13 Search Results for "Kolbe, Benedikt"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.131
Persistence Meets Resistance: Doubling down on Hardness

Authors: Benedikt Kolbe and Tim Mayr

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract
We present results on the approximate computation of stable invariants for filtrations of finite metric spaces in the context of persistent homology. We establish novel approximation algorithms in the setting of n-point metric spaces where the growth of the doubling dimension is in o(log n) and the diameter is bounded. In the 1-parameter case, by revisiting known techniques (greedy permutations) in a new way, we derive the first linear-time algorithms for the problem of computing additive ε-approximations of any stable barcode. By deriving bounds on the convergence rate and the approximation quality of uniform samples, we extend the approach to selected multiparameter filtrations. We show that for normalized measure bifiltrations, including the multicover and subdivision-Rips bifiltration, any stable invariant can be probabilistically approximated in time constant in n. The constants in the running times of our algorithms depend on the doubling dimension, the diameter and the success probability. We further study the problem through the lens of fine-grained complexity and show that computing the rank of a matrix reduces to that of approximating the barcode of the Vietoris-Rips or Čech filtration. We present two variants of the reduction, one for sufficiently good additive approximations and the other for any constant factor multiplicative approximations.
Cite as

Benedikt Kolbe and Tim Mayr. Persistence Meets Resistance: Doubling down on Hardness. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 131:1-131:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kolbe_et_al:LIPIcs.ICALP.2026.131,
  author =	{Kolbe, Benedikt and Mayr, Tim},
  title =	{{Persistence Meets Resistance: Doubling down on Hardness}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{131:1--131:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael 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.2026.131},
  URN =		{urn:nbn:de:0030-drops-265209},
  doi =		{10.4230/LIPIcs.ICALP.2026.131},
  annote =	{Keywords: Persistent homology, approximations, lower bounds, matrix rank, doubling dimension, Vietoris-Rips complex, \v{C}ech complex, measure bifiltration, subdivision-Rips bifiltration, Rhomboid filtration}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.35
On Computing the (Exact) Fréchet Distance with a Frog

Authors: Jacobus Conradi, Ivor van der Hoog, and Eva Rotenberg

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The continuous Fréchet distance 𝒟_F(π,σ) between two polygonal curves π and σ is classically computed by exploring the free space diagram over the two curves. [SoCG'25] recently proposed a radically different approach: they approximate 𝒟_F(π,σ) by computing paths in a discrete graph that models a joint traversal of π and σ, recursively bisecting edges until the discrete distance converges to the continuous one. They implement their "frog-based" technique, and claim that it yields substantial practical speedups compared to the state-of-the-art implementations. In this paper, we revisit this technique. We observe that, in its current form, it has three limitations: (i) it does not use exact arithmetic, (ii) its recursive bisection introduces the required monotonicity events to realise the Fréchet distance only in the limit, and (iii) it applies a heuristic simplification technique which is overly conservative. Motivated by theoretical interest, we develop new techniques that guarantee exactness, polynomial-time convergence and near-optimal lossless simplifications. We provide an open-source C++ implementation of our variant. Our primary contribution is an extensive empirical evaluation on a broad, publically available, suite of real-world and synthetic data sets. Among the frog-based variants, exact computation indeed introduces overhead and increases median runtime. Yet, our new approach is often faster in the worst case, worst ten percent, or even the average runtime due to its worst-case convergence guarantees. More surprisingly, the implementation of [SoCG'19] dominates all frog-based implementations in performance - this finding contrasts previously published claims. These results provide a much-needed nuanced perspective on the capabilities and limitations of frog-based techniques: we showcase its theoretical appeal, but highlight its limited practical feasibility.
Cite as

