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LIPIcs, Volume 367
42nd International Symposium on Computational Geometry (SoCG 2026)
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Leibniz International Proceedings in Informatics (LIPIcs)
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- published at: 2026-05-27
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-418-5
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Complete Volume
LIPIcs, Volume 367, SoCG 2026, Complete Volume
Abstract
LIPIcs, Volume 367, SoCG 2026, Complete Volume
Cite as
42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 1-1724, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{ahn_et_al:LIPIcs.SoCG.2026,
title = {{LIPIcs, Volume 367, SoCG 2026, Complete Volume}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {1--1724},
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},
URN = {urn:nbn:de:0030-drops-261313},
doi = {10.4230/LIPIcs.SoCG.2026},
annote = {Keywords: LIPIcs, Volume 367, SoCG 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 0:i-0:xxiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ahn_et_al:LIPIcs.SoCG.2026.0,
author = {Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {0:i--0:xxiv},
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.2026.0},
URN = {urn:nbn:de:0030-drops-261298},
doi = {10.4230/LIPIcs.SoCG.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Covering and Partitioning Complex Objects with Small Pieces
Abstract
We study the problems of covering or partitioning a polygon P (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to write P as a union of small pieces, and in partitioning, we furthermore require the pieces to be pairwise interior-disjoint. We show that these problems are in fact equivalent: Optimum covers and partitions have the same number of pieces.
For covering, a natural local search algorithm repeatedly attempts to replace k pieces from a candidate cover with k-1 pieces. In two dimensions and for sufficiently large k, we show that when no such swap is possible, the cover is a 1+ O(1/√k) approximation, hence obtaining the first PTAS for the problem. Prior to our work, the only known algorithm was a 13-approximation that only works for polygons without holes [Abrahamsen and Rasmussen, SODA 2025]. In contrast, in the three dimensional version of the problem, for a polyhedron P of complexity n, we show that it is NP-hard to approximate an optimal cover or partition to within a factor that is logarithmic in n, even if P is simple, i.e., has genus 0 and no holes.
Cite as
Anders Aamand, Mikkel Abrahamsen, Reilly Browne, Mayank Goswami, Prahlad Narasimhan Kasthurirangan, Linda Kleist, Joseph S. B. Mitchell, Valentin Polishchuk, and Jack Stade. Covering and Partitioning Complex Objects with Small Pieces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{aamand_et_al:LIPIcs.SoCG.2026.1,
author = {Aamand, Anders and Abrahamsen, Mikkel and Browne, Reilly and Goswami, Mayank and Kasthurirangan, Prahlad Narasimhan and Kleist, Linda and Mitchell, Joseph S. B. and Polishchuk, Valentin and Stade, Jack},
title = {{Covering and Partitioning Complex Objects with Small Pieces}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {1:1--1:16},
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.2026.1},
URN = {urn:nbn:de:0030-drops-258077},
doi = {10.4230/LIPIcs.SoCG.2026.1},
annote = {Keywords: Covering, partitioning, polygon, small piece, PTAS}
}
Document
On the Maximum Number of Tangencies Among 1-Intersecting Curves
Abstract
According to a conjecture of Pach, there are O(n) tangent pairs among any family of n Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two special cases, however, for the general case the currently best upper bound is only O(n^{7/4}). This is also the best known bound on the number of tangencies in the relaxed case where every pair of arcs has at most one common point. We improve the bounds for the latter and former cases to O(n^{5/3}) and O(n^{3/2}), respectively. We also consider a few other variants of these questions, for example, we show that if the arcs are x-monotone, each pair intersects at most once and their left endpoints lie on a common vertical line, then the maximum number of tangencies is Θ(n^{4/3}). Without this last condition the number of tangencies is O(n^{4/3}(log n)^{1/3}), improving a previous bound of Pach and Sharir. Along the way we prove a graph-theoretic theorem which extends a result of Erdős and Simonovits and may be of independent interest.
Cite as
Eyal Ackerman and Balázs Keszegh. On the Maximum Number of Tangencies Among 1-Intersecting Curves. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ackerman_et_al:LIPIcs.SoCG.2026.2,
author = {Ackerman, Eyal and Keszegh, Bal\'{a}zs},
title = {{On the Maximum Number of Tangencies Among 1-Intersecting Curves}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {2:1--2:15},
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.2026.2},
URN = {urn:nbn:de:0030-drops-258085},
doi = {10.4230/LIPIcs.SoCG.2026.2},
annote = {Keywords: tangency graph, forbidden subgraph, extremal graph}
}
Document
Lower Bounding the Gromov-Hausdorff Distance in Metric Graphs
Abstract
Let G be a finite, connected metric graph and let X be a subset of G. If X is sufficiently dense in G, we show that the Gromov-Hausdorff distance matches the Hausdorff distance, namely d_GH(G,X) = d_H(G,X). When the metric graph is the circle G = S¹ with circumference 2π, a recent study established the equality d_GH(S¹,X) = d_H(S¹,X) whenever d_GH(S¹,X) < π/6. Our results relax this hypothesis to d_GH(S¹,X) < π/3, and furthermore, we show that the constant π/3 is the best possible. We lower bound the Gromov-Hausdorff distance d_GH(G,X) by the Hausdorff distance d_H(G,X) via a simple topological obstruction: the existence of a possibly discontinuous function f: G → X with too small distortion contradicts the connectedness of G.
Cite as
Henry Adams, Sushovan Majhi, Fedor Manin, Žiga Virk, and Nicolò Zava. Lower Bounding the Gromov-Hausdorff Distance in Metric Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{adams_et_al:LIPIcs.SoCG.2026.3,
author = {Adams, Henry and Majhi, Sushovan and Manin, Fedor and Virk, \v{Z}iga and Zava, Nicol\`{o}},
title = {{Lower Bounding the Gromov-Hausdorff Distance in Metric Graphs}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {3:1--3:16},
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.2026.3},
URN = {urn:nbn:de:0030-drops-258099},
doi = {10.4230/LIPIcs.SoCG.2026.3},
annote = {Keywords: Gromov-Hausdorff distance, distortion, connectedness, Borsuk-Ulam theorem}
}
Document
Dynamic Nearest-Neighbor Searching Under General Metrics in ℝ³ and Its Applications
Abstract
Let K be a compact, centrally-symmetric, strictly-convex region in ℝ³, which is a semi-algebraic set of constant complexity, i.e. the unit ball of a corresponding metric, denoted as ‖⋅‖_K. Let 𝒦 be a set of n homothetic copies of K. This paper contains two main sets of results:
(i) For a storage parameter s ∈ [n,n³], 𝒦 can be preprocessed in O^*(s) expected time into a data structure of size O^*(s), so that for a query homothet K₀ of K, an intersection-detection query (determine whether K₀ intersects any member of 𝒦, and if so, report such a member) or a nearest-neighbor query (return the member of 𝒦 whose ‖⋅‖_K-distance from K₀ is smallest) can be answered in O^*(n/s^{1/3}) time; all k homothets of 𝒦 intersecting K₀ can be reported in additional O(k) time. In addition, the data structure supports insertions/deletions in O^*(s/n) amortized expected time per operation. Here the O^*(⋅) notation hides factors of the form n^ε, where ε > 0 is an arbitrarily small constant, and the constant of proportionality depends on ε.
(ii) Let 𝒢(𝒦) denote the intersection graph of 𝒦. Using the above data structure, breadth-first or depth-first search on 𝒢(𝒦) can be performed in O^*(n^{3/2}) expected time. Combining this result with the so-called shrink-and-bifurcate technique, the reverse-shortest-path problem in a suitably defined proximity graph of 𝒦 can be solved in O^*(n^{62/39}) expected time. Dijkstra’s shortest-path algorithm, as well as Prim’s MST algorithm, on a ‖⋅‖_K-proximity graph on n points in ℝ³, with edges weighted by ‖⋅‖_K, can also be performed in O^*(n^{3/2}) time.
Cite as
Pankaj K. Agarwal, Matthew J. Katz, and Micha Sharir. Dynamic Nearest-Neighbor Searching Under General Metrics in ℝ³ and Its Applications. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2026.4,
author = {Agarwal, Pankaj K. and Katz, Matthew J. and Sharir, Micha},
title = {{Dynamic Nearest-Neighbor Searching Under General Metrics in \mathbb{R}³ and Its Applications}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {4:1--4:15},
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.2026.4},
URN = {urn:nbn:de:0030-drops-258102},
doi = {10.4230/LIPIcs.SoCG.2026.4},
annote = {Keywords: Homothets, Minkowski metric, Shallow cuttings, Nearest-neighbor searching, Intersection and proximity graphs, Reverse-shortest-path problem}
}
Document
Bifunction and Interlevel Delaunay Trifiltrations
Abstract
A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an ℝ-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an ℝ²-valued function, also satisfying an analogous weak equivalence. For a point cloud X ⊂ ℝ^d, our trifiltration has size O(|X|^{⌈(d+1)/2⌉+1}). We present an algorithm that computes this trifiltration in time O(|X|^{⌈d/2⌉+2}), together with an implementation. Our experiments demonstrate that the implementation can handle thousands of points in ℝ³, with memory growth that is nearly linear.
Cite as
Ángel Javier Alonso, Michael Kerber, Tung Lam, Michael Lesnick, and Abhishek Rathod. Bifunction and Interlevel Delaunay Trifiltrations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alonso_et_al:LIPIcs.SoCG.2026.5,
author = {Alonso, \'{A}ngel Javier and Kerber, Michael and Lam, Tung and Lesnick, Michael and Rathod, Abhishek},
title = {{Bifunction and Interlevel Delaunay Trifiltrations}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {5:1--5:20},
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.2026.5},
URN = {urn:nbn:de:0030-drops-258118},
doi = {10.4230/LIPIcs.SoCG.2026.5},
annote = {Keywords: Delaunay triangulation, Multiparameter persistent homology, Interlevel, Bowyer-Watson}
}
Document
Estimating the Persistent Homology of ℝⁿ-Valued Functions Using Function-Geometric Multifiltrations
Abstract
Given an unknown ℝⁿ-valued function f on a metric space X, can we approximate the persistent homology of f from a finite sampling of X with known pairwise distances and function values? This question has been answered in the case n = 1, assuming f is Lipschitz continuous and X is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. In this paper we answer the question for arbitrary n, under similar assumptions and using function-geometric multifiltrations. Our analysis offers a different view on these multifiltrations by focusing on their approximation properties rather than on their stability properties. We also leverage the multiparameter setting to provide insight into the influence of the scale parameter, whose choice is central to this type of approach. From a practical standpoint, we show that our approximation results are robust to input noise, and that function-geometric multifiltrations have good statistical convergence properties. We also provide an algorithm to compute our estimators, and we use its implementation to conduct extensive experiments, on both synthetic and real biological data, in order to validate our theoretical results.
Cite as
Ethan André, Jingyi Li, David Loiseaux, and Steve Oudot. Estimating the Persistent Homology of ℝⁿ-Valued Functions Using Function-Geometric Multifiltrations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{andre_et_al:LIPIcs.SoCG.2026.6,
author = {Andr\'{e}, Ethan and Li, Jingyi and Loiseaux, David and Oudot, Steve},
title = {{Estimating the Persistent Homology of \mathbb{R}ⁿ-Valued Functions Using Function-Geometric Multifiltrations}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {6:1--6:18},
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.2026.6},
URN = {urn:nbn:de:0030-drops-258120},
doi = {10.4230/LIPIcs.SoCG.2026.6},
annote = {Keywords: Topological data analysis, multi-parameter persistent homology, function-Rips multifiltration}
}
Document
Computing L_∞ Hausdorff Distances Under Translations: The Interplay of Dimensionality, Symmetry and Discreteness
Abstract
To measure the similarity of the shape of point sets, rather than their mere closeness in space, various notions of a Hausdorff distance under translation have been investigated. Specifically, let P and Q denote point sets of n and m points, respectively, in ℝ^d. We consider the task of computing the minimum distance d(P,Q+τ) over an admissible set of translations τ ∈ T, where d(⋅, ⋅) denotes the Hausdorff distance under the L_∞-norm. As variants, we distinguish between continuous (T = ℝ^d) or discrete (T is a given finite set of t translations) as well as directed or undirected (choosing the directed or undirected Hausdorff distance for d(⋅, ⋅)).
We seek to apply the paradigm of fine-grained complexity to understand the complexity of these variants, and in particular: How is the running time influenced by the dimension d, the relationship between n and m, and the specific choice of variant? As our main results, we obtain:
- The asymmetric definition of the most studied variant, the continuous directed Hausdorff distance, results in an intrinsically asymmetric time complexity: While (Chan, SoCG'23) established a symmetric Õ((nm)^{d/2}) upper bound for all d ≥ 3 and proved it to be conditionally optimal for combinatorial algorithms whenever m ≤ n, we show that this lower bound does not hold for the case n ≪ m, by providing a combinatorial, almost-linear-time algorithm for d = 3 and n = m^{o(1)}. We further prove general, i.e., non-combinatorial, conditional lower bounds for d ≥ 3, in particular: (1) m^{⌊d/2⌋ - o(1)} for small n and (2) n^{d/2 - o(1)} for d = 3 and small m.
- We observe that the directed and undirected case is closely related, in particular, all our lower bounds for d ≥ 3 hold for both the directed and undirected variant. A remarkable exception is the case of d = 1 for which we provide a conditional separation. Specifically, in contrast to the undirected variants being solvable in near-linear time (Rote, IPL'91), we show that the directed variants are at least as hard as the additive problem MaxConv LowerBound introduced in (Cygan, Mucha, Wegrzycki and Wlodarczyk, TALG'19).
- We show that the discrete variants reduce to a variant of 3SUM for d ≤ 3. This gives a barrier in proving a tight lower bound of these variants under the Orthogonal Vectors Hypothesis (OVH); in contrast, the continuous variants admit a tight conditional lower bound under OVH in d = 2 (Bringmann, Nusser, JoCG'21). These results reveal an intricate interplay of dimensionality, symmetry and discreteness in determining the fine-grained complexity of computing Hausdorff distances under translation.
Cite as
Sebastian Angrick, Kevin Buchin, Geri Gokaj, and Marvin Künnemann. Computing L_∞ Hausdorff Distances Under Translations: The Interplay of Dimensionality, Symmetry and Discreteness. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{angrick_et_al:LIPIcs.SoCG.2026.7,
author = {Angrick, Sebastian and Buchin, Kevin and Gokaj, Geri and K\"{u}nnemann, Marvin},
title = {{Computing L\underline∞ Hausdorff Distances Under Translations: The Interplay of Dimensionality, Symmetry and Discreteness}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {7:1--7:15},
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.2026.7},
URN = {urn:nbn:de:0030-drops-258131},
doi = {10.4230/LIPIcs.SoCG.2026.7},
annote = {Keywords: Hausdorff Distance, Fine-Grained Complexity, Computational Geometry, Translation-Invariant Similarity Measures}
}
Document
Cauchy’s Surface Area Formula in the Funk Geometry
Abstract
Cauchy’s surface area formula expresses the surface area of a convex body as the average area of its orthogonal projections over all directions. While this tool is fundamental in Euclidean geometry, with applications ranging from geometric tomography to approximation theory, extensions to non-Euclidean settings remain less explored. In this paper, we establish an analog of Cauchy’s formula for the Funk geometry induced by a convex body K in ℝ^d, for the Holmes-Thompson surface area. The formula is based on central projections to boundary points of K. We show that when K is a convex polytope, the formula reduces to a weighted sum of contributions associated with the vertices of K. Finally, as a consequence of our analysis, we derive a generalization of Crofton’s formula for surface areas in the Funk geometry. By viewing Euclidean, Minkowski, Hilbert, and hyperbolic geometries as limiting or special cases of the Funk setting, our results provide a unified framework for these classical surface area formulas.
Cite as
Sunil Arya and David M. Mount. Cauchy’s Surface Area Formula in the Funk Geometry. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{arya_et_al:LIPIcs.SoCG.2026.8,
author = {Arya, Sunil and Mount, David M.},
title = {{Cauchy’s Surface Area Formula in the Funk Geometry}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {8:1--8:18},
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.2026.8},
URN = {urn:nbn:de:0030-drops-258140},
doi = {10.4230/LIPIcs.SoCG.2026.8},
annote = {Keywords: Convexity, Cauchy’s formula, Funk geometry, Hilbert geometry, Crofton’s formula, Holmes-Thompson surface area}
}
Document
Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions
Abstract
A theorem of Matoušek asserts that for any k ≥ 2, any set system whose shatter function is o(n^k) enjoys a fractional Helly theorem of order k: in the k-wise intersection hypergraph, positive density implies a linear-size clique. Kalai and Meshulam conjectured a generalization of that phenomenon to homological shatter functions. It was verified for set systems with bounded homological shatter functions and whose ground set has a forbidden homological minor (which includes ℝ^d by a homological analogue of the van Kampen-Flores theorem). We present two contributions to this line of research:
- We study homological minors in certain manifolds (possibly with boundary), for which we prove analogues of the van Kampen-Flores theorem and of the Hanani-Tutte theorem.
- We introduce graded analogues of the Radon and Helly numbers of set systems and relate their growth rate to the original parameters. This allows to extend the verification of the Kalai-Meshulam conjecture to sufficiently slowly growing homological shatter functions.
Cite as
Sergey Avvakumov, Marguerite Bin, and Xavier Goaoc. Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{avvakumov_et_al:LIPIcs.SoCG.2026.9,
author = {Avvakumov, Sergey and Bin, Marguerite and Goaoc, Xavier},
title = {{Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {9:1--9:16},
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.2026.9},
URN = {urn:nbn:de:0030-drops-258152},
doi = {10.4230/LIPIcs.SoCG.2026.9},
annote = {Keywords: Fractional Helly theorem, homological minor, combinatorial convexity}
}
Document
Fast Free Resolutions of Bifiltered Chain Complexes
Abstract
In a k-critical bifiltration, every simplex enters along a staircase with at most k steps. Examples with k > 1 include degree-Rips bifiltrations and models of the multicover bifiltration. We consider the problem of converting a k-critical bifiltration into a 1-critical (i.e. free) chain complex with equivalent homology. This is known as computing a free resolution of the underlying chain complex and is a first step toward post-processing such bifiltrations.
We present two algorithms. The first one computes free resolutions corresponding to path graphs and assembles them to a chain complex by computing additional maps. The simple combinatorial structure of path graphs leads to good performance in practice, as demonstrated by extensive experiments. However, its worst-case bound is quadratic in the input size because long paths might yield dense boundary matrices in the output. Our second algorithm replaces the simplex-wise path graphs with ones that maintain short paths which leads to almost linear runtime and output size.
We demonstrate that pre-computing a free resolution speeds up the task of computing a minimal presentation of the homology of a k-critical bifiltration in a fixed dimension. Furthermore, our findings show that a chain complex that is minimal in terms of generators can be asymptotically larger than the non-minimal output complex of our second algorithm in terms of description size.
Cite as
Ulrich Bauer, Tamal K. Dey, Michael Kerber, Florian Russold, and Matthias Söls. Fast Free Resolutions of Bifiltered Chain Complexes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bauer_et_al:LIPIcs.SoCG.2026.10,
author = {Bauer, Ulrich and Dey, Tamal K. and Kerber, Michael and Russold, Florian and S\"{o}ls, Matthias},
title = {{Fast Free Resolutions of Bifiltered Chain Complexes}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {10:1--10:21},
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.2026.10},
URN = {urn:nbn:de:0030-drops-258161},
doi = {10.4230/LIPIcs.SoCG.2026.10},
annote = {Keywords: Topological Data Analysis, Multi-Parameter Persistence, Multi-Critical Bifiltrations}
}
Document
Locally Correct Interleavings Between Merge Trees
Abstract
Merge trees are typically used as a topological summary of scalar fields. To analyze e.g. a time-varying scalar field via its merge tree representation, one needs a suitable method to match and compare two merge trees. We consider the interleaving distance as a method to match and compare merge trees. An interleaving between two merge trees consists of two maps, one in each direction. These maps must satisfy ancestor relations and hence introduce a "shift" between points and their image. An optimal interleaving minimizes the maximum shift; the interleaving distance is the value of this shift. However, to study the evolution of merge trees, we need not only a number but also a meaningful matching between the two trees. The two maps of an optimal interleaving induce a matching, but due to the bottleneck nature of the interleaving distance, this matching fails to capture local similarities between the trees. In this paper we hence propose a notion of local optimality for interleavings. To do so, we define the residual interleaving distance, a generalization of the interleaving distance that allows additional constraints on the maps. This allows us to define locally correct interleavings, which use a range of shifts across the two merge trees that reflect the local similarity well. We give a constructive proof that a locally correct interleaving always exists.
