Discrete Fréchet Distance Under Translation

Authors: Karl Bringmann, Marvin Künnemann, and André Nusser

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
Consider the natural question of how to measure the similarity of curves in the plane by a quantity that is invariant under translations of the curves. Such a measure is justified whenever we aim to quantify the similarity of the curves' shapes rather than their positioning in the plane, e.g., to compare the similarity of handwritten characters. Perhaps the most natural such notion is the (discrete) Fréchet distance under translation. Unfortunately, the algorithmic literature on this problem yields a very pessimistic view: On polygonal curves with n vertices, the fastest algorithm runs in time 𝒪(n^4.667) and cannot be improved below n^{4-o(1)} unless the Strong Exponential Time Hypothesis fails. Can we still obtain an implementation that is efficient on realistic datasets? Spurred by the surprising performance of recent implementations for the Fréchet distance, we perform algorithm engineering for the Fréchet distance under translation. Our solution combines fast, but inexact tools from continuous optimization (specifically, branch-and-bound algorithms for global Lipschitz optimization) with exact, but expensive algorithms from computational geometry (specifically, problem-specific algorithms based on an arrangement construction). We combine these two ingredients to obtain an exact decision algorithm for the Fréchet distance under translation. For the related task of computing the distance value up to a desired precision, we engineer and compare different methods. On a benchmark set involving handwritten characters and route trajectories, our implementation answers a typical query for either task in the range of a few milliseconds up to a second on standard desktop hardware. We believe that our implementation will enable, for the first time, the use of the Fréchet distance under translation in applications, whereas previous algorithmic approaches would have been computationally infeasible. Furthermore, we hope that our combination of continuous optimization and computational geometry will inspire similar approaches for further algorithmic questions.
Cite as

Karl Bringmann, Marvin Künnemann, and André Nusser. When Lipschitz Walks Your Dog: Algorithm Engineering of the Discrete Fréchet Distance Under Translation. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bringmann_et_al:LIPIcs.ESA.2020.25,
  author =	{Bringmann, Karl and K\"{u}nnemann, Marvin and Nusser, Andr\'{e}},
  title =	{{When Lipschitz Walks Your Dog: Algorithm Engineering of the Discrete Fr\'{e}chet Distance Under Translation}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.25},
  URN =		{urn:nbn:de:0030-drops-128912},
  doi =		{10.4230/LIPIcs.ESA.2020.25},
  annote =	{Keywords: Fr\'{e}chet Distance, Computational Geometry, Continuous Optimization, Algorithm Engineering}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.17
Walking the Dog Fast in Practice: Algorithm Engineering of the Fréchet Distance

Authors: Karl Bringmann, Marvin Künnemann, and André Nusser

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
The Fréchet distance provides a natural and intuitive measure for the popular task of computing the similarity of two (polygonal) curves. While a simple algorithm computes it in near-quadratic time, a strongly subquadratic algorithm cannot exist unless the Strong Exponential Time Hypothesis fails. Still, fast practical implementations of the Fréchet distance, in particular for realistic input curves, are highly desirable. This has even lead to a designated competition, the ACM SIGSPATIAL GIS Cup 2017: Here, the challenge was to implement a near-neighbor data structure under the Fréchet distance. The bottleneck of the top three implementations turned out to be precisely the decision procedure for the Fréchet distance. In this work, we present a fast, certifying implementation for deciding the Fréchet distance, in order to (1) complement its pessimistic worst-case hardness by an empirical analysis on realistic input data and to (2) improve the state of the art for the GIS Cup challenge. We experimentally evaluate our implementation on a large benchmark consisting of several data sets (including handwritten characters and GPS trajectories). Compared to the winning implementation of the GIS Cup, we obtain running time improvements of up to more than two orders of magnitude for the decision procedure and of up to a factor of 30 for queries to the near-neighbor data structure.
Cite as

Karl Bringmann, Marvin Künnemann, and André Nusser. Walking the Dog Fast in Practice: Algorithm Engineering of the Fréchet Distance. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2019.17,
  author =	{Bringmann, Karl and K\"{u}nnemann, Marvin and Nusser, Andr\'{e}},
  title =	{{Walking the Dog Fast in Practice: Algorithm Engineering of the Fr\'{e}chet Distance}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{17:1--17:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.17},
  URN =		{urn:nbn:de:0030-drops-104219},
  doi =		{10.4230/LIPIcs.SoCG.2019.17},
  annote =	{Keywords: curve simplification, Fr\'{e}chet distance, algorithm engineering}
}
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