Jacobus Conradi, Ivor van der Hoog, and Eva Rotenberg. On Computing the (Exact) Fréchet Distance with a Frog. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{conradi_et_al:LIPIcs.SoCG.2026.35,
  author =	{Conradi, Jacobus and van der Hoog, Ivor and Rotenberg, Eva},
  title =	{{On Computing the (Exact) Fr\'{e}chet Distance with a Frog}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{35:1--35:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.35},
  URN =		{urn:nbn:de:0030-drops-258414},
  doi =		{10.4230/LIPIcs.SoCG.2026.35},
  annote =	{Keywords: Algorithms engineering, Fr\'{e}chet distance}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.65
Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces

Authors: Sándor Kisfaludi-Bak and Geert van Wordragen

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We consider Steiner spanners in Euclidean and non-Euclidean geometries. In the Euclidean setting, a recent line of work initiated by Le and Solomon [FOCS'19] and further improved by Chang et al. [SoCG'24] obtained Steiner (1+ε)-spanners of size O_d(ε^{(1-d)/2} log(1/ε) n), nearly matching the lower bound Ω_d(ε^{(1-d)/2} n) of Bhore and Tóth [SIDMA'22]. We obtain Steiner (1+ε)-spanners of size O_d(ε^{(1-d)/2} log(1/ε)n) not only in d-dimensional Euclidean space, but also in d-dimensional spherical and hyperbolic space. For any fixed dimension d, the obtained edge count is optimal up to an O(log(1/ε)) factor in each of these spaces. Unlike earlier constructions, our Steiner spanners are based on simple quadtrees, and they can be dynamically maintained, leading to efficient data structures for dynamic approximate nearest neighbours and bichromatic closest pair. In the hyperbolic setting, we also show that 2-spanners in the hyperbolic plane must have Ω(nlog n) edges, and we obtain a 2-spanner of size O_d(nlog n) in d-dimensional hyperbolic space, matching our lower bound for any constant d. Finally, we give a Steiner spanner with additive error ε in hyperbolic space with O_d(ε^{(1-d)/2} log(α(n)/ε)n) edges, where α(n) is the inverse Ackermann function. Our techniques generalize to closed orientable surfaces of constant curvature as well as to some other quotient spaces.
Cite as

Sándor Kisfaludi-Bak and Geert van Wordragen. Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kisfaludibak_et_al:LIPIcs.SoCG.2026.65,
  author =	{Kisfaludi-Bak, S\'{a}ndor and van Wordragen, Geert},
  title =	{{Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{65:1--65:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.65},
  URN =		{urn:nbn:de:0030-drops-258728},
  doi =		{10.4230/LIPIcs.SoCG.2026.65},
  annote =	{Keywords: hyperbolic geometry, Steiner spanner, dynamic approximate nearest neighbours}
}
Document
CG Challenge
DOI: 10.4230/LIPIcs.SoCG.2026.106
Engineering Greedy Heuristics and Simulated Annealing Methods for the Median Triangulation Under the Parallel Flip Distance (CG Challenge)

Authors: Jacobus Conradi, Benedikt Kolbe, Philip Mayer, Jonas Sauer, and Jack Spalding-Jamieson

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We present our approach for the CG:SHOP 2026 challenge. In this international challenge, the goal was to find a median triangulation for a set of triangulations in the parallel flip reconfiguration graph of all triangulations of an underlying point set. Our simulated-annealing-based approach makes use of two ingredients: a heuristic edge selection for approximating the parallel flip distance of two given triangulations, and a heuristic procedure to generate good initial triangulations.
Cite as

Jacobus Conradi, Benedikt Kolbe, Philip Mayer, Jonas Sauer, and Jack Spalding-Jamieson. Engineering Greedy Heuristics and Simulated Annealing Methods for the Median Triangulation Under the Parallel Flip Distance (CG Challenge). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 106:1-106:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{conradi_et_al:LIPIcs.SoCG.2026.106,
  author =	{Conradi, Jacobus and Kolbe, Benedikt and Mayer, Philip and Sauer, Jonas and Spalding-Jamieson, Jack},
  title =	{{Engineering Greedy Heuristics and Simulated Annealing Methods for the Median Triangulation Under the Parallel Flip Distance}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{106:1--106:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.106},
  URN =		{urn:nbn:de:0030-drops-259125},
  doi =		{10.4230/LIPIcs.SoCG.2026.106},
  annote =	{Keywords: triangulation, flip distance, parallel flip distance, heuristic, competition}
}
Document
CG Challenge
DOI: 10.4230/LIPIcs.SoCG.2026.107
Shadoks Approach to Parallel Reconfiguration of Triangulations (CG Challenge)