Cite as
Thijs Beurskens, Tim Ophelders, Bettina Speckmann, and Kevin Verbeek. Locally Correct Interleavings Between Merge Trees. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{beurskens_et_al:LIPIcs.SoCG.2026.11,
author = {Beurskens, Thijs and Ophelders, Tim and Speckmann, Bettina and Verbeek, Kevin},
title = {{Locally Correct Interleavings Between Merge Trees}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {11:1--11:15},
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.2026.11},
URN = {urn:nbn:de:0030-drops-258173},
doi = {10.4230/LIPIcs.SoCG.2026.11},
annote = {Keywords: Interleaving distance, merge trees, local correctness, matchings, topological data analysis}
}
Document
Dynamic and Streaming Algorithms for Union Volume Estimation
Abstract
The union volume estimation problem asks to (1±ε)-approximate the volume of the union of n given objects X₁,…,X_n ⊂ ℝ^d. In their seminal work in 1989, Karp, Luby, and Madras solved this problem in time O(n/ε²) in an oracle model where each object X_i can be accessed via three types of queries: obtain the volume of X_i, sample a random point from X_i, and test whether X_i contains a given point x. This running time was recently shown to be optimal [Bringmann, Larsen, Nusser, Rotenberg, and Wang, SoCG'25]. In another line of work, Meel, Vinodchandran, and Chakraborty [PODS'21] designed algorithms that read the objects in one pass using polylogarithmic time per object and polylogarithmic space; this can be phrased as a dynamic algorithm supporting insertions of objects for union volume estimation in the oracle model.
In this paper, we study algorithms for union volume estimation in the oracle model that support both insertions and deletions of objects. We obtain the following results:
1) an algorithm supporting insertions and deletions in polylogarithmic update and query time and linear space (this is the first such dynamic algorithm, even for 2D triangles);
2) an algorithm supporting insertions and suffix queries (which generalizes the sliding window setting) in polylogarithmic update and query time and space;
3) an algorithm supporting insertions and deletions of convex bodies of constant dimension in polylogarithmic update and query time and space.
Cite as
Sujoy Bhore, Karl Bringmann, Timothy M. Chan, and Yanheng Wang. Dynamic and Streaming Algorithms for Union Volume Estimation. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.12,
author = {Bhore, Sujoy and Bringmann, Karl and Chan, Timothy M. and Wang, Yanheng},
title = {{Dynamic and Streaming Algorithms for Union Volume Estimation}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {12:1--12:15},
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.2026.12},
URN = {urn:nbn:de:0030-drops-258180},
doi = {10.4230/LIPIcs.SoCG.2026.12},
annote = {Keywords: union volume estimation, dynamic algorithms, streaming algorithms}
}
Document
Dynamic Light Spanners in Doubling Metrics
Abstract
A t-spanner of a point set X in a metric space (𝒳, δ) is a graph G with vertex set P such that, for any pair of points u,v ∈ X, the distance between u and v in G is at most t times δ(u,v). We study the problem of maintaining a spanner for a dynamic point set X - that is, when X undergoes a sequence of insertions and deletions - in a metric space of constant doubling dimension. For any constant ε > 0, we maintain a (1+ε)-spanner of P whose total weight remains within a constant factor of the weight of the minimum spanning tree of X. Each update (insertion or deletion) can be performed in poly(log Φ) time, where Φ denotes the aspect ratio of X. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.
Cite as
Sujoy Bhore, Jonathan Conroy, and Arnold Filtser. Dynamic Light Spanners in Doubling Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.13,
author = {Bhore, Sujoy and Conroy, Jonathan and Filtser, Arnold},
title = {{Dynamic Light Spanners in Doubling Metrics}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {13:1--13:16},
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.2026.13},
URN = {urn:nbn:de:0030-drops-258193},
doi = {10.4230/LIPIcs.SoCG.2026.13},
annote = {Keywords: Dynamic data structures, spanners, light-weight, Euclidean metrics, doubling metrics}
}
Document
Improved Online Hitting Set Algorithms for Structured and Geometric Set Systems
Abstract
In the online hitting set problem, sets arrive over time, and the algorithm has to maintain a subset of elements that hit all the sets seen so far. Alon, Awerbuch, Azar, Buchbinder, and Naor (SICOMP 2009) gave an algorithm with competitive ratio O(log n log m) for the (general) online hitting set and set cover problems for m sets and n elements; this is known to be tight for efficient online algorithms. Given this barrier for general set systems, we ask: can we break this double-logarithmic phenomenon for online hitting set/set cover on structured and geometric set systems?
We provide an O(log n log log n)-competitive algorithm for the weighted online hitting set problem on set systems with linear shallow-cell complexity, replacing the double-logarithmic factor in the general result by effectively a single logarithmic term. As a consequence of our results we obtain the first bounds for weighted online hitting set for natural geometric set families, thereby answering open questions regarding the gap between general and geometric weighted online hitting set problems.
Cite as
Sujoy Bhore, Anupam Gupta, and Amit Kumar. Improved Online Hitting Set Algorithms for Structured and Geometric Set Systems. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.14,
author = {Bhore, Sujoy and Gupta, Anupam and Kumar, Amit},
title = {{Improved Online Hitting Set Algorithms for Structured and Geometric Set Systems}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {14:1--14:14},
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.2026.14},
URN = {urn:nbn:de:0030-drops-258206},
doi = {10.4230/LIPIcs.SoCG.2026.14},
annote = {Keywords: Hitting Set, Online Algorithms, Shallow-Cell Complexity, VC-Dimension}
}
Document
Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity
Abstract
A Euclidean noncrossing Steiner (1+ε)-spanner for a point set P ⊂ ℝ² is a planar straight-line graph that, for any two points a, b ∈ P, contains a path whose length is at most 1+ε times the Euclidean distance between a and b. We construct a Euclidean noncrossing Steiner (1+ε)-spanner with O(n/ε^{3/2}) edges for any set of n points in the plane. This result improves upon the previous best upper bound of O(n/ε⁴) obtained nearly three decades ago. We also establish an almost matching lower bound: There exist n points in the plane for which any Euclidean noncrossing Steiner (1+ε)-spanner has Ω_μ(n/ε^{3/2-μ}) edges for any μ > 0. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.
Cite as
Sujoy Bhore, Sándor Kisfaludi‑Bak, Lazar Milenković, Csaba D. Tóth, Karol Węgrzycki, and Sampson Wong. Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.15,
author = {Bhore, Sujoy and Kisfaludi‑Bak, S\'{a}ndor and Milenkovi\'{c}, Lazar and T\'{o}th, Csaba D. and W\k{e}grzycki, Karol and Wong, Sampson},
title = {{Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {15:1--15:18},
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.2026.15},
URN = {urn:nbn:de:0030-drops-258210},
doi = {10.4230/LIPIcs.SoCG.2026.15},
annote = {Keywords: geometric network design, spanners, crossing number, incidences}
}
Document
Flip Distance of Non-Crossing Spanning Trees: NP-Hardness and Improved Bounds
Abstract
We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set P of n points in general position in the plane, the flip graph ℱ(P) has a vertex for each non-crossing spanning tree on P and an edge between any two spanning trees that can be transformed into each other by the exchange of a single edge (coined a flip). This flip graph has been intensively studied, lately with an emphasis on determining its diameter diam(ℱ(P)) for sets P of n points in convex position. For this case, the current best bounds are 14/9⋅n - O(1) ≤ diam(ℱ(P)) < 15/9⋅n - 3, obtained in a recent breakthrough work [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]. The crucial tool for both the upper and lower bound are so-called conflict graphs, which the authors stated might be the key ingredient for determining the diameter (up to lower-order terms).
In this paper, we pick up the concept of conflict graphs from the above-mentioned work and show that this tool is even more versatile than previously hoped. As our first main result, we use conflict graphs to show that computing the flip distance between two non-crossing spanning trees is NP-hard, even for point sets in convex position. Interestingly, the result still holds for more constrained flip operations, concretely, compatible flips (where the removed and the added edge do not cross) and rotations (where the removed and the added edge share an endpoint).
Additionally, we present new insights on the diameter of the flip graph, by this directly extending the line of research from [BKUV SODA25]. Their lower bound is based on a constant-size pair of trees, one of which is of a type we refer to as stacked. We show that if one of the trees is stacked, then the lower bound is indeed optimal up to a constant term, that is, there exists a flip sequence of length at most 14/9⋅(n-1) to any other tree.
Lastly, we improve the lower bound on the diameter of the flip graph ℱ(P) for n points in convex position to 11/7⋅n-o(n).
Cite as
Håvard Bakke Bjerkevik, Joseph Dorfer, Linda Kleist, Torsten Ueckerdt, and Birgit Vogtenhuber. Flip Distance of Non-Crossing Spanning Trees: NP-Hardness and Improved Bounds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bjerkevik_et_al:LIPIcs.SoCG.2026.16,
author = {Bjerkevik, H\r{a}vard Bakke and Dorfer, Joseph and Kleist, Linda and Ueckerdt, Torsten and Vogtenhuber, Birgit},
title = {{Flip Distance of Non-Crossing Spanning Trees: NP-Hardness and Improved Bounds}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {16:1--16:18},
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.2026.16},
URN = {urn:nbn:de:0030-drops-258225},
doi = {10.4230/LIPIcs.SoCG.2026.16},
annote = {Keywords: Non-crossing, spanning tree, plane graph, flip graph, reconfiguration, diameter, complexity, NP-hard, edge exchange, compatible flip, rotation, happy edge property}
}
Document
Fréchet Distance in the Imbalanced Case
Abstract
Given two polygonal curves P and Q defined by n and m vertices with m ≤ n, we show that the discrete Fréchet distance in 1D cannot be approximated within a factor of 2-ε in 𝒪((nm)^{1-δ}) time for any ε, δ > 0 unless OVH fails. Using a similar construction, we extend this bound for curves in 2D under the continuous or discrete Fréchet distance and increase the approximation factor to 1+√2-ε (resp. 3-ε) if the curves lie in the Euclidean space (resp. in the L_∞-space). This strengthens the lower bound by Buchin, Ophelders, and Speckmann to the case where m = n^α for α ∈ (0,1) and increases the approximation factor of 1.001 by Bringmann. For the discrete Fréchet distance in 1D, we provide an approximation algorithm with optimal approximation factor and almost optimal running time. Further, for curves in any dimension embedded in any L_p space, we present a (3+ε)-approximation algorithm for the continuous and discrete Fréchet distance using 𝒪((n+m²)log n) time, which almost matches the approximation factor of the lower bound for the L_∞ metric.
Cite as
Lotte Blank. Fréchet Distance in the Imbalanced Case. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blank:LIPIcs.SoCG.2026.17,
author = {Blank, Lotte},
title = {{Fr\'{e}chet Distance in the Imbalanced Case}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {17:1--17:17},
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.2026.17},
URN = {urn:nbn:de:0030-drops-258232},
doi = {10.4230/LIPIcs.SoCG.2026.17},
annote = {Keywords: Fr\'{e}chet distance, SETH, Orthogonal Vectors, Lower Bounds, distance oracle, data structures}
}
Document
Product Structure and Treewidth of Hyperbolic Uniform Disk Graphs
Abstract
Hyperbolic uniform disk graphs (HUDGs) are intersection graphs of disks with some radius r in the hyperbolic plane, where r may be constant or depend on the number of vertices in a family of HUDGs. We show that HUDGs with constant clique number do not admit product structure, i.e., that there is no constant c such that every such graph is a subgraph of H ⊠ P for some graph H of treewidth at most c. This justifies that HUDGs are described as not having a grid-like structure in the literature, and is in contrast to unit disk graphs in the Euclidean plane, whose grid-like structure is evident from the fact that they are subgraphs of the strong product of two paths and a clique of constant size [Dvořák et al., '21, MATRIX Annals]. By allowing H to be any graph of constant treewidth instead of a path-like graph, we reject the possibility of a grid-like structure not merely by the maximum degree (which is unbounded for HUDGs) but due to their global structure. We complement this by showing that for every (sub-)constant r, HUDGs admit product structure, whereas the typical hyperbolic behavior is observed if r grows with the number of vertices.
Our proof involves a family of n-vertex HUDGs with radius log n that has bounded clique number but unbounded treewidth, and one for which the ratio of treewidth and clique number is log n / log log n. Up to a log log n factor, this negatively answers a question raised by Bläsius et al. [SoCG '25] asking whether balanced separators of HUDGs with radius log n can be covered by less than log n cliques. Our results also imply that the local and layered tree-independence number of HUDGs are both unbounded, answering an open question of Dallard et al. [arXiv '25].
Cite as
Thomas Bläsius, Emil Dohse, Deborah Haun, and Laura Merker. Product Structure and Treewidth of Hyperbolic Uniform Disk Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blasius_et_al:LIPIcs.SoCG.2026.18,
author = {Bl\"{a}sius, Thomas and Dohse, Emil and Haun, Deborah and Merker, Laura},
title = {{Product Structure and Treewidth of Hyperbolic Uniform Disk Graphs}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {18:1--18:17},
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.2026.18},
URN = {urn:nbn:de:0030-drops-258249},
doi = {10.4230/LIPIcs.SoCG.2026.18},
annote = {Keywords: hyperbolic uniform disk graphs, product structure, treewidth}
}
Document
Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements
Abstract
The famous Ham-Sandwich theorem states that any d point sets in ℝ^d can be simultaneously bisected by a single hyperplane. The α-Ham-Sandwich theorem gives a sufficient condition for the existence of biased cuts, i.e., hyperplanes that do not cut off half but some prescribed fraction of each point set. We give two new proofs for this theorem. The first proof is completely combinatorial and highlights a strong connection between the α-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincaré-Miranda theorem and allows us to generalize the result to and beyond oriented matroids. For this we introduce a new concept of rainbow arrangements, generalizing colored pseudo-hyperplane arrangements. Along the way, we also show that the realizability problem for rainbow arrangements is ∃ℝ-complete, which also implies that the realizability problem for grid Unique Sink Orientations is ∃ℝ-complete.
Cite as
Michaela Borzechowski, Sebastian Haslebacher, Hung P. Hoang, Patrick Schnider, and Simon Weber. Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{borzechowski_et_al:LIPIcs.SoCG.2026.19,
author = {Borzechowski, Michaela and Haslebacher, Sebastian and Hoang, Hung P. and Schnider, Patrick and Weber, Simon},
title = {{Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {19:1--19:16},
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.2026.19},
URN = {urn:nbn:de:0030-drops-258250},
doi = {10.4230/LIPIcs.SoCG.2026.19},
annote = {Keywords: \alpha-Ham-Sandwich Theorem, Pseudo-Hyperplanes, Arrangements, Unique Sink Orientations, Oriented Matroids}
}
Document
The Spanning Ratio of the Directed Θ₆-Graph Is 5
Abstract
Given a finite set P ⊂ ℝ², the directed Theta-6 graph, denoted Θ₆(P), is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The Θ₆(P)-graph is defined as follows: the plane around each point u ∈ P is partitioned into 6 equiangular cones with apex u, and in each cone, u is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the Θ₆(P)-graph contains an edge from u to v exactly when the interior of ∇_u^v is disjoint from P, where ∇_u^v is the unique equilateral triangle containing u on a corner, v on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the Θ₆(P)-graph is between 4 and 7 in the worst case (Akitaya, Biniaz, and Bose Comput. Geom., 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any Θ_k(P)-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.
Cite as
Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill. The Spanning Ratio of the Directed Θ₆-Graph Is 5. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bose_et_al:LIPIcs.SoCG.2026.20,
author = {Bose, Prosenjit and De Carufel, Jean-Lou and Stuart, John and Hill, Darryl},
title = {{The Spanning Ratio of the Directed \Theta₆-Graph Is 5}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {20:1--20:18},
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.2026.20},
URN = {urn:nbn:de:0030-drops-258268},
doi = {10.4230/LIPIcs.SoCG.2026.20},
annote = {Keywords: Geometric Spanners, Theta Graphs, Directed Theta Graphs, Spanning Ratio, Computational Geometry}
}
Document
On Minimum Venn Diagrams
Abstract
An n-Venn diagram is a diagram in the plane consisting of n simple closed curves that intersect only finitely many times such that each of the 2ⁿ possible intersections of their interiors is represented by a single connected region. An n-Venn diagram has at most 2ⁿ-2 crossings, and if this maximum number of crossings is attained, then only two curves intersect in every crossing. To complement this, Bultena and Ruskey considered n-Venn diagrams that minimize the number of crossings, which implies that many curves intersect in every crossing. Specifically, they proved that the total number of crossings in any n-Venn diagram is at least L_n≔⌈(2ⁿ-2)/(n-1)⌉, and if this lower bound is attained, then essentially all n curves intersect in every crossing. Diagrams achieving this bound are called minimum Venn diagrams, and are known only for n ≤ 7. Bultena and Ruskey conjectured that they exist for all n ≥ 8. In this work, we establish an asymptotic version of their conjecture. For n = 8 we construct a diagram with 40 crossings, only 3 more than the lower bound L₈ = 37. Furthermore, for every n of the form n = 2^k for some integer k ≥ 4, we construct an n-Venn diagram with at most (1+33/8n)L_n = (1+o(1))L_n many crossings. Via a doubling trick this also gives (n+m)-Venn diagrams for all 0 ≤ m < n with at most 40⋅ 2^m crossings for n = 8 and at most (1+33/8n) (n+m)/n L_{n+m} = (2+o(1))L_{n+m} many crossings for k ≥ 4. In particular, we obtain n-Venn diagrams with the smallest known number of crossings for all n ≥ 8. Our constructions are based on partitions of the hypercube into isometric paths and cycles, using a result of Ramras.
Cite as
Sofia Brenner, Petr Gregor, Torsten Mütze, and Francesco Verciani. On Minimum Venn Diagrams. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brenner_et_al:LIPIcs.SoCG.2026.21,
author = {Brenner, Sofia and Gregor, Petr and M\"{u}tze, Torsten and Verciani, Francesco},
title = {{On Minimum Venn Diagrams}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {21:1--21:18},
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.2026.21},
URN = {urn:nbn:de:0030-drops-258278},
doi = {10.4230/LIPIcs.SoCG.2026.21},
annote = {Keywords: Venn diagram, crossing, conjecture, hypercube, partition}
}
Document
Disproving Two Conjectures on the Hamiltonicity of Venn Diagrams
Abstract
In 1984, Winkler conjectured that every simple Venn diagram with n curves can be extended to a simple Venn diagram with n+1 curves. This conjecture is equivalent to the statement that the dual graph of any simple Venn diagram has a Hamilton cycle. In this work, we construct counterexamples to Winkler’s conjecture for all n ≥ 6. As part of this proof, we computed all 3.430.404 simple Venn diagrams with n = 6 curves (even their number was not previously known), among which we found 72 counterexamples.
We also disprove another conjecture about the Hamiltonicity of the arrangement graph of a Venn diagram. Specifically, while working on Winkler’s conjecture, Pruesse and Ruskey proved that this graph has a Hamilton cycle for every simple Venn diagram with n curves, and conjectured that this also holds for non-simple diagrams. We construct counterexamples to this conjecture for all n ≥ 4.
Cite as
Sofia Brenner, Linda Kleist, Torsten Mütze, Christian Rieck, and Francesco Verciani. Disproving Two Conjectures on the Hamiltonicity of Venn Diagrams. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brenner_et_al:LIPIcs.SoCG.2026.22,
author = {Brenner, Sofia and Kleist, Linda and M\"{u}tze, Torsten and Rieck, Christian and Verciani, Francesco},
title = {{Disproving Two Conjectures on the Hamiltonicity of Venn Diagrams}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {22:1--22:16},
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.2026.22},
URN = {urn:nbn:de:0030-drops-258285},
doi = {10.4230/LIPIcs.SoCG.2026.22},
annote = {Keywords: Venn diagram, Winkler’s conjecture, Hamilton cycle, perfect matching, hypercube}
}
Document
Shortest Paths in Geodesic Unit-Disk Graphs
Abstract
Let S be a set of n points in a polygon P with m vertices. The geodesic unit-disk graph G(S) induced by S has vertex set S and contains an edge between two vertices whenever their geodesic distance in P is at most one. In the weighted version, each edge is assigned weight equal to the geodesic distance between its endpoints; in the unweighted version, every edge has weight 1. Given a source point s ∈ S, we study the problem of computing shortest paths from s to all vertices of G(S). To the best of our knowledge, this problem has not been investigated previously. A naive approach constructs G(S) explicitly and then applies a standard shortest path algorithm for general graphs, but this requires quadratic time in the worst case, since G(S) may contain Ω(n²) edges. In this paper, we give the first subquadratic-time algorithms for this problem. For the weighted case, when P is a simple polygon, we obtain an O(m + n log³ n log² m)-time algorithm. For the unweighted case, we provide an O(m + n log n log² m)-time algorithm for simple polygons, and an O(√n (n+m)log(n+m))-time algorithm for polygons with holes.
Cite as
Bruce W. Brewer and Haitao Wang. Shortest Paths in Geodesic Unit-Disk Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brewer_et_al:LIPIcs.SoCG.2026.23,
author = {Brewer, Bruce W. and Wang, Haitao},
title = {{Shortest Paths in Geodesic Unit-Disk Graphs}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {23:1--23:15},
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.2026.23},
URN = {urn:nbn:de:0030-drops-258297},
doi = {10.4230/LIPIcs.SoCG.2026.23},
annote = {Keywords: unit-disk graph, geodesic distance, shortest paths, geodesic Voronoi diagrams, range emptiness queries, dynamic data structures}
}
Document
Single-Criteria Metric r-Dominating Set Problem via Minor-Preserving Support
Abstract
Given an unweighted graph G, the minimum r-dominating set problem asks for a subset of vertices S of the smallest cardinality, such that every vertex in G is within radius r to some vertex in S. While the r-dominating set problem on planar graph admits PTAS from Baker’s shifting/layering technique when r is a constant, the problem becomes significantly harder when r can depend on n. In fact, under Exponential-Time Hypothesis, Fox-Epstein ηl [SODA 2019] observed that no efficient PTAS can exist for the unbounded r-dominating set problem on planar graphs. One may consider even harder weighted-variant known as the vertex-weighted metric r-dominating set, where edges are associated with lengths, and every vertex is associated with a positive-valued weight, and the goal is to compute an r-dominating set with minimum total weight. As a result, people resorted to bicriteria algorithms by allowing the returned solution to use radius-(1+ε)r balls instead, in addition to the total weight being a 1+ε approximation to the optimal value.
We establish the first single-criteria polynomial-time O(1)-approximation algorithm for the vertex-weighted metric r-dominating set problem on planar graphs when r is part of the input, and can be arbitrarily large compared to n. Our new (single-criteria) O(1)-approximation algorithm uses the quasi-uniformity sampling technique of Chan et al. [SODA 2012] by bounding the shallow cell complexity of the (unbounded) radius-r ball system to be linear in n. To this end we have two technical innovations:
1) The discrete ball system on planar graphs are neither pseudodisks nor have well-defined boundaries for standard union-complexity arguments. We construct a support graph for arbitrary distance ball systems as contractions of Voronoi cells; the sparseness comes as a byproduct.
2) We present an assignment of each depth-(≥3) cell to a unique 3-tuple of ball centers. This allows us to use standard Clarkson-Shor techniques to reduce the counting to cells of depth exactly 3, which we prove to be size O(n) by a novel geometric argument based on our support being a Voronoi contraction.
Cite as
Reilly Browne and Hsien-Chih Chang. Single-Criteria Metric r-Dominating Set Problem via Minor-Preserving Support. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{browne_et_al:LIPIcs.SoCG.2026.24,
author = {Browne, Reilly and Chang, Hsien-Chih},
title = {{Single-Criteria Metric r-Dominating Set Problem via Minor-Preserving Support}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {24:1--24:17},
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.2026.24},
URN = {urn:nbn:de:0030-drops-258300},
doi = {10.4230/LIPIcs.SoCG.2026.24},
annote = {Keywords: Minimum dominating set, planar graphs, shallow cell complexity}
}
Document
The Typical Algebraic Shifting of Graphs and Surfaces
Abstract
We initiate a statistical study of Kalai’s exterior algebraic shifting, focusing on concentration phenomena for random triangulations of a fixed space. First, for a uniform n-vertex refinement of any given graph G, we show that asymptotically almost-surely (a.a.s.) its exterior algebraic shifting is an explicit shifted graph depending only on n and the Betti numbers of G. Next, for any given compact connected Riemannian surface S, sample n points independently at random according to the volume measure, and consider the resulted a.a.s. unique Delaunay triangulation. We prove that a.a.s. its exterior algebraic shifting is an explicit shifted complex depending only on n and the Euler genus of S, and in particular is area-rigid. In both results the expected shifted complex is a homology lex-segment complex, a notion we define combinatorially and characterize numerically à la Björner-Kalai.
As a tool to prove the result on surfaces, we prove a universality result on edge contractions: for every fixed surface triangulation K, every dense enough point set in the surface yields a Delaunay triangulation that edge contracts to K.
Cite as
Denys Bulavka, Eran Nevo, and Yuval Peled. The Typical Algebraic Shifting of Graphs and Surfaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bulavka_et_al:LIPIcs.SoCG.2026.25,
author = {Bulavka, Denys and Nevo, Eran and Peled, Yuval},
title = {{The Typical Algebraic Shifting of Graphs and Surfaces}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {25:1--25:15},
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.2026.25},
URN = {urn:nbn:de:0030-drops-258312},
doi = {10.4230/LIPIcs.SoCG.2026.25},
annote = {Keywords: Algebraic shifting, Delaunay triangulation, surfaces, random triangulation, area rigidity}
}
Document
Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice
Abstract
We study the problem of finding a fair linear scoring function over (numerical) attributes for top-k selection, ensuring fairness through a proportional representation constraint on the protected group. Existing algorithms do not scale efficiently, particularly in higher dimensions. Our hardness analysis shows that in more than two dimensions, no algorithm is likely to scale efficiently with respect to dataset size, and the computational complexity is likely to grow rapidly with dimensionality. However, the hardness results also provide key insights guiding algorithm design, leading to our two-pronged solution: (1) For small k, our analysis reveals a gap in the hardness barrier. By addressing various engineering challenges, including achieving efficient parallelism, we turn this potential of efficiency into an optimized geometry-based algorithm delivering substantial performance gains. (2) For large k, where the hardness is robust, we employ a practically efficient optimization-based algorithm which, despite being theoretically worse, achieves superior real-world performance. Experimental evaluations on real-world datasets then explore scenarios where worst-case behavior does not manifest, identifying areas critical to practical performance. Our solution achieves speedups of up to several orders of magnitude compared to the state of the art, an efficiency made possible through a tight integration of hardness analysis, algorithm design, practical engineering, and empirical evaluation.
Cite as
Guangya Cai. Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cai:LIPIcs.SoCG.2026.26,
author = {Cai, Guangya},
title = {{Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {26:1--26:17},
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.2026.26},
URN = {urn:nbn:de:0030-drops-258320},
doi = {10.4230/LIPIcs.SoCG.2026.26},
annote = {Keywords: Fairness, Top-k, Integration}
}
Document
The Squishy Grid Problem
Abstract
In this paper we consider the problem of approximating Euclidean distances by the infinite integer grid graph. Although the topology of the graph is fixed, we have control over the edge-weight assignment w : E → ℝ_{≥ 0}, and hope to have grid distances be asymptotically isometric to Euclidean distances, that is: For all grid points u,v, dist_w(u,v) = (1± o(1))‖u-v‖₂. We give three methods for solving this problem, each attractive in its own way.
- Our first construction is based on an embedding of the recursive, non-periodic pinwheel tiling of Radin and Conway [Charles Radin, 1994; Radin and Sadun, 1996; John H. Conway and Charles Radin, 1998] into the integer grid. Distances in the pinwheel graph are asymptotically isometric to Euclidean distances, but no explicit bound on the rate of convergence was known. We prove that the multiplicative distortion of the pinwheel graph is (1 + 1/Θ(log^ξ log D)), where D is the Euclidean distance and ξ = Θ(1). The pinwheel tiling approach is conceptually simple, but can be improved quantitatively.
- Our second construction is based on a hierarchical arrangement of highways. It is simple, achieving stretch (1 + 1/Θ(D^{1/9})), which converges doubly exponentially faster than the pinwheel tiling approach.
- The first two methods are deterministic, with rigorous guarantees. An even simpler approach is to sample the edge weights independently and randomly from a common distribution D. Whether there exists a distribution D^* that makes grid distances Euclidean, asymptotically and in expectation, is major open problem in the theory of first passage percolation. Previous experiments show that when D is a Fisher distribution (which is continuous), grid distances are within 1% of Euclidean distances. We demonstrate experimentally that this level of accuracy can be achieved by a simple 2-point distribution that assigns weights 0.41 or 4.75 with probability 44% and 56%, respectively.
Cite as
Zixi Cai, Kuowen Chen, Shengquan Du, Arnold Filtser, Seth Pettie, and Daniel Skora. The Squishy Grid Problem. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cai_et_al:LIPIcs.SoCG.2026.27,
author = {Cai, Zixi and Chen, Kuowen and Du, Shengquan and Filtser, Arnold and Pettie, Seth and Skora, Daniel},
title = {{The Squishy Grid Problem}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {27:1--27:16},
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.2026.27},
URN = {urn:nbn:de:0030-drops-258333},
doi = {10.4230/LIPIcs.SoCG.2026.27},
annote = {Keywords: grid graph, Euclidean distance, metric embedding, first passage percolation}
}
Document
Triangulating a Polygon with Holes in Optimal (Deterministic) Time
Abstract
We consider the problem of triangulating a polygon with n vertices and h holes, or relatedly the problem of computing the trapezoidal decomposition of a collection of h disjoint simple polygonal chains with n vertices total. Clarkson, Cole, and Tarjan (1992) and Seidel (1991) gave randomized algorithms running in O(nlog^*n + hlog h) time, while Bar-Yehuda and Chazelle (1994) described deterministic algorithms running in O(n+hlog^{1+ε}h) or O((n+hlog h)log log h) time, for an arbitrarily small positive constant ε. No improvements have been reported since. We describe a new O(n+hlog h)-time algorithm, which is optimal and deterministic.
More generally, when the given polygonal chains are not necessarily simple and may intersect each other, we show how to compute their trapezoidal decomposition (and in particular, compute all intersections) in optimal O(n+hlog h) deterministic time when the number of intersections is at most n^{1-ε}.
To obtain these results, Chazelle’s linear-time algorithm for triangulating a simple polygon is used as a black box.
Cite as
Timothy M. Chan. Triangulating a Polygon with Holes in Optimal (Deterministic) Time. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chan:LIPIcs.SoCG.2026.28,
author = {Chan, Timothy M.},
title = {{Triangulating a Polygon with Holes in Optimal (Deterministic) Time}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {28:1--28:13},
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.2026.28},
URN = {urn:nbn:de:0030-drops-258348},
doi = {10.4230/LIPIcs.SoCG.2026.28},
annote = {Keywords: Polygons, triangulation, intersection, derandomization}
}
Document
Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above
Abstract
Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case [Duraj et al., 2024; Hsien-Chih Chang et al., 2024; Chan et al., 2025] where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [Karl Bringmann et al., 2022] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter Δ is at least Ω(log n), hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include:
1) A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest.
2) A truly subquadratic time lower bound for Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be quadratic hard when the diameter is Ω(log n) [Karl Bringmann et al., 2022].
3) A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D [Karl Bringmann et al., 2022].
4) A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.
Cite as
Timothy M. Chan, Hsien-Chih Chang, Jie Gao, Sándor Kisfaludi-Bak, Hung Le, and Da Wei Zheng. Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chan_et_al:LIPIcs.SoCG.2026.29,
author = {Chan, Timothy M. and Chang, Hsien-Chih and Gao, Jie and Kisfaludi-Bak, S\'{a}ndor and Le, Hung and Zheng, Da Wei},
title = {{Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {29:1--29:15},
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.2026.29},
URN = {urn:nbn:de:0030-drops-258357},
doi = {10.4230/LIPIcs.SoCG.2026.29},
annote = {Keywords: Graph Diameter, Geometric Intersection Graphs, Unit Ball Graphs}
}
Document
Computing the Girth of a Segment Intersection Graph
Abstract
We present an algorithm that computes the girth of the intersection graph of n given line segments in the plane in O(n^1.483) expected time. This is the first such algorithm with O(n^{3/2-ε}) running time for a positive constant ε, and makes progress towards an open question posed by Chan (SODA 2023). The main techniques include (i) the usage of recent subcubic algorithms for bounded-difference min-plus matrix multiplication, and (ii) an interesting variant of the planar graph separator theorem. The result extends to intersection graphs of connected algebraic curves or semialgebraic sets of constant description complexity.
Cite as
Timothy M. Chan and Yuancheng Yu. Computing the Girth of a Segment Intersection Graph. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chan_et_al:LIPIcs.SoCG.2026.30,
author = {Chan, Timothy M. and Yu, Yuancheng},
title = {{Computing the Girth of a Segment Intersection Graph}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {30:1--30:16},
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.2026.30},
URN = {urn:nbn:de:0030-drops-258364},
doi = {10.4230/LIPIcs.SoCG.2026.30},
annote = {Keywords: Geometric intersection graphs, girth, shortest paths, graph separators, matrix multiplication}
}
Document
Linear-Time (1+ε)-Approximation Algorithms for Two-Line-Center Problems
Abstract
Given a set S of n points in the plane, we study the two-line-center problem: finding two lines that minimize the maximum distance from each point in S to its closest line. We present a (1+ε)-approximation algorithm for the two-line-center problem that runs in O((n/ε) log (1/ε)) time, which improves the previously best O(nlog n + (n/ε²) log (1/ε) + (1/ε³)log (1/ε))-time algorithm. We also consider three variants of this problem, in which the orientations of the two lines are restricted: (1) the orientation of one of the two lines is fixed, (2) the orientations of both lines are fixed, and (3) the two lines are required to be parallel. For each of these three variants, we give the first (1+ε)-approximation algorithm that runs in linear time. In particular, for the variant where the orientation of one of the two lines is fixed, we also give an improved exact algorithm that runs in O(n log n) time and show that it is optimal.
Cite as
Chaeyoon Chung, Anil Maheshwari, and Michiel Smid. Linear-Time (1+ε)-Approximation Algorithms for Two-Line-Center Problems. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chung_et_al:LIPIcs.SoCG.2026.31,
author = {Chung, Chaeyoon and Maheshwari, Anil and Smid, Michiel},
title = {{Linear-Time (1+\epsilon)-Approximation Algorithms for Two-Line-Center Problems}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {31:1--31:17},
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.2026.31},
URN = {urn:nbn:de:0030-drops-258374},
doi = {10.4230/LIPIcs.SoCG.2026.31},
annote = {Keywords: Approximation algorithm, two-line-center problem, k-line-center problem, projective clustering, \epsilon-certificate, \epsilon-coreset, width of a point set}
}
Document
Tensor Computation of Euler Characteristic Functions and Transforms
Abstract
The weighted Euler characteristic transform (WECT) and Euler characteristic function (ECF) have proven to be useful tools in a variety of applications. However, current methods for computing these functions are either not optimized for GPU computation or do not scale to higher-dimensional settings. In this work, we present a tensor-based framework for computing such topological descriptors which is highly optimized for GPU architectures and works in full generality across simplicial and cubical complexes of arbitrary dimension. Experimentally, the framework demonstrates significant speedups over existing methods when computing the WECT and ECF across a variety of two- and three-dimensional datasets. Computation of these transforms is implemented in a publicly available Python package called pyECT.
Cite as
Jessi Cisewski-Kehe, Brittany Terese Fasy, Alexander McCleary, and Eli Quist. Tensor Computation of Euler Characteristic Functions and Transforms. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cisewskikehe_et_al:LIPIcs.SoCG.2026.32,
author = {Cisewski-Kehe, Jessi and Fasy, Brittany Terese and McCleary, Alexander and Quist, Eli},
title = {{Tensor Computation of Euler Characteristic Functions and Transforms}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {32:1--32:17},
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.2026.32},
URN = {urn:nbn:de:0030-drops-258380},
doi = {10.4230/LIPIcs.SoCG.2026.32},
annote = {Keywords: Topological data analysis, weighted Euler characteristic transform, Euler characteristic function, tensor computation, GPU computation}
}
Document
Near-Optimal Bounds for Parameterized Euclidean k-Means
Abstract
The k-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find k representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of k-means in Euclidean spaces parameterized by the number of clusters, k.
In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a (1+ε)-approximation for high-dimensional Euclidean k-means in time 2^{(k/ε)^O(1)} ⋅ dn^O(1).
In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean k-means by showing that there is no 2^{(k/ε)^{1-o(1)}} ⋅ n^O(1) time algorithm achieving a (1+ε)-approximation for k-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is 2^O(k/ε) + O(ndk).
Furthermore, assuming XXH, we show that the seminal O(n^{kd+1}) runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for k-means is optimal for small values of k.
Cite as
Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Near-Optimal Bounds for Parameterized Euclidean k-Means. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.33,
author = {Cohen-Addad, Vincent and C. S., Karthik and Saulpic, David and Schwiegelshohn, Chris},
title = {{Near-Optimal Bounds for Parameterized Euclidean k-Means}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {33:1--33:17},
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.2026.33},
URN = {urn:nbn:de:0030-drops-258391},
doi = {10.4230/LIPIcs.SoCG.2026.33},
annote = {Keywords: k-means clustering, Euclidean space, Fine-Grained Complexity}
}
Document
Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces
Abstract
The k-median and k-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the k-median (resp. k-means) problem is to find k representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a (1+ε)-factor approximation in low-dimensional Euclidean metric for both the k-median and k-means problems in near-linear time 2^{(1/ε)^O(d²)} n ⋅ polylog(n) (where d is the dimension and n is the number of input points).
We improve this running time to 2^{O(1/ε)^{d-1}} ⋅ n ⋅ polylog(n), and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no 2^o(1/ε^{d-1}) n^O(1) algorithm achieving a (1+ε)-approximation for k-means.
Cite as
Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.34,
author = {Cohen-Addad, Vincent and Karthik C. S. and Saulpic, David and Schwiegelshohn, Chris},
title = {{Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {34:1--34:16},
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.2026.34},
URN = {urn:nbn:de:0030-drops-258404},
doi = {10.4230/LIPIcs.SoCG.2026.34},
annote = {Keywords: k-means clustering, k-median clustering, Euclidean space, Fine-Grained Complexity}
}
Document
On Computing the (Exact) Fréchet Distance with a Frog
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
Upward Book Embeddings of Partitioned Digraphs
Abstract
In 1999, Heath, Pemmaraju, and Trenk [SIAM J. Comput. 28(4), 1999] extended the classic notion of book embeddings to digraphs, introducing the concept of upward book embeddings, in which the vertices must appear along the spine in a topological order and the edges are partitioned into pages, so that no two edges in the same page cross. For a partitioned digraph G = (V, ⋃^k_{i=1} E_i), that is, a digraph whose edge set is partitioned into k subsets, an upward book embedding is required to assign edges to pages as prescribed by the given partition. In a companion paper, Heath and Pemmaraju [SIAM J. Comput. 28(5), 1999] proved that the problem of testing the existence of an upward book embedding of a partitioned digraph is linear-time solvable for k = 1 and recently Akitaya, Demaine, Hesterberg, and Liu [GD, 2017] have shown the problem NP-complete for k ≥ 3. In this paper, we study upward book embeddings of partitioned digraphs and focus on the unsolved case k = 2. Our first main result is a novel characterization of the upward embeddings that support an upward book embedding in two pages. We exploit this characterization in several ways, and obtain a rich picture of the complexity landscape of the problem. First, we show that the problem remains NP-complete when k = 2, thus closing the complexity gap for the problem. Second, we show that, for an n-vertex partitioned digraph with a prescribed planar embedding, the existence of an upward book embedding that respects the given planar embedding can be tested in O(n log³ n) time. Finally, leveraging the SPQ(R)-tree decomposition of biconnected graphs into triconnected components, we present a cubic-time testing algorithm for biconnected directed partial 2-trees.
Cite as
Giordano Da Lozzo, Fabrizio Frati, and Ignaz Rutter. Upward Book Embeddings of Partitioned Digraphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dalozzo_et_al:LIPIcs.SoCG.2026.36,
author = {Da Lozzo, Giordano and Frati, Fabrizio and Rutter, Ignaz},
title = {{Upward Book Embeddings of Partitioned Digraphs}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {36:1--36:18},
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.2026.36},
URN = {urn:nbn:de:0030-drops-258424},
doi = {10.4230/LIPIcs.SoCG.2026.36},
annote = {Keywords: upward book embeddings, partitioned digraphs, SPQ-trees, 2-trees}
}
Document
A Free Lunch: Manifolds of Positive Reach Can Be Smoothed Without Decreasing the Reach
Abstract
Assumptions on the reach are crucial for ensuring the correctness of many geometric and topological algorithms, including triangulation, manifold reconstruction and learning, homotopy reconstruction, and methods for estimating curvature or reach. However, these assumptions are often coupled with the requirement that the manifold be smooth, typically at least C².
In this paper, we prove that any manifold with positive reach can be approximated arbitrarily well by a C^∞ manifold without significantly reducing the reach. More precisely, given a manifold with reach R, we construct a manifold that is ε-close to it in the C¹ sense (both the manifold and its tangent spaces are close), and has reach at least R-ε. The proof employs techniques from differential topology - partitions of unity and smoothing using convolution kernels.
This result implies that nearly all theorems established for C² or manifolds with a certain reach naturally extend to manifolds with the same reach, even if they are not C², for free!
Cite as
Hana Dal Poz Kouřimská, André Lieutier, and Mathijs Wintraecken. A Free Lunch: Manifolds of Positive Reach Can Be Smoothed Without Decreasing the Reach. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 37:1-37:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dalpozkourimska_et_al:LIPIcs.SoCG.2026.37,
author = {Dal Poz Kou\v{r}imsk\'{a}, Hana and Lieutier, Andr\'{e} and Wintraecken, Mathijs},
title = {{A Free Lunch: Manifolds of Positive Reach Can Be Smoothed Without Decreasing the Reach}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {37:1--37:19},
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.2026.37},
URN = {urn:nbn:de:0030-drops-258434},
doi = {10.4230/LIPIcs.SoCG.2026.37},
annote = {Keywords: Reach, Manifolds, Smoothing, Differentiability, Differential topology}
}
Document
On the Size of k-Irreducible Triangulations
Abstract
A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of an orientable surface of genus g has O(k²g) triangles, which is optimal. This is an improvement over the previous best bound k^O(k) g² of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].