Authors: Guilherme D. da Fonseca, Fabien Feschet, and Yan Gerard

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We describe the methods used by Team Shadoks to win the CG:SHOP 2026 Challenge on parallel reconfiguration of planar triangulations. Our approach combines exact methods based on SAT with several greedy heuristics, and also makes use of SAT and MaxSAT for solution improvement.
Cite as

Guilherme D. da Fonseca, Fabien Feschet, and Yan Gerard. Shadoks Approach to Parallel Reconfiguration of Triangulations (CG Challenge). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 107:1-107:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dafonseca_et_al:LIPIcs.SoCG.2026.107,
  author =	{da Fonseca, Guilherme D. and Feschet, Fabien and Gerard, Yan},
  title =	{{Shadoks Approach to Parallel Reconfiguration of Triangulations}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{107:1--107:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.107},
  URN =		{urn:nbn:de:0030-drops-259130},
  doi =		{10.4230/LIPIcs.SoCG.2026.107},
  annote =	{Keywords: Exact algorithm, SAT, MaxSAT, heuristic, computational geometry}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.82
Dimension Reduction for Clustering: The Curious Case of Discrete Centers

Authors: Shaofeng H.-C. Jiang, Robert Krauthgamer, Shay Sapir, Sandeep Silwal, and Di Yue

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The Johnson-Lindenstrauss transform is a fundamental method for dimension reduction in Euclidean spaces, that can map any dataset of n points into dimension O(log n) with low distortion of their distances. This dimension bound is tight in general, but one can bypass it for specific problems. Indeed, tremendous progress has been made for clustering problems, especially in the continuous setting where centers can be picked from the ambient space ℝ^d. Most notably, for k-median and k-means, the dimension bound was improved to O(log k) [Makarychev, Makarychev and Razenshteyn, STOC 2019]. We explore dimension reduction for clustering in the discrete setting, where centers can only be picked from the dataset, and present two results that are both parameterized by the doubling dimension of the dataset, denoted as ddim. The first result shows that dimension O_{ε}(ddim + log k + log log n) suffices, and is moreover tight, to guarantee that the cost is preserved within factor 1±ε for every set of centers. Our second result eliminates the log log n term in the dimension through a relaxation of the guarantee (namely, preserving the cost only for all approximately-optimal sets of centers), which maintains its usefulness for downstream applications. Overall, we achieve strong dimension reduction in the discrete setting, and find that it differs from the continuous setting not only in the dimension bound, which depends on the doubling dimension, but also in the guarantees beyond preserving the optimal value, such as which clusterings are preserved.
Cite as

Shaofeng H.-C. Jiang, Robert Krauthgamer, Shay Sapir, Sandeep Silwal, and Di Yue. Dimension Reduction for Clustering: The Curious Case of Discrete Centers. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 82:1-82:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{jiang_et_al:LIPIcs.ITCS.2026.82,
  author =	{Jiang, Shaofeng H.-C. and Krauthgamer, Robert and Sapir, Shay and Silwal, Sandeep and Yue, Di},
  title =	{{Dimension Reduction for Clustering: The Curious Case of Discrete Centers}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{82:1--82:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.82},
  URN =		{urn:nbn:de:0030-drops-253698},
  doi =		{10.4230/LIPIcs.ITCS.2026.82},
  annote =	{Keywords: dimension reduction, clustering, k-median, k-means, doubling dimension}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.61
ε-Net Algorithm Implementation on Hyperbolic Surfaces