Cite as
Vincent Delecroix, Oscar Fontaine, and Arnaud de Mesmay. On the Size of k-Irreducible Triangulations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{delecroix_et_al:LIPIcs.SoCG.2026.38,
author = {Delecroix, Vincent and Fontaine, Oscar and de Mesmay, Arnaud},
title = {{On the Size of k-Irreducible Triangulations}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {38:1--38:16},
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.2026.38},
URN = {urn:nbn:de:0030-drops-258446},
doi = {10.4230/LIPIcs.SoCG.2026.38},
annote = {Keywords: surface, irreducible triangulation, system of curves, minimal position, systolic geometry}
}
Document
Locality Sensitive Hashing in Hyperbolic Space
Abstract
For a metric space (X, d), a family ℋ of locality sensitive hash functions is called (r, cr, p₁, p₂) sensitive if a randomly chosen function h ∈ ℋ has probability at least p₁ (at most p₂) to map any a, b ∈ X in the same hash bucket if d(a, b) ≤ r (or d(a, b) ≥ cr). Locality Sensitive Hashing (LSH) is one of the most popular techniques for approximate nearest-neighbor search in high-dimensional spaces, and has been studied extensively for Hamming, Euclidean, and spherical geometries. An (r, cr, p₁, p₂)-sensitive hash function enables approximate nearest neighbor search (i.e., returning a point within distance cr from a query q if there exists a point within distance r from q) with space O(n^{1+ρ}) and query time O(n^ρ) where ρ = (log 1/p₁)/(log 1/p₂). But LSH for hyperbolic spaces ℍ^d remains largely unexplored. In this work, we present the first LSH construction native to hyperbolic space. For the hyperbolic plane (d = 2), we show a construction achieving ρ ≤ 1/c, based on the hyperplane rounding scheme. For general hyperbolic spaces (d ≥ 3), we use dimension reduction from ℍ^d to ℍ² and the 2D hyperbolic LSH to get ρ ≤ 1.59/c. On the lower bound side, we show that the lower bound on ρ of Euclidean LSH extends to the hyperbolic setting via local isometry, therefore giving ρ ≥ 1/c².
Cite as
Chengyuan Deng, Jie Gao, Kevin Lu, Feng Luo, and Cheng Xin. Locality Sensitive Hashing in Hyperbolic Space. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{deng_et_al:LIPIcs.SoCG.2026.39,
author = {Deng, Chengyuan and Gao, Jie and Lu, Kevin and Luo, Feng and Xin, Cheng},
title = {{Locality Sensitive Hashing in Hyperbolic Space}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {39:1--39:19},
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.2026.39},
URN = {urn:nbn:de:0030-drops-258454},
doi = {10.4230/LIPIcs.SoCG.2026.39},
annote = {Keywords: Locality Sensitive Hashing, Hyperbolic Geometry, Dimension Reduction, Approximate Nearest Neighbor Search}
}
Document
Computing the Intrinsic Delaunay Triangulation of a Closed Polyhedral Surface
Abstract
Every surface that is intrinsically polyhedral can be represented by a portalgon: a collection of polygons in the Euclidean plane with some pairs of equally long edges abstractly identified. While this representation is arguably simpler than meshes (flat polygons in ℝ³ forming a surface), it has unbounded happiness: a shortest path in the surface may visit the same polygon arbitrarily many times. This pathological behavior is an obstacle towards efficient algorithms. On the other hand, Löffler, Ophelders, Staals, and Silveira [SoCG 2023] recently proved that the (intrinsic) Delaunay triangulations have bounded happiness.
In this paper, given a closed polyhedral surface S, represented by a triangular portalgon T, we provide an algorithm to compute the Delaunay triangulation of S whose vertices are the singularities of S (the points whose surrounding angle is distinct from 2π). The time complexity of our algorithm is polynomial in the number of triangles and in the logarithm of the aspect ratio r of T. Within our model of computation, we show that the dependency in log r is unavoidable. Our algorithm can be used to pre-process a triangular portalgon before computing shortest paths on its surface, and to determine whether the surfaces of two triangular portalgons are isometric.
Cite as
Loïc Dubois. Computing the Intrinsic Delaunay Triangulation of a Closed Polyhedral Surface. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dubois:LIPIcs.SoCG.2026.40,
author = {Dubois, Lo\"{i}c},
title = {{Computing the Intrinsic Delaunay Triangulation of a Closed Polyhedral Surface}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {40:1--40:15},
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.2026.40},
URN = {urn:nbn:de:0030-drops-258460},
doi = {10.4230/LIPIcs.SoCG.2026.40},
annote = {Keywords: Polyhedral surface, intrinsic Delaunay triangulation, algorithmic complexity}
}
Document
The Depth Poset Under Transpositions in the Filter
Abstract
The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.
Cite as
Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, Manuel Soriano-Trigueros, and Fedor Zimin. The Depth Poset Under Transpositions in the Filter. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2026.41,
author = {Edelsbrunner, Herbert and Lipi\'{n}ski, Micha{\l} and Mrozek, Marian and Soriano-Trigueros, Manuel and Zimin, Fedor},
title = {{The Depth Poset Under Transpositions in the Filter}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {41:1--41:18},
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.2026.41},
URN = {urn:nbn:de:0030-drops-258479},
doi = {10.4230/LIPIcs.SoCG.2026.41},
annote = {Keywords: Algebraic topology, Lefschetz complexes, persistent homology, vines and vineyards, birth-death pairs, shallow pairs, relations, partial orders, transpositions}
}
Document
Compressed Data Structures for Heegaard Splitting
Abstract
Heegaard splittings provide a natural representation of closed 3-manifolds by gluing handlebodies along a common surface. These splittings can be equivalently given by two finite sets of meridians lying on the surface, which define a Heegaard diagram. We present a data structure to effectively represent Heegaard diagrams as normal curves with respect to triangulations of a surface of complexity measured by the space required to express the normal coordinates' vectors in binary. This structure can be significantly more compressed than triangulations of 3-manifolds, giving exponential gains for some families. Even with this succinct definition of complexity, we establish polynomial-time algorithms for comparing and manipulating diagrams, performing stabilizations, detecting trivial stabilizations and reductions, and computing topological invariants of the underlying manifolds, such as their fundamental and homology groups. We also contrast early implementations of our techniques with standard software programs for 3-manifolds, achieving faster algorithms for the average cases and exponential gains in speed for some particular presentations of the inputs.
Cite as
Henrique Ennes and Clément Maria. Compressed Data Structures for Heegaard Splitting. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ennes_et_al:LIPIcs.SoCG.2026.42,
author = {Ennes, Henrique and Maria, Cl\'{e}ment},
title = {{Compressed Data Structures for Heegaard Splitting}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {42:1--42:15},
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.2026.42},
URN = {urn:nbn:de:0030-drops-258484},
doi = {10.4230/LIPIcs.SoCG.2026.42},
annote = {Keywords: 3-manifold, Heegaard splitting, curves on surfaces, surface theory, data structure, computational topology}
}
Document
Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-Offs
Abstract
We study unlabeled multi-robot motion planning for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Solovey et al. (RSS'15) provide a near-optimal solution assuming that start/target positions must have pairwise distance at least 4, and at least √5≈2.236 from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments.
In this paper we present a generalized algorithm that achieve different trade-offs on the robots-separation and obstacles-separation bounds, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) the robots-separation is 2 2/3 and the obstacles-separation is 1 2/3, or (ii) the robots-separation is ≈3.291 and the obstacles-separation ≈1.354. Additionally, we introduce a different strategy yielding a polynomial-time solution when the robots-separation is only 2, and the obstacles-separation is 3. Finally, we show that without any robots-separation assumption, obstacles-separation of at least 1.5 may be necessary for a solution to exist.
Cite as
Tsuri Farhana, Omrit Filtser, and Shalev Goldshtein. Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-Offs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{farhana_et_al:LIPIcs.SoCG.2026.43,
author = {Farhana, Tsuri and Filtser, Omrit and Goldshtein, Shalev},
title = {{Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-Offs}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {43:1--43:16},
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.2026.43},
URN = {urn:nbn:de:0030-drops-258495},
doi = {10.4230/LIPIcs.SoCG.2026.43},
annote = {Keywords: multi-robot motion planning}
}
Document
Tilt Automata: Gathering Particles with Uniform External Control
Abstract
Motivated by targeted drug delivery, we investigate the gathering of particles in the full tilt model of externally controlled motion planning: A set of particles is located at the tiles of a polyomino with all particles reacting uniformly to an external force by moving as far as possible in one of the four axis-parallel directions until they hit the boundary. The goal is to choose a sequence of directions that moves all particles to a common position. Our results include a polynomial-time algorithm for gathering in a completely filled polyomino as well as hardness reductions for approximating shortest gathering sequences and for determining whether the particles in a partially filled polyomino can be gathered. We pay special attention to the impact of restricted geometry, particularly polyominoes without holes. As a corollary, we make progress on an open question from [Balanza-Martinez et al., SODA 2020] by showing that deciding whether a given position can be occupied remains NP-hard in polyominoes without holes. Our results build on a connection we establish between tilt models and the theory of synchronizing automata.
Cite as
Sándor P. Fekete, Jonas Friemel, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, and Christian Scheffer. Tilt Automata: Gathering Particles with Uniform External Control. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fekete_et_al:LIPIcs.SoCG.2026.44,
author = {Fekete, S\'{a}ndor P. and Friemel, Jonas and Kramer, Peter and Reinhardt, Jan-Marc and Rieck, Christian and Scheffer, Christian},
title = {{Tilt Automata: Gathering Particles with Uniform External Control}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {44:1--44:19},
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.2026.44},
URN = {urn:nbn:de:0030-drops-258508},
doi = {10.4230/LIPIcs.SoCG.2026.44},
annote = {Keywords: Uniform control, gathering, full tilt, polyominoes, synchronizing automata}
}
Document
Line Segment Visibility in Simple Polygons: Exact, Robust, Scalable Computation and Applications
Abstract
The weak visibility polygon of a line segment s inside a simple polygon P, denoted by V_P(s), is the region of the polygon that is visible from at least one point on s. Given its fundamental nature in computational geometry, several algorithms have been proposed to compute weak visibility polygons efficiently, each with different trade-offs in terms of preprocessing time, query time, and space complexity. Although there are many applications that require computing these polygons such as computer graphics, robot motion planning, and network communication systems, there is a lack of any implementations of these algorithms in the literature - not to mention one that is exact, robust, and scalable. Furthermore, weak segment visibility polygons are used as basic building blocks in several other algorithms, such as in minimum-link path computation.
In this work, we present an implementation of an optimal linear-time algorithm for computing the weak visibility polygon of a segment inside a triangulated simple polygon. Our implementation provides exact, robust geometric primitives and optimizations to handle large inputs with more than 18,000,000 vertices. We demonstrate two concrete applications: (1) construction of window partitions, a standard data structure in visibility algorithms, and (2) support for optimal minimum-link path queries between two points in a simple polygon, the latter serving as a direct use case of the former. Experimental results on a variety of polygon families confirm that the end-to-end running time scales linearly with the size of the polygon and is dominated by the cost of computing the triangulation, validating the practicality and scalability of the approach. The implementation is released as open source in the format of a CGAL package to support reproducibility and further research.
Cite as
Sándor P. Fekete, Prahlad Narasimhan Kasthurirangan, Phillip Keldenich, Fabian Kollhoff, Chek-Manh Loi, and Michael Perk. Line Segment Visibility in Simple Polygons: Exact, Robust, Scalable Computation and Applications. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fekete_et_al:LIPIcs.SoCG.2026.45,
author = {Fekete, S\'{a}ndor P. and Kasthurirangan, Prahlad Narasimhan and Keldenich, Phillip and Kollhoff, Fabian and Loi, Chek-Manh and Perk, Michael},
title = {{Line Segment Visibility in Simple Polygons: Exact, Robust, Scalable Computation and Applications}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {45:1--45:19},
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.2026.45},
URN = {urn:nbn:de:0030-drops-258516},
doi = {10.4230/LIPIcs.SoCG.2026.45},
annote = {Keywords: Visibility, line segments, link distance, window partition, computation, implementation, robustness, scalability, exactness, CGAL}
}
Document
A Branch-And-Bound Algorithm for the Traveling Salesman Problem with Difficult Neighborhoods
Abstract
The Traveling Salesman Problem with Neighborhoods (TSPN) generalizes the classical Traveling Salesman Problem (TSP) by requiring a tour to visit a set of polygonal regions rather than fixed points, a natural goal that arises in various applications. While the geometric TSP allows arbitrarily close approximation and provably optimal solutions for benchmark instances of significant size, the TSPN is considerably more challenging, both in theory (due to APX-hardness) and practice, for which only benchmark instances up to 16 regions have been solved to optimality.
Here we present a branch-and-bound algorithm that combines a spectrum of geometry-based filters (for reducing the number of considered sequences) with Second-Order Cone Programs (SOCP) (for computing optimal tours for a given permutation of neighborhoods). This allows us to solve larger polygonal TSPN instances than before to within an optimality tolerance of 0.1%; moreover, while previous work (both in theory and practice) relied on relatively benign neighborhoods, we can handle non-convex, non-simple neighborhoods of different sizes. In experiments on 490 benchmark instances with up to 50 polygons each, our method achieves a 99.6% optimality rate within 300s, with the remaining two instances solved within 595s. For 68 larger instances of size n = 60, our method still allows solving 86.8% of instances to optimality within 900s, leaving only 3 of the instances with optimality gaps above 3%, with the maximum being 5.53%.
Cite as
Sándor P. Fekete, Rouven Kniep, Dominik Krupke, and Michael Perk. A Branch-And-Bound Algorithm for the Traveling Salesman Problem with Difficult Neighborhoods. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fekete_et_al:LIPIcs.SoCG.2026.46,
author = {Fekete, S\'{a}ndor P. and Kniep, Rouven and Krupke, Dominik and Perk, Michael},
title = {{A Branch-And-Bound Algorithm for the Traveling Salesman Problem with Difficult Neighborhoods}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {46:1--46:20},
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.2026.46},
URN = {urn:nbn:de:0030-drops-258529},
doi = {10.4230/LIPIcs.SoCG.2026.46},
annote = {Keywords: Geometric optimization, geometric covering, TSP with neighborhoods, exact algorithms, algorithm engineering}
}
Document
Computing the Skyscraper Invariant
Abstract
We develop the first algorithms for computing the Skyscraper Invariant [FJNT24]. This is a filtration of the classical rank invariant for multiparameter persistence modules defined by the Harder-Narasimhan filtrations along every central charge supported at a single parameter value.
Cheng’s algorithm [Cheng24] can be used to compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension, but in practice, the large dimension of persistence modules makes this direct approach infeasible. We show that by exploiting the additivity of the HN filtration and the special central charges, one can get away with a brute-force approach. For d-parameter modules, this produces an FPT ε-approximate algorithm with runtime dominated by 𝒪(1/ε^d ⋅ T_dec), where T_dec is the time for decomposition, which we compute with aida [DJK25].
We show that the wall-and-chamber structure of the module can be computed via lower envelopes of degree d - 1 polynomials. This allows for an exact computation of the Skyscraper Invariant roughly in 𝒪(n^d ⋅ T_dec) time for n the size of the presentation and enables a fast hybrid algorithm.
For 2-parameter modules, we have implemented not only our algorithms but also, for the first time, Cheng’s algorithm. We compare all algorithms and, as a proof of concept for data analysis, compute a filtered version of the Multiparameter Landscape for biomedical data.
Cite as
Marc Fersztand and Jan Jendrysiak. Computing the Skyscraper Invariant. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 47:1-47:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fersztand_et_al:LIPIcs.SoCG.2026.47,
author = {Fersztand, Marc and Jendrysiak, Jan},
title = {{Computing the Skyscraper Invariant}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {47:1--47:23},
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.2026.47},
URN = {urn:nbn:de:0030-drops-258535},
doi = {10.4230/LIPIcs.SoCG.2026.47},
annote = {Keywords: Topological Data Analysis, Multiparameter Persistence, Persistence, Harder-Narasimhan Filtration, Skyscraper Invariant}
}
Document
FPT Approximations for Capacitated Sum of Radii and Diameters
Abstract
The Capacitated Sum of Radii problem involves partitioning a set of points P, where each point p ∈ P has capacity U_p, into k clusters that minimize the sum of cluster radii, such that the number of points in the cluster centered at point p is at most U_p. We begin by showing that the problem is APX-hard, and that under gap-ETH there is no parameterized approximation scheme (FPT-AS). We then construct a ≈5.83-approximation algorithm in FPT time (improving a previous ≈7.61 approximation in FPT time). Our results also hold when the objective is a general monotone symmetric norm of radii. We also improve the approximation factors for the uniform capacity case, and for the closely related problem of Capacitated Sum of Diameters.
Cite as
Arnold Filtser and Ameet Gadekar. FPT Approximations for Capacitated Sum of Radii and Diameters. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{filtser_et_al:LIPIcs.SoCG.2026.48,
author = {Filtser, Arnold and Gadekar, Ameet},
title = {{FPT Approximations for Capacitated Sum of Radii and Diameters}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {48:1--48:18},
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.2026.48},
URN = {urn:nbn:de:0030-drops-258545},
doi = {10.4230/LIPIcs.SoCG.2026.48},
annote = {Keywords: clustering, sum of radii, sum of diameter, capacitated clustering, fpt}
}
Document
Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality
Abstract
In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of computational geometry with numerous applications. While classical work on this problem often focuses on exploiting connections to semidefinite programming typically leading to approximation algorithms, we focus on exact algorithms and propose a novel distance-from-triviality parameterization framework to obtain tractability results for d-EDMC.
We identify key structural patterns in the input that capture entry density, including chordal substructures and coverability of specified entries by fully specified principal submatrices. We obtain:
1) The first fixed-parameter algorithm (FPT algorithm) for d-EDMC parameterized by d and the maximum number of unspecified entries per row/column. This is achieved through a novel compression algorithm that reduces a given instance to a submatrix on 𝒪(1) rows (for fixed values of the parameters).
2) The first FPT algorithm for d-EDMC parameterized by d and the minimum number of fully specified principal submatrices whose entries cover all specified entries of the given matrix. This result is also achieved through a compression algorithm.
3) A polynomial-time algorithm for d-EDMC when both d and the minimum fill-in of a natural graph representing the specified entries are fixed constants. This result is achieved by combining tools from distance geometry and algorithms from real algebraic geometry. Our work identifies interesting parallels between EDM completion and graph problems, with our algorithms exploiting techniques from both domains.
Cite as
Fedor V. Fomin, Petr A. Golovach, M. S. Ramanujan, and Saket Saurabh. Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fomin_et_al:LIPIcs.SoCG.2026.49,
author = {Fomin, Fedor V. and Golovach, Petr A. and Ramanujan, M. S. and Saurabh, Saket},
title = {{Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {49:1--49:15},
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.2026.49},
URN = {urn:nbn:de:0030-drops-258552},
doi = {10.4230/LIPIcs.SoCG.2026.49},
annote = {Keywords: Parameterized Complexity, Euclidean Embedding, Polynomial Compression}
}
Document
Separators for Intersection Graphs of Spheres
Abstract
We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension d. One of our results is that if an intersection graph of n spheres in ℝ^d has m edges, then it contains a balanced separator of size O_d(m^{1/d}n^{1-2/d}). This bound is best possible in terms of the parameters involved. The same result holds if the balls and spheres are replaced by fat convex bodies and their boundaries.
Cite as
Jacob Fox and Jonathan Tidor. Separators for Intersection Graphs of Spheres. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fox_et_al:LIPIcs.SoCG.2026.50,
author = {Fox, Jacob and Tidor, Jonathan},
title = {{Separators for Intersection Graphs of Spheres}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {50:1--50:16},
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.2026.50},
URN = {urn:nbn:de:0030-drops-258566},
doi = {10.4230/LIPIcs.SoCG.2026.50},
annote = {Keywords: graph separators, intersection graphs}
}
Document
First-Order Logic and Twin-Width for Some Geometric Graphs
Abstract
For some geometric graph classes, tractability of testing first-order formulas is precisely characterised by the graph parameter twin-width. This was first proved for interval graphs among others in [BCKKLT, IPEC '22], where the equivalence is called delineation, and more generally holds for circle graphs, rooted directed path graphs, and H-graphs when H is a forest. Delineation is based on the key idea that geometric graphs often admit natural vertex orderings, allowing to use the very rich theory of twin-width for ordered graphs.
Answering two questions raised in their work, we prove that delineation holds for intersection graphs of non-degenerate axis-parallel unit segment graphs, but fails for visibility graphs of 1.5D terrains. We also prove delineation for intersection graphs of circular arcs.
Cite as
Colin Geniet, Gunwoo Kim, and Lucas Meijer. First-Order Logic and Twin-Width for Some Geometric Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{geniet_et_al:LIPIcs.SoCG.2026.51,
author = {Geniet, Colin and Kim, Gunwoo and Meijer, Lucas},
title = {{First-Order Logic and Twin-Width for Some Geometric Graphs}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {51:1--51:16},
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.2026.51},
URN = {urn:nbn:de:0030-drops-258575},
doi = {10.4230/LIPIcs.SoCG.2026.51},
annote = {Keywords: Twin-width, axis-parallel unit segment graphs, circular arc graphs, terrain visibility graphs, first-order logic, model checking, FPT}
}
Document
Online Packing of Orthogonal Polygons
Abstract
While rectangular and box-shaped objects dominate the classic discourse of theoretic investigations, a fascinating frontier lies in packing more complex shapes. Given recent insights that convex polygons do not allow for constant competitive online algorithms for diverse variants under translation, we study orthogonal polygons, in particular of small complexity. For translational packings of orthogonal 6-gons, we show that the competitive ratio of any online algorithm that aims to pack the items into a minimal number of unit bins is in Ω(n/(log n)), where n denotes the number of objects. In contrast, we show that constant competitive algorithms exist when the orthogonal 6-gons are symmetric or small. For (orthogonally convex) orthogonal 8-gons, we show that the trivial n-competitive algorithm, which places each item in its own bin, is best-possible, i.e., every online algorithm has an asymptotic competitive ratio of at least n. This implies that for general orthogonal polygons, the trivial algorithm is best possible.
Interestingly, for packing degenerate orthogonal polygons (with thickness 0), called skeletons, the change in complexity is even more drastic. While constant competitive algorithms for 6-skeletons exist, no online algorithm for 8-skeletons achieves a competitive ratio better than n.
For other packing variants of orthogonal 6-gons under translation, our insights imply the following consequences. The asymptotic competitive ratio of any online algorithm is in Ω(n/(log n)) for strip packing, and there exist online algorithms with competitive ratios in O(1) for perimeter packing, or in O(√n) for minimizing the area of the bounding box. Moreover, the critical packing density is positive (if every object individually fits into the interior of a unit bin).
Cite as
Tim Gerlach, Benjamin Hennies, and Linda Kleist. Online Packing of Orthogonal Polygons. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 52:1-52:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gerlach_et_al:LIPIcs.SoCG.2026.52,
author = {Gerlach, Tim and Hennies, Benjamin and Kleist, Linda},
title = {{Online Packing of Orthogonal Polygons}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {52:1--52:17},
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.2026.52},
URN = {urn:nbn:de:0030-drops-258589},
doi = {10.4230/LIPIcs.SoCG.2026.52},
annote = {Keywords: Packing, orthogonal polygon, algorithm, offline, online, competitive ratio, bin packing, strip packing, perimeter packing, critical density, 6-gon, 8-gon, L-shape, Z-shape, skeleton}
}
Document
Constructing Doppelgängers of Greedy Geometric Spanners in Practice
Abstract
Greedy geometric spanners are considered to be the gold standard for their near-optimal guarantees in terms of sparsity and total weight. However, their inefficient construction poses significant challenges for large-scale geometric networks, especially for low values of stretch factors (< 2). We present Θ-Greedy, a simple and practical parallel algorithm engineered for constructing doppelgängers of greedy geometric spanners that empirically resemble the greedy spanners in key structural and performance metrics, including average degree, degree, and lightness. Unlike approximate greedy spanners, doppelgängers of greedy spanners are almost indistinguishable from the actual greedy spanners in practice.
In our experiments, Θ-Greedy consistently produced greedy spanner doppelgängers across a broad range of synthetic and real-world datasets, offering the first practical alternative to the computationally intensive greedy spanners. Θ-Greedy can construct a 1.1-spanner on a 128K-element uniformly distributed point set in well under 5 minutes. In contrast, Bucketing, the most practical greedy spanner algorithm, takes around 3 hours. For million-sized point sets, Θ-Greedy can run to completion in a few hours, making it much faster than Bucketing, which takes days to finish. In extensive experiments on synthetic and real-world datasets, Θ-Greedy delivered speedups of up to 147x over Bucketing while preserving greedy-like sparsity and weight.
For broader uses of the algorithm and reproducibility, we share our engineered C++ code.
Cite as
Anirban Ghosh. Constructing Doppelgängers of Greedy Geometric Spanners in Practice. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ghosh:LIPIcs.SoCG.2026.53,
author = {Ghosh, Anirban},
title = {{Constructing Doppelg\"{a}ngers of Greedy Geometric Spanners in Practice}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {53:1--53:21},
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.2026.53},
URN = {urn:nbn:de:0030-drops-258599},
doi = {10.4230/LIPIcs.SoCG.2026.53},
annote = {Keywords: geometric graph, geometric spanners, greedy spanners, algorithm engineering}
}
Document
Approximating Pareto Sum via Bounded Monotone Min-Plus Convolution
Abstract
The Pareto sum of two-dimensional point sets P and Q in ℝ² is defined as the skyline of the points in their Minkowski sum. The problem of efficiently computing the Pareto sum arises frequently in bi-criteria optimization algorithms. Prior work establishes that computing the Pareto sum of sets P and Q of size n suffers from conditional lower bounds that rule out strongly subquadratic O(n^{2-ε})-time algorithms, even when the output size is Θ(n). Naturally, we ask: How efficiently can we approximate Pareto sums, both in theory and practice? Can we beat the near-quadratic-time state of the art for exact algorithms?
On the theoretical side, we formulate a notion of additively approximate Pareto sets and show that computing an approximate Pareto set is fine-grained equivalent to Bounded Monotone Min-Plus Convolution. Leveraging a remarkable Õ(n^{1.5})-time algorithm for the latter problem (Chi, Duan, Xie, Zhang; STOC '22), we thus obtain a strongly subquadratic (and conditionally optimal) approximation algorithm for computing Pareto sums.
On the practical side, we engineer different algorithmic approaches for approximating Pareto sets on realistic instances. Our implementations enable a granular trade-off between approximation quality and running time/output size compared to the state of the art for exact algorithms established in (Funke, Hespe, Sanders, Storandt, Truschel; Algorithmica '25). Perhaps surprisingly, the (theoretical) connection to Bounded Monotone Min-Plus Convolution remains beneficial even for our implementations: in particular, we implement a simplified, yet still subquadratic version of an algorithm due to Chi, Duan, Xie and Zhang, which on some sufficiently large instances outperforms the competing quadratic-time approaches.
Cite as
Geri Gokaj, Marvin Künnemann, Sabine Storandt, and Carina Truschel. Approximating Pareto Sum via Bounded Monotone Min-Plus Convolution. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 54:1-54:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gokaj_et_al:LIPIcs.SoCG.2026.54,
author = {Gokaj, Geri and K\"{u}nnemann, Marvin and Storandt, Sabine and Truschel, Carina},
title = {{Approximating Pareto Sum via Bounded Monotone Min-Plus Convolution}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {54:1--54:21},
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.2026.54},
URN = {urn:nbn:de:0030-drops-258602},
doi = {10.4230/LIPIcs.SoCG.2026.54},
annote = {Keywords: computational geometry, fine-grained complexity, algorithm engineering}
}
Document
Linear Time Single-Source Shortest Path Algorithms in Euclidean Graph Classes
Abstract
In the celebrated paper of Henzinger, Klein, Rao and Subramanian (1997), it was shown that planar graphs admit a linear time single-source shortest path algorithm. Their algorithm unfortunately does not extend to Euclidean graph classes. We give criteria and prove that any Euclidean graph class satisfying the criteria admits a linear time single-source shortest path algorithm. As a main ingredient, we show that the contracted graphs of these Euclidean graph classes admit sublinear separators.
Cite as
Joachim Gudmundsson, Yuan Sha, and Sampson Wong. Linear Time Single-Source Shortest Path Algorithms in Euclidean Graph Classes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 55:1-55:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gudmundsson_et_al:LIPIcs.SoCG.2026.55,
author = {Gudmundsson, Joachim and Sha, Yuan and Wong, Sampson},
title = {{Linear Time Single-Source Shortest Path Algorithms in Euclidean Graph Classes}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {55:1--55:14},
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.2026.55},
URN = {urn:nbn:de:0030-drops-258618},
doi = {10.4230/LIPIcs.SoCG.2026.55},
annote = {Keywords: Graph algorithms, Single-Source Shortest Path, Euclidean Graphs, Recursive Division}
}
Document
Optimal Randomized Clustering of Matrices
Abstract
If X = (𝖬_n(ℝ),‖⋅‖_X) is a unitarily invariant normed space, i.e., ‖𝖴𝖠𝖵‖_X = ‖𝖠‖_X for every matrix 𝖠 ∈ 𝖬_n(ℝ) and every two orthogonal matrices 𝖴,𝖵 ∈ 𝖬_n(ℝ), then we evaluate up to universal constant factors the smallest σ > 0 for which there is a probability distribution over partitions of X into clusters of diameter at most 1 yet for every two matrices 𝖠,𝖡 ∈ 𝖬_n(ℝ) the probability that they fall into distinct clusters is at most σ times the X-distance between 𝖠 and 𝖡. Specifically, we prove that this infimal σ, which is called the separation modulus of X and is denoted SEP(X), satisfies:
(1) SEP(X) = Θ(√n⋅ ‖𝖨_n‖_X⋅ diam(B_X)),
where 𝖨_n is the n-by-n identity matrix and diam(B_X) is the diameter with respect to the standard Euclidean metric on 𝖬_n(ℝ) of the unit ball B_ X of X. Our proof of (1) proceeds through an asymptotic evaluation of the spectral gap of the Laplacian with Dirichlet boundary conditions on B_ X, which we achieve by exact computations for a Jacobi orthogonal random matrix ensemble. Assuming oracle access to norm evaluations in X, by combining (1) with a new deterministic algorithm for a O(1)-approximation of the diameter of convex bodies in ℝⁿ that are given by a weak membership oracle and are symmetric with respect to coordinate permutations and reflections about the standard axes (this task is famously known to be impossible in the absence of such symmetries), we get an oracle polynomial time algorithm whose output is the separation modulus of X up to universal constant factors. Another example of a consequence of (1) is that for each m ∈ {1,…,n} the separation modulus of the m'th Ky Fan norm on 𝖬_n(ℝ) is bounded from above and from below by universal constant multiples of m√n if m ⩾ √n, and of n if m ⩽ √n. We also deduce from (1) an upper bound on the Lipschitz extension modulus of X that improves over the previously best-known bound even in the special case when X is 𝖬_n(ℝ) equipped with the 𝓁₂ⁿ → 𝓁₂ⁿ operator norm.
Cite as
Mustafa Alper Gunes and Assaf Naor. Optimal Randomized Clustering of Matrices. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 56:1-56:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gunes_et_al:LIPIcs.SoCG.2026.56,
author = {Gunes, Mustafa Alper and Naor, Assaf},
title = {{Optimal Randomized Clustering of Matrices}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {56:1--56:18},
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.2026.56},
URN = {urn:nbn:de:0030-drops-258624},
doi = {10.4230/LIPIcs.SoCG.2026.56},
annote = {Keywords: Clustering, Unitarily Invariant Matrix Norms, Oracle Polynomial Time Approximation Algorithms for Radii of Convex Bodies, Extension of Lipschitz Functions, Random Matrices, Spectrum of the Laplacian with Dirichlet Boundary Conditions, Reverse Isoperimetry}
}
Document
Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps
Abstract
We present an exact and efficient algorithm for computing the Reeb space of a bivariate PL map. The Reeb space is a topological structure that generalizes the Reeb graph to the setting of multiple scalar-valued functions defined over a shared domain, a situation that frequently arises in practical applications. While the Reeb graph has become a standard tool in computer graphics, shape analysis, and scientific visualization, the Reeb space is still in the early stages of adoption. Although several algorithms for computing the Reeb space have been proposed, none offer an implementation that is both exact and efficient, which has substantially limited its practical use. To address this gap, we introduce singular arrange and traverse, a new algorithm built upon the arrange and traverse framework [Hristov et al., 2025]. Our method exploits the fact that, in the bivariate case, only singular edges contribute to the structure of Reeb space, allowing us to ignore many regular edges [Tierny and Carr, 2017]. This observation results in substantial efficiency gains on datasets where most edges are regular, which is common in many numerical simulations of physical systems. We provide an implementation of our method and benchmark it against the original arrange and traverse algorithm, showing performance gains of up to four orders of magnitude on real-world datasets.
Cite as
Petar Hristov, Ingrid Hotz, and Talha Bin Masood. Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hristov_et_al:LIPIcs.SoCG.2026.57,
author = {Hristov, Petar and Hotz, Ingrid and Masood, Talha Bin},
title = {{Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {57:1--57:17},
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.2026.57},
URN = {urn:nbn:de:0030-drops-258644},
doi = {10.4230/LIPIcs.SoCG.2026.57},
annote = {Keywords: Computational topology, Reeb graph, Reeb space, Multivariate data, Multifield, Geometric arrangement}
}
Document
On Sparse Representations of 3‑Manifolds
Abstract
3-manifolds are commonly represented as triangulations, consisting of abstract tetrahedra whose triangular faces are identified in pairs. The combinatorial sparsity of a triangulation, as measured by the treewidth of its dual graph, plays a fundamental role in the design of parameterized algorithms. In this work, we investigate algorithmic procedures that transform or modify a given triangulation while controlling specific sparsity parameters. First, we revisit a standard, linear-time algorithm that converts a given triangulation into a Heegaard diagram of the underlying 3-manifold, showing that the construction preserves treewidth. We apply this construction to exhibit a fixed-parameter tractable framework for computing Kuperberg’s quantum invariants of 3-manifolds. Second, we present a quasi-linear-time algorithm that retriangulates a given triangulation into one with maximum edge valence of at most nine, while only moderately increasing the treewidth of the dual graph. Combining these two algorithms yields a quasi-linear-time algorithm that produces, from a given triangulation, a Heegaard diagram in which every attaching curve intersects at most nine others.
Cite as
Kristóf Huszár and Clément Maria. On Sparse Representations of 3‑Manifolds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{huszar_et_al:LIPIcs.SoCG.2026.58,
author = {Husz\'{a}r, Krist\'{o}f and Maria, Cl\'{e}ment},
title = {{On Sparse Representations of 3‑Manifolds}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {58:1--58:18},
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.2026.58},
URN = {urn:nbn:de:0030-drops-258659},
doi = {10.4230/LIPIcs.SoCG.2026.58},
annote = {Keywords: computational 3-manifold topology, fixed-parameter tractability, Heegaard splittings and diagrams, triangulations, edge valence, treewidth, quantum invariants, tensor networks}
}
Document
Improved Bound for the k-Variate Elekes-Rónyai Theorem
Abstract
Let f ∈ ℝ[x₁,…,x_k], for k ≥ 2. For any finite sets A₁,…,A_k ⊂ ℝ, consider the set f(A₁,…,A_k): = {f(a₁,…,a_k)∣ (a₁,⋯,a_k) ∈ A₁×⋯× A_k}, that is, the image of A₁×⋯×A_k under f. Extending a theorem of Elekes and Rónyai, which deals with the case k = 2, and the result of Raz, Sharir, and De Zeeuw [Raz et al., 2018], dealing with the case k = 3, it is proved in Raz and Shem Tov [Raz and Shem{-}Tov, 2020], that for every choice of finite A₁,…, A_k ⊂ ℝ, each of size n, one has
(1) |f(A₁,…,A_k)| = Ω(n^{3/2}),
unless f has some degenerate special form.
In this paper, we introduce the notion of a rank of a k-variate polynomial f, denoted as rank(f). Letting r = rank(f), we prove that
(2) |f(A₁,…,A_k)| = Ω(n^{(5r-4)/2r-ε}) ,
for every ε > 0, where the constant of proportionality depends on ε and on deg(f). This improves the lower bound (1), for polynomials f for which rank(f) ≥ 3.
We present an application of our main result, to lower bound the number of distinct d-volumes spanned by (d+1)-tuples of points lying on the moment curve in ℝ^d.
Cite as
Yaara Jahn and Orit E. Raz. Improved Bound for the k-Variate Elekes-Rónyai Theorem. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jahn_et_al:LIPIcs.SoCG.2026.59,
author = {Jahn, Yaara and Raz, Orit E.},
title = {{Improved Bound for the k-Variate Elekes-R\'{o}nyai Theorem}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {59:1--59:15},
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.2026.59},
URN = {urn:nbn:de:0030-drops-258663},
doi = {10.4230/LIPIcs.SoCG.2026.59},
annote = {Keywords: Polynomial Expansion, Elekes-R\'{o}nyai theorem}
}
Document
Space-Efficient Approximate Spherical Range Counting in High Dimensions
Abstract
We study the following range searching problem in high-dimensional Euclidean spaces: given a finite set P ⊂ ℝ^d, where each p ∈ P is assigned a weight w_p, and radius r > 0, we need to preprocess P into a data structure such that when a new query point q ∈ ℝ^d arrives, the data structure reports the cumulative weight of points of P within Euclidean distance r from q. Solving the problem exactly seems to require space usage that is exponential to the dimension, a phenomenon known as the curse of dimensionality. Thus, we focus on approximate solutions where points up to (1+ε)r away from q may be taken into account, where ε > 0 is an input parameter known during preprocessing. We build a data structure with near-linear space usage, and query time in n^{1-Θ(ε⁴/log(1/ε))}+t_q^ϱ⋅n^{1-ϱ}, for some ϱ = Θ(ε²), where t_q is the number of points of P in the ambiguity zone, i.e., at distance between r and (1+ε)r from the query q. To the best of our knowledge, this is the first data structure with efficient space usage (subquadratic or near-linear for any ε > 0) and query time that remains sublinear for any sublinear t_q. We supplement our worst-case bounds with a query-driven preprocessing algorithm to build data structures that are well-adapted to the query distribution.
Cite as
Andreas Kalavas and Ioannis Psarros. Space-Efficient Approximate Spherical Range Counting in High Dimensions. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kalavas_et_al:LIPIcs.SoCG.2026.60,
author = {Kalavas, Andreas and Psarros, Ioannis},
title = {{Space-Efficient Approximate Spherical Range Counting in High Dimensions}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {60:1--60:15},
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.2026.60},
URN = {urn:nbn:de:0030-drops-258670},
doi = {10.4230/LIPIcs.SoCG.2026.60},
annote = {Keywords: Approximate range counting, partition trees, high dimensions}
}
Document
Complements of Finite Unions of Convex Sets
Abstract
Finite unions of convex sets are a central object of study in discrete and computational geometry. In this paper we initiate a systematic study of complements of such unions - i.e., sets of the form S = ℝ^d ⧵ (∪_{i=1}^n K_i), where K_i are convex sets. In the first part of the paper we study isolated points in S, whose number is related to the Betti numbers of ∪_{i=1}^n K_i and to its non-convexity properties. We obtain upper bounds on the number of such points, which are sharp for n = 3 and significantly improve previous bounds of Lawrence and Morris (2009) for all n ≪ 2^d/d. In the second part of the paper we study coverings of S by well-behaved sets. We show that S can be covered by at most g(d,n) flats of different dimensions, in such a way that each x ∈ S is covered by a flat whose dimension equals the "local dimension" of S in the neighborhood of x. Furthermore, we determine the structure of a minimum cover that satisfies this property. Then, we study quantitative aspects of this minimum cover and obtain sharp upper bounds on its size in various settings.
Cite as
Chaya Keller and Micha A. Perles. Complements of Finite Unions of Convex Sets. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{keller_et_al:LIPIcs.SoCG.2026.61,
author = {Keller, Chaya and Perles, Micha A.},
title = {{Complements of Finite Unions of Convex Sets}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {61:1--61:18},
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.2026.61},
URN = {urn:nbn:de:0030-drops-258684},
doi = {10.4230/LIPIcs.SoCG.2026.61},
annote = {Keywords: convexity, unions of convex sets}
}
Document
Computing the Bottleneck Distance Between Persistent Homology Transforms
Abstract
The Persistent Homology Transform (PHT) summarizes a shape in ℝ^m by collecting persistence diagrams obtained from linear height filtrations in all directions on 𝕊^{m-1}. It enjoys strong theoretical guarantees, including continuity, stability, and injectivity. A natural way to compare two PHTs is to use the bottleneck distance between their diagrams as the direction varies. Prior work has either compared PHTs by sampling directions or, in 2D, computed the exact integral of bottleneck distance over all angles via a kinetic data structure. We improve the integral objective to Õ(n⁵) in place of the earlier Õ(n⁶) bound, where n denotes the number of simplices. For the max objective, we give an Õ(n³) expected-time algorithm in ℝ² and an Õ(n⁵) expected-time algorithm in ℝ³.