Authors: Vincent Despré, Camille Lanuel, Marc Pouget, and Monique Teillaud

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We propose an implementation, using the CGAL library, of an algorithm to compute ε-nets on hyperbolic surfaces proposed by Despré, Lanuel and Teillaud [Despré et al., 2024]. We describe the data structure, detail the implemented algorithm and report experimental results on hyperbolic surfaces of genus 2. The implementation differs from the cited algorithm on several aspects. In particular, we use a different data structure, based on combinatorial maps, to represent a triangulation of a surface. We explain how to generate fundamental polygons to represent our input hyperbolic surfaces and the arithmetic issues related to the number type of the coordinates of their vertices.
Cite as

Vincent Despré, Camille Lanuel, Marc Pouget, and Monique Teillaud. ε-Net Algorithm Implementation on Hyperbolic Surfaces. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{despre_et_al:LIPIcs.ESA.2025.61,
  author =	{Despr\'{e}, Vincent and Lanuel, Camille and Pouget, Marc and Teillaud, Monique},
  title =	{{\epsilon-Net Algorithm Implementation on Hyperbolic Surfaces}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{61:1--61:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.61},
  URN =		{urn:nbn:de:0030-drops-245296},
  doi =		{10.4230/LIPIcs.ESA.2025.61},
  annote =	{Keywords: Hyperbolic surface, Delaunay triangulation, Data structure, Combinatorial map, Implementation, CGAL}
}
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}
}
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
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}
}
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.2024.42
Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping

Authors: Jacobus Conradi, Benedikt Kolbe, Ioannis Psarros, and Dennis Rohde

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

Abstract
We present algorithms for the computation of ε-coresets for k-median clustering of point sequences in ℝ^d under the p-dynamic time warping (DTW) distance. Coresets under DTW have not been investigated before, and the analysis is not directly accessible to existing methods as DTW is not a metric. The three main ingredients that allow our construction of coresets are the adaptation of the ε-coreset framework of sensitivity sampling, bounds on the VC dimension of approximations to the range spaces of balls under DTW, and new approximation algorithms for the k-median problem under DTW. We achieve our results by investigating approximations of DTW that provide a trade-off between the provided accuracy and amenability to known techniques. In particular, we observe that given n curves under DTW, one can directly construct a metric that approximates DTW on this set, permitting the use of the wealth of results on metric spaces for clustering purposes. The resulting approximations are the first with polynomial running time and achieve a very similar approximation factor as state-of-the-art techniques. We apply our results to produce a practical algorithm approximating (k,𝓁)-median clustering under DTW.
Cite as

Jacobus Conradi, Benedikt Kolbe, Ioannis Psarros, and Dennis Rohde. Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{conradi_et_al:LIPIcs.SoCG.2024.42,
  author =	{Conradi, Jacobus and Kolbe, Benedikt and Psarros, Ioannis and Rohde, Dennis},
  title =	{{Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{42:1--42:17},
  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.2024.42},
  URN =		{urn:nbn:de:0030-drops-199875},
  doi =		{10.4230/LIPIcs.SoCG.2024.42},
  annote =	{Keywords: Dynamic time warping, coreset, median clustering, approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.27
Computing a Dirichlet Domain for a Hyperbolic Surface

Authors: Vincent Despré, Benedikt Kolbe, Hugo Parlier, and Monique Teillaud

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

Abstract
This paper exhibits and analyzes an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm runs in polynomial time, in terms of the initial perimeter and the genus of the surface.
Cite as

Vincent Despré, Benedikt Kolbe, Hugo Parlier, and Monique Teillaud. Computing a Dirichlet Domain for a Hyperbolic Surface. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{despre_et_al:LIPIcs.SoCG.2023.27,
  author =	{Despr\'{e}, Vincent and Kolbe, Benedikt and Parlier, Hugo and Teillaud, Monique},
  title =	{{Computing a Dirichlet Domain for a Hyperbolic Surface}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{27:1--27:15},
  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.2023.27},
  URN =		{urn:nbn:de:0030-drops-178771},
  doi =		{10.4230/LIPIcs.SoCG.2023.27},
  annote =	{Keywords: Hyperbolic geometry, Topology, Voronoi diagram, Algorithm}
}
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