Cite as
Michael Kerber and Elena Xinyi Wang. Computing the Bottleneck Distance Between Persistent Homology Transforms. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kerber_et_al:LIPIcs.SoCG.2026.62,
author = {Kerber, Michael and Wang, Elena Xinyi},
title = {{Computing the Bottleneck Distance Between Persistent Homology Transforms}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {62:1--62:15},
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.2026.62},
URN = {urn:nbn:de:0030-drops-258693},
doi = {10.4230/LIPIcs.SoCG.2026.62},
annote = {Keywords: Kinetic data structure, bottleneck distance, persistent homology transform, vineyards}
}
Document
Unavoidable Patterns and Plane Paths in Dense Topological Graphs
Abstract
Let C_{s,t} be the complete bipartite geometric graph, with s and t vertices on two distinct parallel lines respectively, and all s t straight-line edges drawn between them. In this paper, we show that every complete bipartite simple topological graph, with parts of size 2(k-1)⁴ + 1 and 2^{k^{5k}}, contains a topological subgraph weakly isomorphic to C_{k,k}. As a corollary, every n-vertex simple topological graph not containing a plane path of length k has at most O_k(n^{2 - 8/k⁴}) edges. When k = 3, we obtain a stronger bound by showing that every n-vertex simple topological graph not containing a plane path of length 3 has at most O(n^{4/3}) edges. We also prove that x-monotone simple topological graphs not containing a plane path of length 3 have at most a linear number of edges.
Cite as
Balázs Keszegh, Andrew Suk, Gábor Tardos, and Ji Zeng. Unavoidable Patterns and Plane Paths in Dense Topological Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{keszegh_et_al:LIPIcs.SoCG.2026.63,
author = {Keszegh, Bal\'{a}zs and Suk, Andrew and Tardos, G\'{a}bor and Zeng, Ji},
title = {{Unavoidable Patterns and Plane Paths in Dense Topological Graphs}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {63:1--63:15},
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.2026.63},
URN = {urn:nbn:de:0030-drops-258706},
doi = {10.4230/LIPIcs.SoCG.2026.63},
annote = {Keywords: graph drawing, topological graph, bipartite geometric graph, forbidden subgraph, extremal graph, thrackle}
}
Document
Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree
Abstract
We give an approximation scheme for the TSP in d-dimensional hyperbolic space that has optimal dependence on ε under Gap-ETH. For any fixed dimension d ≥ 2 and for any ε > 0 our randomized algorithm gives a (1+ε)-approximation in time 2^O(1/ε^{d-1}) n^{1+o(1)}. We also provide an algorithm for the hyperbolic Steiner tree problem with the same running time.
Our algorithm is an Arora-style dynamic program based on a randomly shifted hierarchical decomposition. However, we introduce a new hierarchical decomposition called the hybrid hyperbolic quadtree to achieve the desired large-scale structure, which deviates significantly from the recently proposed hyperbolic quadtree of Kisfaludi-Bak and Van Wordragen (JoCG'25). Moreover, we have a new non-uniform portal placement, and our structure theorem employs a new weighted crossing analysis. We believe that these techniques could form the basis for further developments in geometric optimization in curved spaces.
Cite as
Sándor Kisfaludi-Bak, Saeed Odak, Satyam Singh, and Geert van Wordragen. Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 64:1-64:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kisfaludibak_et_al:LIPIcs.SoCG.2026.64,
author = {Kisfaludi-Bak, S\'{a}ndor and Odak, Saeed and Singh, Satyam and van Wordragen, Geert},
title = {{Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {64:1--64:17},
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.2026.64},
URN = {urn:nbn:de:0030-drops-258710},
doi = {10.4230/LIPIcs.SoCG.2026.64},
annote = {Keywords: Hyperbolic traveling salesman problem, TSP, Hyperbolic Steiner tree problem, Approximation scheme, Banyan, Hyperbolic geometry}
}
Document
Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces
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
Fast Nearest Neighbor Search for 𝓁_p Metrics
Abstract
The Nearest Neighbor Search (NNS) problem asks to design a data structure that preprocesses an n-point dataset X lying in a metric space ℳ, so that given a query point q ∈ ℳ, one can quickly return a point of X minimizing the distance to q. The efficiency of such a data structure is evaluated primarily by the amount of space it uses and the time required to answer a query. We focus on the fast query-time regime, which is crucial for modern large-scale applications, where datasets are massive and queries must be processed online, and is often modeled by query time poly(d log n) when ℳ is a d-dimensional normed space. Our main result is such a randomized data structure for NNS in 𝓁_p^d spaces, p > 2, that achieves p^{O(1) + log log p} approximation with fast query time and poly(dn) space. Our data structure improves, or is incomparable to, the state-of-the-art for the fast query-time regime from [Bartal and Gottlieb, TCS 2019] and [Krauthgamer, Petruschka and Sapir, FOCS 2025].
Cite as
Robert Krauthgamer and Nir Petruschka. Fast Nearest Neighbor Search for 𝓁_p Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 66:1-66:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{krauthgamer_et_al:LIPIcs.SoCG.2026.66,
author = {Krauthgamer, Robert and Petruschka, Nir},
title = {{Fast Nearest Neighbor Search for 𝓁\underlinep Metrics}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {66:1--66:9},
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.2026.66},
URN = {urn:nbn:de:0030-drops-258737},
doi = {10.4230/LIPIcs.SoCG.2026.66},
annote = {Keywords: Nearest neighbor search, metric embeddings, 𝓁\underlinep norm}
}
Document
Non-Dissective Coverings by Planks
Abstract
A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viewed as the converse of Tarski’s plank problem (Bang’s theorem): Is it true that if the total width of a collection of planks is sufficiently large, then the planks can be individually translated to cover a unit ball B?
A translative covering of B by planks is said to be non-dissective if the planks can be added one by one, in some order, such that the uncovered part remains connected at each step and is empty at the end. Improving a classical result of Groemer, we show that every set of C/ε^{7/4} planks of width ε admits a non-dissective translative covering of a 3-dimensional ball B³, provided C is large enough. Our proof yields a low-complexity algorithm. We also show that c/ε^{4/3} planks are, in general, insufficient for a non-dissective covering of B³. This provides the first non-trivial lower bound for this problem.
Cite as
Andrey Kupavskii and János Pach. Non-Dissective Coverings by Planks. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 67:1-67:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kupavskii_et_al:LIPIcs.SoCG.2026.67,
author = {Kupavskii, Andrey and Pach, J\'{a}nos},
title = {{Non-Dissective Coverings by Planks}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {67:1--67:11},
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.2026.67},
URN = {urn:nbn:de:0030-drops-258743},
doi = {10.4230/LIPIcs.SoCG.2026.67},
annote = {Keywords: Tarski’s plank problem, translative cover, non-dissective cover}
}
Document
Optimal Bounds for Spanners and Tree Covers in Doubling Metrics
Abstract
It is known that any n-point set in the d-dimensional Euclidean space ℝ^d, for d = O(1), admits:
1) A (1+ε)-spanner with maximum degree Õ(ε^{-d+1}) and with lightness Õ(ε^{-d}), for any ε > 0.
2) A (1+ε)-tree cover with Õ(n ⋅ ε^{-d+1}) trees and maximum degree of O(1) in each tree. Moreover, all the parameters in these constructions are optimal: For any 2 ≤ d = O(1), there exists an n-point set in ℝ^d, for which any (1+ε)-spanner has Ω̃(n⋅ε^{-d+1}) edges and lightness Ω̃(ε^{-d}).
The upper bounds for Euclidean spanners rely heavily on the spatial property of cone partitioning in ℝ^d, which does not seem to extend to the wider family of doubling metrics, i.e., metric spaces of constant doubling dimension. In doubling metrics, a simple spanner construction from two decades ago, the net-tree spanner, has Õ(n⋅ε^{-d}) edges, and it could be transformed into a spanner of maximum degree Õ(ε^{-d}) and lightness Õ(n⋅ε^{-(d+1)}) by pruning redundant edges. Moreover, a careful refinement of the net-tree spanner yields a (1+ε)-tree cover with Õ(ε^{-d}) trees.
Despite a large body of work, the problem of obtaining tight bounds for spanners and tree covers in the wider family of doubling metrics has remained elusive. We resolve this problem by presenting:
1) A surprisingly simple and tight lower bound, which shows that the net-tree spanner and its pruned version are optimal with respect to all the involved parameters.
2) A new construction of (1+ε)-tree covers with Õ(n⋅ε^{-d}) trees, with maximum degree O(1) in each tree. This construction is optimal with respect to the number of trees and maximum degree.
Cite as
An La, Hung Le, Shay Solomon, Cuong Than, Vinayak, Shuang Yang, and Tianyi Zhang. Optimal Bounds for Spanners and Tree Covers in Doubling Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 68:1-68:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{la_et_al:LIPIcs.SoCG.2026.68,
author = {La, An and Le, Hung and Solomon, Shay and Than, Cuong and Vinayak and Yang, Shuang and Zhang, Tianyi},
title = {{Optimal Bounds for Spanners and Tree Covers in Doubling Metrics}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {68:1--68:16},
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.2026.68},
URN = {urn:nbn:de:0030-drops-258756},
doi = {10.4230/LIPIcs.SoCG.2026.68},
annote = {Keywords: doubling metrics, doubling spanners, Euclidean spanners, tree cover}
}
Document
Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain
Abstract
We present efficient data structures for approximate nearest neighbor searching and approximate 2-point shortest path queries in a two-dimensional polygonal domain P with n vertices. Our goal is to store a dynamic set of m point sites S in P so that we can efficiently find a site s ∈ S closest to an arbitrary query point q. We will allow both insertions and deletions in the set of sites S. However, as even just computing the distance between an arbitrary pair of points q,s ∈ P requires a substantial amount of space, we allow for approximating the distances. Given a parameter ε > 0, we build an O(n/(ε)log n) space data structure that can compute a 1+ε-approximation of the distance between q and s in O((1/ε²)log n) time. Building on this, we then obtain an O((n+m)/ε log n + m/ε log m) space data structure that allows us to report a site s ∈ S so that the distance between query point q and s is at most (1+ε)-times the distance between q and its true nearest neighbor in O((1/ε²)log n + 1/(ε)log n log m + (1/ε)log² m) time. Our data structure supports updates in O((1/ε²)log n + (1/ε)log n log m + (1/ε)log² m) amortized time.
Cite as
Joost van der Laan, Frank Staals, and Lorenzo Theunissen. Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 69:1-69:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vanderlaan_et_al:LIPIcs.SoCG.2026.69,
author = {van der Laan, Joost and Staals, Frank and Theunissen, Lorenzo},
title = {{Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {69:1--69:16},
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.2026.69},
URN = {urn:nbn:de:0030-drops-258769},
doi = {10.4230/LIPIcs.SoCG.2026.69},
annote = {Keywords: dynamic data structure, nearest neighbor search, polygonal domain}
}
Document
Tree-Like Shortcuttings of Trees
Abstract
Sparse shortcuttings of trees - equivalently, sparse 1-spanners for tree metrics with bounded hop-diameter - have been studied extensively (under different names and settings), since the pioneering works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994], initially motivated by applications to range queries, online tree product, and MST verification, to name a few. These constructions were also lifted from trees to other graph families using known low-distortion embedding results. The works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994] establish a tight tradeoff between hop-diameter and sparsity (or average degree) for tree shortcuttings and imply constant-hop shortcuttings for n-node trees with sparsity O(log^* n). Despite their small sparsity, all known constant-hop shortcuttings contain dense subgraphs (of sparsity Ω(log n)), which is a significant drawback for many applications.
We initiate a systematic study of constant-hop tree shortcuttings that are "tree-like". We focus on two well-studied graph parameters that measure how far a graph is from a tree: arboricity and treewidth. Our contribution is twofold.
- New upper and lower bounds for tree-like shortcuttings of trees, including an optimal tradeoff between hop-diameter and treewidth for all hop-diameter up to O(log log n). We also provide a lower bound for larger values of k, which together yield hop-diameter× treewidth = Ω((log log n)²) for all values of hop-diameter, resolving an open question of [Arnold Filtser and Hung Le, 2022; H. Le, 2023].
- Applications of these bounds, focusing on low-dimensional Euclidean and doubling metrics. A seminal work of Arya et al. [S. Arya et al., 1995] presented a (1+ε)-spanner with constant hop-diameter and sparsity O(log^* n), but with large arboricity. We show that constant hop-diameter is sufficient to achieve arboricity O(log^*{n}). Furthermore, we present a (1+ε)-stretch routing scheme in the fixed-port model with 3 hops and a local memory of O(log²n / log log n) bits, resolving an open question of [Omri Kahalon et al., 2022].
Cite as
Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Tree-Like Shortcuttings of Trees. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 70:1-70:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{le_et_al:LIPIcs.SoCG.2026.70,
author = {Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay and Than, Cuong},
title = {{Tree-Like Shortcuttings of Trees}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {70:1--70:15},
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.2026.70},
URN = {urn:nbn:de:0030-drops-258776},
doi = {10.4230/LIPIcs.SoCG.2026.70},
annote = {Keywords: spanner, tree shortcutting, arboricity, treewidth}
}
Document
Approximating Euclidean Shallow-Light Trees
Abstract
For a weighted graph G = (V, E, w) and a designated source vertex s ∈ V, a spanning tree that simultaneously approximates a shortest-path tree w.r.t. source s and a minimum spanning tree is called a shallow-light tree (SLT). Specifically, an (α, β)-SLT of G w.r.t. s ∈ V is a spanning tree of G with root-stretch α (preserving all distances between s and all other vertices up to a factor of α) and lightness β (its weight is at most β times the weight of a minimum spanning tree of G).
It was shown in the early 1990s that (1) for any graph, any source, and any ε > 0, there is a (1 + ε, O(1/ε))-SLT, and (2) there exist graphs for which β = Ω(1/ε) for any (1+ε,β)-SLT.
The focus of this work is on SLTs in low-dimensional Euclidean spaces, which are of special interest for some applications of SLTs, in geometric network optimization problems. The aforementioned existential lower bound applies to Euclidean plane, as well. It was shown more than a decade ago that (1) by using Steiner points, one can reduce the lightness bound from O(1/ε) to O(√{1/ε}), and (2) there exist point sets in the plane for which β = Ω(√{1/ε}) for any Steiner (1+ε,β)-SLT.
These tight existential bounds for the Euclidean case yield approximation factors of O(1/ε) and O(√{1/ε}) on the minimum weight of any non-Steiner and Steiner tree with root-stretch 1+ε, respectively. Despite the large body of work on SLTs, the basic question of whether a better approximation algorithm exists was left untouched to date, and this holds in any graph family. This paper makes a first nontrivial step towards resolving this question by presenting two bicriteria approximation algorithms. For any ε > 0, a set P of n points in constant-dimensional Euclidean space and a source s ∈ P, our first (respectively, second) algorithm returns, in O(n log n ⋅ polylog(ε^{-1})) time, a non-Steiner (resp., Steiner) tree with root-stretch 1+O(ε log ε^{-1}) and weight at most O(opt_ε ⋅ log² ε^{-1}) (resp., O(opt_ε ⋅ log ε^{-1})), where opt_ε denotes the minimum weight of a non-Steiner (resp., Steiner) tree with root-stretch 1+ε.
Cite as
Hung Le, Shay Solomon, Cuong Than, Csaba D. Tóth, and Tianyi Zhang. Approximating Euclidean Shallow-Light Trees. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 71:1-71:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{le_et_al:LIPIcs.SoCG.2026.71,
author = {Le, Hung and Solomon, Shay and Than, Cuong and T\'{o}th, Csaba D. and Zhang, Tianyi},
title = {{Approximating Euclidean Shallow-Light Trees}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {71:1--71:16},
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.2026.71},
URN = {urn:nbn:de:0030-drops-258789},
doi = {10.4230/LIPIcs.SoCG.2026.71},
annote = {Keywords: geometric network design, optimization, shallow-light tree, Steiner point}
}
Document
Topological Simplification Guided by Forbidden Regions
Abstract
Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c⋅n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.
Cite as
Jakub Leśkiewicz, Bartosz Furmanek, Michał Lipiński, and Dmitriy Morozov. Topological Simplification Guided by Forbidden Regions. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{leskiewicz_et_al:LIPIcs.SoCG.2026.72,
author = {Le\'{s}kiewicz, Jakub and Furmanek, Bartosz and Lipi\'{n}ski, Micha{\l} and Morozov, Dmitriy},
title = {{Topological Simplification Guided by Forbidden Regions}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {72:1--72:17},
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.2026.72},
URN = {urn:nbn:de:0030-drops-258797},
doi = {10.4230/LIPIcs.SoCG.2026.72},
annote = {Keywords: persistent homology, topological simplification, depth posets}
}
Document
The Complete 10-Tetrahedra Census of Orientable Cusped Hyperbolic 3-Manifolds
Abstract
We extend the complete census of orientable cusped hyperbolic 3-manifolds to 10 tetrahedra, giving the next 150,730 manifolds and their 496,638 minimal ideal triangulations. As applications, we find the precisely 439,898 exceptional Dehn fillings on them, revealing the next 1,849 simplest hyperbolic knot exteriors in S³. We also give the simplest example of an orientable cusped hyperbolic 3-manifold containing a closed totally geodesic surface.
Cite as
Shana Yunsheng Li. The Complete 10-Tetrahedra Census of Orientable Cusped Hyperbolic 3-Manifolds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{li:LIPIcs.SoCG.2026.73,
author = {Li, Shana Yunsheng},
title = {{The Complete 10-Tetrahedra Census of Orientable Cusped Hyperbolic 3-Manifolds}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {73:1--73:15},
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.2026.73},
URN = {urn:nbn:de:0030-drops-258800},
doi = {10.4230/LIPIcs.SoCG.2026.73},
annote = {Keywords: hyperbolic manifolds, 3-manifolds, triangulation, census, tabulation, exact computation, computational topology, low-dimensional topology}
}
Document
Manifolds of Positive Reach, Differentiability, Tangent Variation, and Attaining the Reach
Abstract
This paper contains three main results.
Firstly, we give an elementary proof of the following statement: Let ℳ be a topological manifold without boundary embedded in R^d. If ℳ has positive reach, then ℳ can locally be written as the graph of a C^{1,1} function from the tangent space to the normal space. Conversely if ℳ can locally be written as the graph of a C^{1,1} function from the tangent space to the normal space, then ℳ has positive reach. The result was hinted at by Federer when he introduced the reach, and proved by Lytchak. Lytchak’s proof relies heavily on CAT(k)-theory. The proof presented here uses only basic results on homology.
Secondly, we give optimal Lipschitz-constants for the derivative, in other words we give an optimal bound for the angle between tangent spaces in term of the distance between the points. We stress that Lytchak did not provide any bound, let alone an optimal one, making his proof, although interesting from a mathematical perspective, ineffectual in an algorithmic setting. To provide precise and optimal bounds on the angle between tangent spaces, we formally introduce the local reach for sets of positive reach, based on Aamari et al.’s discussion for C² manifolds. We prove that the local reach of a manifold is completely characterized by the variation of tangent spaces. This improves earlier results, that were either suboptimal or assumed that the manifold was C².
Thirdly, we show that the value of the reach is equals minimum of the local reach of the set and a global bottleneck for any set. This generalizes a result by Aamari et al. which explains how the reach is attained for C² manifolds.
Cite as
André Lieutier and Mathijs Wintraecken. Manifolds of Positive Reach, Differentiability, Tangent Variation, and Attaining the Reach. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 74:1-74:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lieutier_et_al:LIPIcs.SoCG.2026.74,
author = {Lieutier, Andr\'{e} and Wintraecken, Mathijs},
title = {{Manifolds of Positive Reach, Differentiability, Tangent Variation, and Attaining the Reach}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {74:1--74:16},
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.2026.74},
URN = {urn:nbn:de:0030-drops-258812},
doi = {10.4230/LIPIcs.SoCG.2026.74},
annote = {Keywords: Reach, Manifolds, Differentiability class, Lipschitz continuity, Tangent space}
}
Document
Geodesics of Length Less Than πR in a Set of Reach R Are Unique and Continuous with Respect to the Endpoints
Abstract
Positive reach underpins many results in computational geometry and topology. It is used for triangulation criteria, topological inference, and manifold learning. The geometric properties of these sets have therefore been studied intensely. Here we focus on the shortest paths or minimizing geodesics in these sets. Our main result states that minimizing geodesics of length strictly less than π R in a set of reach R are unique. This in turn implies that such minimizing geodesics are continuous with respect to the endpoints.
Cite as
André Lieutier and Mathijs Wintraecken. Geodesics of Length Less Than πR in a Set of Reach R Are Unique and Continuous with Respect to the Endpoints. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 75:1-75:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lieutier_et_al:LIPIcs.SoCG.2026.75,
author = {Lieutier, Andr\'{e} and Wintraecken, Mathijs},
title = {{Geodesics of Length Less Than \piR in a Set of Reach R Are Unique and Continuous with Respect to the Endpoints}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {75:1--75:12},
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.2026.75},
URN = {urn:nbn:de:0030-drops-258823},
doi = {10.4230/LIPIcs.SoCG.2026.75},
annote = {Keywords: Reach, geodesics, metric geometry}
}
Document
A Fast Algorithm for the Hecke Representation of the Braid Group, and Applications to the Computation of the HOMFLY-PT Polynomial and the Search for Interesting Braids
Abstract
Knot theory is an active field of mathematics, in which combinatorial and computational methods play an important role. One side of computational knot theory, that has gained interest in recent years, both for complexity analysis and practical algorithms, is quantum topology and the computation of topological invariants issued from the theory.
In this article, we leverage the rigidity brought by the representation-theoretic origins of the quantum invariants for algorithmic purposes. We do so by exploiting braids and the algebraic properties of the braid group to describe, analyze, and implement a fast algorithm to compute the Hecke representation of the braid group. We apply this construction to design a parameterized algorithm to compute the HOMFLY-PT polynomial of knots, and demonstrate its interest experimentally. Finally, we combine our fast Hecke representation algorithm with Garside theory, to implement a reservoir sampling search and find non-trivial braids with trivial Hecke representations with coefficients in ℤ/pℤ. We find explicitly several such braids, for the 4-strand and 5-strand braid groups.
Cite as
Clément Maria and Hoel Queffelec. A Fast Algorithm for the Hecke Representation of the Braid Group, and Applications to the Computation of the HOMFLY-PT Polynomial and the Search for Interesting Braids. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 76:1-76:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{maria_et_al:LIPIcs.SoCG.2026.76,
author = {Maria, Cl\'{e}ment and Queffelec, Hoel},
title = {{A Fast Algorithm for the Hecke Representation of the Braid Group, and Applications to the Computation of the HOMFLY-PT Polynomial and the Search for Interesting Braids}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {76:1--76:18},
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.2026.76},
URN = {urn:nbn:de:0030-drops-258838},
doi = {10.4230/LIPIcs.SoCG.2026.76},
annote = {Keywords: Hecke representation of the braid group, parameterized algorithm, HOMFLY-PT polynomial of knots, reservoir sampling, faithfulness of Hecke representation}
}
Document
Robust Algorithms for Path and Cycle Problems in Geometric Intersection Graphs
Abstract
We study the design of robust subexponential algorithms for classical connectivity problems on intersection graphs of similarly sized fat objects in ℝ^d. In this setting, each vertex corresponds to a geometric object, and two vertices are adjacent if and only if their objects intersect. We introduce a new tool for designing such algorithms, which we call a λ-linked partition. This is a partition of the vertex set into groups of highly connected vertices. Crucially, such a partition can be computed in polynomial time and does not require access to the geometric representation of the graph.
We apply this framework to problems related to paths and cycles in graphs. First, we obtain the first robust ETH-tight algorithms for Hamiltonian Path and Hamiltonian Cycle, running in time 2^O(n^{1-1/d}) on intersection graphs of similarly sized fat objects in ℝ^d. This resolves an open problem of de Berg et al. [STOC 2018] and completes the study of these problems on geometric intersection graphs from the viewpoint of ETH-tight exact algorithms.
We further extend our approach to the parameterized setting and design the first robust subexponential parameterized algorithm for Long Path in any fixed dimension d. More precisely, we obtain a randomized robust algorithm running in time 2^O(k^{1-1/d} log² k) n^O(1) on intersection graphs of similarly sized fat objects in ℝ^d, where k is the natural parameter. Besides λ-linked partitions, our algorithm also relies on a low-treewidth pattern covering theorem that we establish for geometric intersection graphs, which may be viewed as a refinement of a result of Marx-Pilipczuk [ESA 2017]. This structural result may be of independent interest.
Cite as
Malory Marin, Jean-Florent Raymond, and Rémi Watrigant. Robust Algorithms for Path and Cycle Problems in Geometric Intersection Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{marin_et_al:LIPIcs.SoCG.2026.77,
author = {Marin, Malory and Raymond, Jean-Florent and Watrigant, R\'{e}mi},
title = {{Robust Algorithms for Path and Cycle Problems in Geometric Intersection Graphs}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {77:1--77:14},
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.2026.77},
URN = {urn:nbn:de:0030-drops-258842},
doi = {10.4230/LIPIcs.SoCG.2026.77},
annote = {Keywords: Robust algorithms, geometric intersection graphs, subexponential FPT algorithms}
}
Document
Mapping Chemical Space: Topological Data Analysis of Chemical Latent Space with Mapper
Abstract
The vast chemical space, encompassing virtually innumerable molecules and materials, presents both immense opportunities and significant challenges. The design and discovery of novel drugs and functional materials may be viewed as a search within this space; however, the sheer scale of potential candidates renders exhaustive exploration infeasible. To address this, we introduce Chemical Mapper, a framework that integrates topological data analysis with deep learning to enable the visual exploration and analysis of chemical latent spaces. At its core, Chemical Mapper employs mapper, a widely used tool in topological data analysis, to investigate the organizational principles of chemical latent spaces defined by molecular representations learned by geometric deep learning models. In doing so, Chemical Mapper not only highlights groups of molecular representations but also uncovers the relationships among them through linkages and branching structures. Our results show that Chemical Mapper reveals intrinsic patterns associated with molecular scaffolds, functional groups, and chemical properties, as well as the structural and functional evolutions of the molecules.
Cite as
Dhruv Meduri, Chuan-Shen Hu, Cong Shen, Kelin Xia, and Bei Wang. Mapping Chemical Space: Topological Data Analysis of Chemical Latent Space with Mapper. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 78:1-78:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{meduri_et_al:LIPIcs.SoCG.2026.78,
author = {Meduri, Dhruv and Hu, Chuan-Shen and Shen, Cong and Xia, Kelin and Wang, Bei},
title = {{Mapping Chemical Space: Topological Data Analysis of Chemical Latent Space with Mapper}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {78:1--78:20},
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.2026.78},
URN = {urn:nbn:de:0030-drops-258854},
doi = {10.4230/LIPIcs.SoCG.2026.78},
annote = {Keywords: Practice of computational topology, topological data analysis, applications in chemistry, mapper algorithm, high-dimensional data analysis, chemical spaces, geometric deep learning, latent space geometry}
}
Document
D-GRIL: End-To-End Topological Learning with 2-Parameter Persistence
Abstract
End-to-end topological learning using 1-parameter persistence is well-known. We show that the framework can be enhanced using 2-parameter persistence by adopting a recently introduced 2-parameter persistence based vectorization technique called Gril. We establish a theory for gradient descent on Gril producing D-Gril. We show that D-Gril can be used to learn a bifiltration function on benchmark graph datasets. Further, we exhibit that this framework can be applied in the context of bio-activity prediction in drug discovery.
Cite as
Soham Mukherjee, Shreyas N. Samaga, Cheng Xin, Steve Oudot, and Tamal K. Dey. D-GRIL: End-To-End Topological Learning with 2-Parameter Persistence. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 79:1-79:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mukherjee_et_al:LIPIcs.SoCG.2026.79,
author = {Mukherjee, Soham and Samaga, Shreyas N. and Xin, Cheng and Oudot, Steve and Dey, Tamal K.},
title = {{D-GRIL: End-To-End Topological Learning with 2-Parameter Persistence}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {79:1--79:17},
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.2026.79},
URN = {urn:nbn:de:0030-drops-258865},
doi = {10.4230/LIPIcs.SoCG.2026.79},
annote = {Keywords: Topological Data Analysis, Persistent Homology, Multiparameter Persistence, Graph Learning, Graph Neural Networks}
}
Document
Hardness of High-Dimensional Linear Classification
Abstract
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only O(n^d) and respectively Õ(1/ε^d) upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and k-Sum problems. Our reductions yield matching lower bounds of Ω̃(n^d) and respectively Ω̃(1/ε^d) based on Affine Degeneracy testing, and Ω̃(n^{d/2}) and respectively Ω̃(1/ε^{d/2}) conditioned on k-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.
Cite as
Alexander Munteanu, Simon Omlor, and Jeff M. Phillips. Hardness of High-Dimensional Linear Classification. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 80:1-80:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{munteanu_et_al:LIPIcs.SoCG.2026.80,
author = {Munteanu, Alexander and Omlor, Simon and Phillips, Jeff M.},
title = {{Hardness of High-Dimensional Linear Classification}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {80:1--80:16},
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.2026.80},
URN = {urn:nbn:de:0030-drops-258871},
doi = {10.4230/LIPIcs.SoCG.2026.80},
annote = {Keywords: Conditional Hardness, k-Sum, Affine Degeneracy, Halfspace Discrepancy, Classification}
}
Document
Optimal-Cost Construction of Shallow Cuttings for 3-D Dominance Ranges in the I/O-Model
Abstract
Shallow cuttings are a fundamental tool in computational geometry and spatial databases for solving offline and online range searching problems. For a set P of N points in 3-D, at SODA'14, Afshani and Tsakalidis designed an optimal O(N log₂N) time algorithm that constructs shallow cuttings for 3-D dominance ranges in internal memory. Even though shallow cuttings are used in the I/O-model to design space and query efficient range searching data structures, an efficient construction of them is not known till now. In this paper, we design an optimal-cost algorithm to construct shallow cuttings for 3-D dominance ranges. The number of I/Os performed by the algorithm is O (N/B log_{M/B}(N/B)), where B is the block size and M is the memory size.
As two applications of the optimal-cost construction algorithm, we design fast algorithms for offline 3-D dominance reporting and offline 3-D approximate dominance counting. We believe that our algorithm will find further applications in offline 3-D range searching problems and in improving construction cost of data structures for 3-D range searching problems.
Cite as
Yakov Nekrich and Saladi Rahul. Optimal-Cost Construction of Shallow Cuttings for 3-D Dominance Ranges in the I/O-Model. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 81:1-81:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{nekrich_et_al:LIPIcs.SoCG.2026.81,
author = {Nekrich, Yakov and Rahul, Saladi},
title = {{Optimal-Cost Construction of Shallow Cuttings for 3-D Dominance Ranges in the I/O-Model}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {81:1--81:15},
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.2026.81},
URN = {urn:nbn:de:0030-drops-258884},
doi = {10.4230/LIPIcs.SoCG.2026.81},
annote = {Keywords: Data Structures, I/O-efficient algorithms, Orthogonal Range Searching}
}
Document
A Persistent Version of Latschev’s Theorem
Abstract
Latschev’s theorem provides sufficient conditions on a metric space M and δ > 0 for the homotopy type of M to agree with that of the Vietoris-Rips complex ℛ^δ(𝕄) of any nearby space 𝕄 in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair (M, f:M → ℝ^N) and δ > 0 for the persistent homotopy type of the sublevel set filtration of (M, f) to be interleaved with that of the function-Rips complex ℛ^δ(𝕄^•) of any nearby pair (𝕄, 𝕗). In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.
Cite as
Steve Oudot and Lukas Waas. A Persistent Version of Latschev’s Theorem. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 82:1-82:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{oudot_et_al:LIPIcs.SoCG.2026.82,
author = {Oudot, Steve and Waas, Lukas},
title = {{A Persistent Version of Latschev’s Theorem}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {82:1--82:17},
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.2026.82},
URN = {urn:nbn:de:0030-drops-258891},
doi = {10.4230/LIPIcs.SoCG.2026.82},
annote = {Keywords: Topological data analysis (TDA), metric geometry, Vietoris-Rips complex, homotopy theory, multi-parameter persistent homology}
}
Document
Erdős’s Unit Distance Problem and Rigidity
Abstract
According to a classical result of Spencer, Szemerédi, and Trotter (1984), the maximum number of times the unit distance can occur among n points in the plane is O(n^{4/3}). This is far from Erdős’s lower bound, n^{1+O(1/log log n)}, which is conjectured to be optimal. We prove a structural result for point sets with nearly n^{4/3} unit distances and use it to reduce the problem to a conjecture on rigid frameworks. This conjecture, if true, would yield the first improvement on the bound of Spencer et al. A weaker version of this conjecture has been established by Raz and Solymosi.
Cite as
János Pach, Orit E. Raz, and József Solymosi. Erdős’s Unit Distance Problem and Rigidity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 83:1-83:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{pach_et_al:LIPIcs.SoCG.2026.83,
author = {Pach, J\'{a}nos and Raz, Orit E. and Solymosi, J\'{o}zsef},
title = {{Erd\H{o}s’s Unit Distance Problem and Rigidity}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {83:1--83:9},
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.2026.83},
URN = {urn:nbn:de:0030-drops-258906},
doi = {10.4230/LIPIcs.SoCG.2026.83},
annote = {Keywords: Unit distance problem, Erd\H{o}s, graph rigidity, incidences, polynomial partitioning technique}
}
Document
The Voronoi Diagram of Four Lines in ℝ³
Abstract
We consider the Voronoi diagram of lines in ℝ³ under the Euclidean metric, and give a full classification of its structure in the base case of four lines in general position. We first show that the number of vertices in the Voronoi diagram of four lines in general position is always even, between 0 and 8, and all such numbers can be realized. We identify a key structure for the diagram formation, called a twist, which is a pair of consecutive intersections among trisector branches; only two types of twists are possible, so-called full and partial twists. A full twist is a purely local structure, which can be inserted or removed without affecting the rest of the diagram. Assuming no full twists, the nearest and the farthest Voronoi diagrams of four lines, each have 15 distinct topologies, which are in one-to-one correspondence; the two-dimensional faces are all unbounded, and the total number of vertices is at most six. The unbounded features of the farthest diagram, encoded in a two-dimensional spherical map, are also in one-to-one correspondence. The identified topologies are all realizable. Any Voronoi diagram of four lines in general position in ℝ³ can be obtained from one of these topologies by inserting full twists; each twist induces a bounded face of exactly two vertices in both the nearest and farthest diagrams. We obtain the classification by an exhaustive search algorithm using some new structural and combinatorial observations of line Voronoi diagrams.
Cite as
Evanthia Papadopoulou and Zeyu Wang. The Voronoi Diagram of Four Lines in ℝ³. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 84:1-84:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{papadopoulou_et_al:LIPIcs.SoCG.2026.84,
author = {Papadopoulou, Evanthia and Wang, Zeyu},
title = {{The Voronoi Diagram of Four Lines in \mathbb{R}³}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {84:1--84:17},
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.2026.84},
URN = {urn:nbn:de:0030-drops-258916},
doi = {10.4230/LIPIcs.SoCG.2026.84},
annote = {Keywords: Voronoi diagram, lines, three dimensions, structural properties}
}
Document
ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes
Abstract
The Optimal Morse Matching (OMM) problem asks for a discrete gradient vector field on a simplicial complex that minimizes the number of critical simplices. It is NP-hard and has been studied extensively in heuristic, approximation, and parameterized complexity settings. Parameterized by treewidth k, OMM has long been known to be solvable on triangulations of 3-manifolds in 2^O(k²) n^O(1) time and in FPT time for triangulations of arbitrary manifolds, but the exact dependence on k has remained an open question. We resolve this by giving a new 2^O(k log k) n-time algorithm for any finite regular CW complex, and show that no 2^o(k log k) n^O(1)-time algorithm exists unless the Exponential Time Hypothesis (ETH) fails.
Cite as
Geevarghese Philip and Erlend Raa Vågset. ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 85:1-85:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{philip_et_al:LIPIcs.SoCG.2026.85,
author = {Philip, Geevarghese and V\r{a}gset, Erlend Raa},
title = {{ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {85:1--85:19},
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.2026.85},
URN = {urn:nbn:de:0030-drops-258926},
doi = {10.4230/LIPIcs.SoCG.2026.85},
annote = {Keywords: Discrete Morse Theory, Simplicial Complexes, Optimal Morse Matching, Treewidth, Parameterized Algorithms, Computational Topology, Dynamic Programming, Exponential Time Hypothesis, Topological Data Analysis}
}
Document
Expansion of Trivariate Polynomials Using Proximity
Abstract
We extend the proximity technique of Solymosi and Zahl [J. Solymosi and J. Zahl, 2024] to the setting of trivariate polynomials. In particular, we prove the following result: Let f(x,y,z) = (x-y)²+(φ(x)-z)², where φ(x) ∈ ℝ[x] has degree at least 3. Then, for every finite A,B,C ⊂ ℝ each of size n, one has |f(A,B,C)| = Ω(n^{5/3-ε}), for every ε > 0, where the constant of proportionality depends on ε and on deg(φ). This improves the previous exponent 3/2, due to Raz, Sharir, and De Zeeuw [O. E. Raz M. Sharir and F. de Zeeuw, 2018]. To the best of our knowledge, prior to this work no trivariate polynomial was known to have expansion exceeding Ω(n^{3/2}).
Cite as
Orit E. Raz. Expansion of Trivariate Polynomials Using Proximity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 86:1-86:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{raz:LIPIcs.SoCG.2026.86,
author = {Raz, Orit E.},
title = {{Expansion of Trivariate Polynomials Using Proximity}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {86:1--86:10},
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.2026.86},
URN = {urn:nbn:de:0030-drops-258936},
doi = {10.4230/LIPIcs.SoCG.2026.86},
annote = {Keywords: Polynomial Expansion, Elekes-R\'{o}nyai theorem}
}
Document
Robustness of Persistent Topological Features and Minimum Homological Cuts
Abstract
Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the Čech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers. This lack of robustness has been frequently addressed in the literature. In this paper, we take a novel perspective by asking the following question: When can we guarantee that an observed persistent feature (a bar) is inherent to the underlying data in the presence of a limited number of unknown, arbitrary outliers. We formalize this question by introducing the notion of adversarial robustness, and study the problem of deciding whether a given bar in the barcode of a filtered simplicial complex is adversarially robust. We show that this problem is essentially equivalent to a homological variant of the minimum cut problem in simplicial complexes, which we believe to be of independent interest. As our main technical contribution, we provide the first computational complexity results for this problem, consisting of an efficient algorithm in 0-dimensional homology, NP-hardness for the general problem, and an efficient algorithm for codimension-1 in n-dimensional complexes embedded in ℝⁿ. We also analyze its natural linear programming relaxation, whose dual defines a homological analog of the max-flow problem in graphs. We show that a max-flow/min-cut theorem does not hold in our setting, implying that the LP relaxation is not tight in general. Finally, in the special case of the Rips filtration, we provide a global heuristic based on the Hausdorff distance that guarantees adversarial robustness of sufficiently long bars. This connects adversarial robustness to standard stability theorems in persistent homology.
Cite as
Pepijn Roos Hoefgeest and Lucas Slot. Robustness of Persistent Topological Features and Minimum Homological Cuts. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 87:1-87:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{rooshoefgeest_et_al:LIPIcs.SoCG.2026.87,
author = {Roos Hoefgeest, Pepijn and Slot, Lucas},
title = {{Robustness of Persistent Topological Features and Minimum Homological Cuts}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {87:1--87:15},
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.2026.87},
URN = {urn:nbn:de:0030-drops-258636},
doi = {10.4230/LIPIcs.SoCG.2026.87},
annote = {Keywords: Topological Data Analysis, Persistent Homology, Min-cut Max-flow, Robustness, Vietoris-Rips Filtration}
}
Document
Integer Points in Dilates of Polytopes
Abstract
In this paper we study how the number of integer points in a polytope grows as we dilate the polytope. We prove new and essentially tight bounds on this quantity by specifically studying dilates of the Hadamard polytope.
The motivation for studying this question comes from the question of understanding the maximal number of monomials in a factor of a multivariate polynomial of s monomials. A recent result by Bhargava, Saraf and Volkovich [Bhargava et al., 2020] showed that if f is an n-variate polynomial, where each variable has degree d, and f has s monomials, then any factor of f has at most s^{O(d² log n)} monomials. The key technical ingredient of their proof was to show that any polytope with s vertices, where each vertex lies in {0,..,d}ⁿ can have at most s^{O(d² log n)} integer points. The precise dependence on d of the number of integer points was left open. We show that this bound, and in particular the dependence on d is essentially tight by studying dilates of the Hadamard polytope and proving new lower bounds on its number of integer points.
Cite as
Shubhangi Saraf and Narmada Varadarajan. Integer Points in Dilates of Polytopes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 88:1-88:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{saraf_et_al:LIPIcs.SoCG.2026.88,
author = {Saraf, Shubhangi and Varadarajan, Narmada},
title = {{Integer Points in Dilates of Polytopes}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {88:1--88:9},
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.2026.88},
URN = {urn:nbn:de:0030-drops-258948},
doi = {10.4230/LIPIcs.SoCG.2026.88},
annote = {Keywords: Computational geometry, Complexity theory, Integer polytopes}
}
Document
Approximating Convex Hulls via Range Queries
Abstract
Recently, motivated by the rapid increase of the data size in various applications, Monemizadeh [APPROX'23] and Driemel, Monemizadeh, Oh, Staals, and Woodruff [SoCG'25] studied geometric problems in the setting where the only access to the input point set is via querying a range-search oracle. Algorithms in this setting are evaluated on two criteria: (i) the number of queries to the oracle and (ii) the error of the output. In this paper, we continue this line of research and investigate one of the most fundamental geometric problems in the oracle setting, i.e., the convex hull problem.
Let P be an unknown set of points in [0,1]^d equipped with a range-emptiness oracle. Via querying the oracle, the algorithm is supposed to output a convex polygon C ⊆ [0,1]^d as an estimation of the convex hull CH(P) of P. The error of the output is defined as the volume of the symmetric difference C ⊕ CH(P) = (C∖CH(P)) ∪ (CH(P)∖C). We prove tight and near-tight tradeoffs between the number of queries and the error of the output for different variants of the problem, depending on the type of the range-emptiness queries and whether the queries are non-adaptive or adaptive.
- Orthogonal emptiness queries in d-dimensional space: We show that the minimum error a deterministic algorithm can achieve with q queries is Θ(q^{-1/d}) if the queries are non-adaptive, and Θ(q^{-1/(d-1)}) if the queries are adaptive. In particular, in 2D, the bounds are Θ(1/√q) and Θ(1/q) for non-adaptive and adaptive queries, respectively.
- Halfplane emptiness queries in 2D: We show that the minimum error a deterministic algorithm can achieve with q queries is Θ(1/√q) if the queries are non-adaptive, and Θ̃(1/q²) if the queries are adaptive. Here Θ̃(⋅) hides logarithmic factors.
Cite as
Thomas Schibler, Jie Xue, and Jiumu Zhu. Approximating Convex Hulls via Range Queries. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 89:1-89:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{schibler_et_al:LIPIcs.SoCG.2026.89,
author = {Schibler, Thomas and Xue, Jie and Zhu, Jiumu},
title = {{Approximating Convex Hulls via Range Queries}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {89:1--89:15},
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.2026.89},
URN = {urn:nbn:de:0030-drops-258956},
doi = {10.4230/LIPIcs.SoCG.2026.89},
annote = {Keywords: convex hull, range searching}
}
Document
Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells
Abstract
The restricted Delaunay triangulation of a closed surface Σ and a finite point set V ⊂ Σ is a subcomplex of the Delaunay tetrahedralization of V whose triangles approximate Σ. It is well known that if V is a sufficiently dense sample of a smooth Σ, then the union of the restricted Delaunay triangles is homeomorphic to Σ. We show that an ε-sample with ε ≤ 0.3245 suffices. By comparison, Dey proves it for a 0.18-sample; our improved sampling bound reduces the number of sample points required by a factor of 3.25. More importantly, we improve a related sampling bound of Cheng et al. for Delaunay surface meshing, reducing the number of sample points required by a factor of 21. The first step of our homeomorphism proof is particularly interesting: we show that for a 0.44-sample, the restricted Voronoi cell of each site v ∈ V is homeomorphic to a disk, and the orthogonal projection of the cell onto T_vΣ (the plane tangent to Σ at v) is star-shaped.
Cite as
Jonathan Richard Shewchuk. Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 90:1-90:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{shewchuk:LIPIcs.SoCG.2026.90,
author = {Shewchuk, Jonathan Richard},
title = {{Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {90:1--90:16},
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.2026.90},
URN = {urn:nbn:de:0030-drops-258961},
doi = {10.4230/LIPIcs.SoCG.2026.90},
annote = {Keywords: Restricted Delaunay triangulation, restricted Voronoi diagram, surface sampling, surface mesh generation, surface reconstruction, \epsilon-sample, homeomorphism}
}
Document
The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks
Abstract
A linear neural network computes a linear transformation of its input vector. Given a fully-connected linear network, the set of all weight vectors for which the network computes the same linear transformation is an algebraic variety in weight space, called a fiber under the matrix multiplication map. Sometimes this variety is a manifold, but usually not. The rank stratification of a fiber is a natural partition of the fiber into manifolds of various dimensions called strata. We characterize how these strata are connected to each other. They satisfy the frontier condition: if a stratum intersects the closure of another stratum, then the former stratum is a subset of the closure of the latter stratum. This subset relationship can be expressed as a partial order with a single minimal element. Our main result describes the relationship between this partial order and the ranks of certain matrices in the network. Each stratum represents a different pattern of information flow through the network, expressed as a barcode. Connections among the strata are best understood through simple transformations of the barcodes called barcode moves.
Cite as
Jonathan Richard Shewchuk and Sagnik Bhattacharya. The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 91:1-91:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{shewchuk_et_al:LIPIcs.SoCG.2026.91,
author = {Shewchuk, Jonathan Richard and Bhattacharya, Sagnik},
title = {{The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {91:1--91:19},
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.2026.91},
URN = {urn:nbn:de:0030-drops-258971},
doi = {10.4230/LIPIcs.SoCG.2026.91},
annote = {Keywords: Linear neural network, real algebraic variety, stratification, multilinear algebra, product of matrices, persistence barcode, real algebraic geometry, discrete geometry}
}
Document
Estimation of Conformal Metrics
Abstract
We study deformations of the geodesic distances on a domain of ℝ^N induced by a function called conformal factor. We show that under a positive reach assumption on the domain (not necessarily a submanifold) and mild assumptions on the conformal factor, geodesics for the conformal metric have good regularity properties in the form of a lower bounded reach. This regularity allows for efficient estimation of the conformal metric from a random point cloud with a relative error proportional to the Hausdorff distance between the point cloud and the original domain. We then establish convergence rates of order n^{-1/d} that are close to sharp when the intrinsic dimension d of the domain is large, for an estimator that can be computed in O(n²) time. Finally, this paper includes a useful equivalence result between ball graphs and nearest-neighbors graphs when assuming Ahlfors regularity of the sampling measure, allowing to transpose results from one setting to another.
Cite as
Jérôme Taupin. Estimation of Conformal Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 92:1-92:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{taupin:LIPIcs.SoCG.2026.92,
author = {Taupin, J\'{e}r\^{o}me},
title = {{Estimation of Conformal Metrics}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {92:1--92:15},
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.2026.92},
URN = {urn:nbn:de:0030-drops-258986},
doi = {10.4230/LIPIcs.SoCG.2026.92},
annote = {Keywords: Geometric inference, metric estimation, conformal metric, geodesics, sets of positive reach}
}
Document
Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations
Abstract
Simplicial approximation provides a framework for constructing simplicial complexes that are homotopy equivalent to a given manifold, provided a CW structure is explicitly known. However, its conventional implementation quickly becomes intractable on a computer: barycentric subdivision produces poorly shaped simplices, and the star condition introduces many vertices. To address these limitations, this article develops a subdivision scheme based on spherical Delaunay triangulations, which attains better refinement properties than barycentric subdivisions. Moreover, the star condition is reframed as two independent problems, one geometric and the other combinatorial, respectively tackled in the language of locally equiconnected spaces and the list homomorphism problem, allowing an exponential reduction in the number of vertices. Via a prototype implementation, we obtain simplicial complexes homotopy equivalent to Grassmannians and Stiefel manifolds up to dimension 5.
Cite as
Raphaël Tinarrage. Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 93:1-93:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{tinarrage:LIPIcs.SoCG.2026.93,
author = {Tinarrage, Rapha\"{e}l},
title = {{Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {93:1--93:22},
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.2026.93},
URN = {urn:nbn:de:0030-drops-258991},
doi = {10.4230/LIPIcs.SoCG.2026.93},
annote = {Keywords: Triangulation of manifolds, Simplicial approximation, CW complexes, Delaunay complexes, List homomorphism problem, Topological Data Analysis}
}
Document
Mixup Barcodes: Quantifying Geometric-Topological Interactions Between Point Clouds
Abstract
We propose a novel geometric-topological descriptor called a mixup barcode. Intuitively, it characterizes the shape of a point cloud as well as its spatial relationship with another point cloud embedded in the same ambient space. More technically, it enriches a standard persistence barcode with information on the image persistent homology. In three dimensions it captures natural spatial relationships like overlap and surrounding; in higher dimensions more intricate spatial relationships are captured. We provide a theoretical setup and a simple algorithm for mixup barcodes.
As a proof of concept, we explore data arising in a geometric-topological problem from machine learning. Specifically, we take first steps towards verifying a hypothesis stating that geometric-topological relationships within intermediate point cloud representations in an artificial neural network can hinder its training. More broadly, our experiments suggest that mixup barcodes are useful for characterizing spatial relationships and spatial interactions (i.e. the evolution of spatial relationships) that are hard to directly visualize or capture using standard methods.
Cite as
Hubert Wagner, Nickolas Arustamyan, Matthew Wheeler, and Peter Bubenik. Mixup Barcodes: Quantifying Geometric-Topological Interactions Between Point Clouds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 94:1-94:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wagner_et_al:LIPIcs.SoCG.2026.94,
author = {Wagner, Hubert and Arustamyan, Nickolas and Wheeler, Matthew and Bubenik, Peter},
title = {{Mixup Barcodes: Quantifying Geometric-Topological Interactions Between Point Clouds}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {94:1--94:19},
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.2026.94},
URN = {urn:nbn:de:0030-drops-259009},
doi = {10.4230/LIPIcs.SoCG.2026.94},
annote = {Keywords: mixup barcode, persistent homology, persistence barcode, persistence diagram, image persistent homology, image persistence, deep learning, multilayer perceptron, topology of neural network embeddings, disentanglement}
}
Document
An Optimal Algorithm for Computing Many Faces in Line Arrangements
Abstract
Given a set of m points and a set of n lines in the plane, we consider the classical problem of computing the faces of the arrangement of the lines that contain at least one point. We present an algorithm of O(m^{2/3} n^{2/3} + (n+m)log n) time for the problem. We also prove that this matches the lower bound under the algebraic decision tree model and thus our algorithm is optimal. In particular, when m = n, the runtime is O(n^{4/3}), which matches the worst case combinatorial complexity Ω(n^{4/3}) of all output faces. This is the first optimal algorithm since the problem was first studied more than three decades ago [Edelsbrunner, Guibas, and Sharir, SoCG 1988].
Cite as
Haitao Wang. An Optimal Algorithm for Computing Many Faces in Line Arrangements. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 95:1-95:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wang:LIPIcs.SoCG.2026.95,
author = {Wang, Haitao},
title = {{An Optimal Algorithm for Computing Many Faces in Line Arrangements}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {95:1--95:14},
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.2026.95},
URN = {urn:nbn:de:0030-drops-259012},
doi = {10.4230/LIPIcs.SoCG.2026.95},
annote = {Keywords: Many faces, line arrangements, cuttings, \Gamma-algorithms, decision tree complexities}
}
Document
Media Exposition
Sliding Cubes in Parallel (Media Exposition)
Abstract
The sliding cubes model serves as a well-established theoretical framework for formalizing and analyzing reconfiguration algorithms in modular robotic systems built from face-connected cubic modules. We extend the parallel sliding cubes model from two to three dimensions, presenting new algorithms, surprising complexity results, and a generalization of the best known bounds from two to three dimensions. A companion video visualizes and explains our results.
Cite as
Hugo A. Akitaya, Joseph Dorfer, Peter Kramer, Christian Rieck, Soham Samanta, Gabriel Shahrouzi, and Frederick Stock. Sliding Cubes in Parallel (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 96:1-96:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{a.akitaya_et_al:LIPIcs.SoCG.2026.96,
author = {A. Akitaya, Hugo and Dorfer, Joseph and Kramer, Peter and Rieck, Christian and Samanta, Soham and Shahrouzi, Gabriel and Stock, Frederick},
title = {{Sliding Cubes in Parallel}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {96:1--96:5},
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.2026.96},
URN = {urn:nbn:de:0030-drops-259020},
doi = {10.4230/LIPIcs.SoCG.2026.96},
annote = {Keywords: Sliding squares, parallel motion, reconfigurability, three dimensions, constant makespan, log-APX hardness, NP-hardness, worst-case optimality}
}
Document
Media Exposition
"Visualizing" the CG Community (Media Exposition)
Abstract
We analyze and visualize collaboration within the Computational Geometry community by modeling co-authorship relations as a graph, where nodes correspond to individual researchers and edges represent shared publications. By aggregating and time-slicing conference data, we construct a dynamic representation of the community that supports both interactive visualization and structured search.
Cite as
Oswin Aichholzer, Hugo A. Akitaya, Anna Brötzner, Peter Kramer, Christian Rieck, and Frederick Stock. "Visualizing" the CG Community (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 97:1-97:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2026.97,
author = {Aichholzer, Oswin and A. Akitaya, Hugo and Br\"{o}tzner, Anna and Kramer, Peter and Rieck, Christian and Stock, Frederick},
title = {{"Visualizing" the CG Community}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {97:1--97:4},
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.2026.97},
URN = {urn:nbn:de:0030-drops-259039},
doi = {10.4230/LIPIcs.SoCG.2026.97},
annote = {Keywords: CG community, visualization, graph parameters, web application}
}
Document
Media Exposition
Interactive Uniform Floodlight Illumination and Rotating Rays Voronoi Diagrams (Media Exposition)
Abstract
Floodlight illumination problems are art-gallery variants, where a target domain needs to be illuminated by guards, each associated with a field of view. The rotating rays Voronoi diagram is a Voronoi diagram with rays as sites under the angular distance. There is a natural connection of this Voronoi structure with the problem of finding the minimum aperture such that a given set of uniform aperture floodlights illuminates a target domain. In this work we present an interactive visualization software for such problems, supporting different angular distances, namely, oriented and unoriented versions, and for different domains, namely, the plane and simple polygons.
Cite as
Carlos Alegría, Ioannis Mantas, Marko Savić, and Martin Suderland. Interactive Uniform Floodlight Illumination and Rotating Rays Voronoi Diagrams (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 98:1-98:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alegria_et_al:LIPIcs.SoCG.2026.98,
author = {Alegr{\'\i}a, Carlos and Mantas, Ioannis and Savi\'{c}, Marko and Suderland, Martin},
title = {{Interactive Uniform Floodlight Illumination and Rotating Rays Voronoi Diagrams}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {98:1--98:7},
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.2026.98},
URN = {urn:nbn:de:0030-drops-259048},
doi = {10.4230/LIPIcs.SoCG.2026.98},
annote = {Keywords: rotating rays Voronoi diagram, oriented angular distance, unoriented angular distance, Brocard angle, floodlight illumination, coverage problems, visualization software}
}
Document
Media Exposition
Proximity Alert: Ipelets for Neighborhood Graphs and Clustering (Media Exposition)
Abstract
Neighborhood graphs and clustering algorithms are fundamental structures in both computational geometry and data analysis. Visualizing them can help build insight into their behavior and properties. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating figures. One particular aspect of Ipe is the ability to add Ipelets, which extend its functionality. Here we showcase a set of Ipelets designed to help visualize neighborhood graphs and clustering algorithms. These include: ε-neighbor graphs, furthest-neighbor graphs, Gabriel graphs, k-nearest neighbor graphs, k-th-nearest neighbor graphs, k-mutual neighbor graphs, k-th-mutual neighbor graphs, asymmetric k-nearest neighbor graphs, asymmetric k-th-nearest neighbor graphs, relative-neighbor graphs, sphere-of-influence graphs, Urquhart graphs, Yao graphs, and clustering algorithms including complete-linkage, DBSCAN, HDBSCAN, k-means, k-means++, k-medoids, mean shift, and single-linkage. Our Ipelets are all programmed in Lua and are freely available.
Cite as
Gitan Balogh, June Cagan, Bea Fatima, Auguste H. Gezalyan, Danesh Sivakumar, Arushi Srinivasan, Yixuan Sun, Vahe Zaprosyan, and David M. Mount. Proximity Alert: Ipelets for Neighborhood Graphs and Clustering (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 99:1-99:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{balogh_et_al:LIPIcs.SoCG.2026.99,
author = {Balogh, Gitan and Cagan, June and Fatima, Bea and Gezalyan, Auguste H. and Sivakumar, Danesh and Srinivasan, Arushi and Sun, Yixuan and Zaprosyan, Vahe and Mount, David M.},
title = {{Proximity Alert: Ipelets for Neighborhood Graphs and Clustering}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {99:1--99:8},
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.2026.99},
URN = {urn:nbn:de:0030-drops-259058},
doi = {10.4230/LIPIcs.SoCG.2026.99},
annote = {Keywords: neighborhood graphs, clustering, proximity graphs, Ipelets, visualization}
}
Document
Media Exposition
Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry (Media Exposition)
Abstract
Higher-order Voronoi diagrams and Delaunay mosaics in polygonal metrics have only recently been studied, yet no tools exist for visualizing them. We introduce a tool that fills this gap, providing dynamic interactive software for visualizing higher-order Voronoi diagrams and Delaunay mosaics along with clustering and tools for exploring overlap and outer regions in the Hilbert polygonal metric. We prove that k-th order Voronoi cells are not always star-shaped and establish complexity bounds for our algorithm, which generates all order Voronoi diagrams at once. Our software unifies and extends previous tools for visualizing the Hilbert, Funk, and Thompson geometries.
Cite as
Hridhaan Banerjee, Soren Brown, June Cagan, Auguste H. Gezalyan, Megan Hunleth, Veena Kailad, Chaewoon Kyoung, Rowan Shigeno, Yasmine Tajeddin, Andrew Wagger, Kelin Zhu, and David M. Mount. Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 100:1-100:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{banerjee_et_al:LIPIcs.SoCG.2026.100,
author = {Banerjee, Hridhaan and Brown, Soren and Cagan, June and Gezalyan, Auguste H. and Hunleth, Megan and Kailad, Veena and Kyoung, Chaewoon and Shigeno, Rowan and Tajeddin, Yasmine and Wagger, Andrew and Zhu, Kelin and Mount, David M.},
title = {{Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {100:1--100:6},
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.2026.100},
URN = {urn:nbn:de:0030-drops-259062},
doi = {10.4230/LIPIcs.SoCG.2026.100},
annote = {Keywords: Hilbert metric, Funk metric, Voronoi diagrams}
}
Document
Media Exposition
From Chaos to Continents: Voronoi-Based Procedural Terrain Generation with Hydrology and 3D Visualization (Media Exposition)
Abstract
Procedural content generation often employs grid-based methods to create virtual environments. We present a pipeline that utilizes Voronoi diagrams and Lloyd’s Relaxation to construct an irregular mesh for terrain generation. We implement a customizable "Land Anchor" system combined with Perlin noise to determine landmass shapes, distinct from standard radial distribution methods. Furthermore, we simulate hydrology using priority-flood routing on the Voronoi edges and assign biomes via a Gaussian-smoothed Whittaker classification. The full pipeline is exposed through an interactive application that enables real-time parameter tuning and terrain export, and resulting geometric data is extruded in Blender to produce a 3D terrain model.
Cite as
Batsambuu Batbold and Lori Ziegelmeier. From Chaos to Continents: Voronoi-Based Procedural Terrain Generation with Hydrology and 3D Visualization (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 101:1-101:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{batbold_et_al:LIPIcs.SoCG.2026.101,
author = {Batbold, Batsambuu and Ziegelmeier, Lori},
title = {{From Chaos to Continents: Voronoi-Based Procedural Terrain Generation with Hydrology and 3D Visualization}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {101:1--101:7},
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.2026.101},
URN = {urn:nbn:de:0030-drops-259077},
doi = {10.4230/LIPIcs.SoCG.2026.101},
annote = {Keywords: Procedural Content Generation, Voronoi Diagrams, Lloyd’s Relaxation, Perlin Noise, Blender}
}
Document
Media Exposition
Tracking a Set of Moving Objects with Minimal Peak Power (Media Exposition)
Abstract
A common sensing problem is to use a set of stationary tracking locations to monitor a collection of moving devices. Given n objects that need to be tracked, each following its own trajectory, and m stationary traffic control stations, each with a sensing region that can be changed over time; how should we adjust the individual sensor ranges in order to optimize energy consumption? We illustrate how to combine geometric insights with mathematical optimization to find optimal solutions for the min max variant of the problem, which aims at minimizing peak power consumption. Instances with 500 moving objects and 25 stations can be solved in the order of seconds for scenarios that take minutes to play out in the real world, demonstrating real-time capability of our methods.
Cite as
Sándor P. Fekete, Malte Hoffmann, Chek-Manh Loi, and Michael Perk. Tracking a Set of Moving Objects with Minimal Peak Power (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 102:1-102:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fekete_et_al:LIPIcs.SoCG.2026.102,
author = {Fekete, S\'{a}ndor P. and Hoffmann, Malte and Loi, Chek-Manh and Perk, Michael},
title = {{Tracking a Set of Moving Objects with Minimal Peak Power}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {102:1--102:7},
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.2026.102},
URN = {urn:nbn:de:0030-drops-259087},
doi = {10.4230/LIPIcs.SoCG.2026.102},
annote = {Keywords: Set cover, kinetic problems, geometric optimization, exact optimization}
}
Document
Media Exposition
Scalable Algorithmic Methods for Simulating Heavy-Rain Events (Media Exposition)
Abstract
We motivate and demonstrate simulation and evaluation of large-scale, fine-grained hydrodynamic flows, triggered by heavy-rain events. We show significant progress for simulating time-dependent, high-resolution runoff in large-scale heavy-rain scenarios, based on different geometry-based algorithmic speedup techniques. This enables us to address a second challenge: How can we deal with the instability of precipitation events, which are notoriously difficult to predict with good accuracy?
Cite as
Sándor P. Fekete, Phillip Keldenich, Michael Perk, and Tobias Wallner. Scalable Algorithmic Methods for Simulating Heavy-Rain Events (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 103:1-103:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fekete_et_al:LIPIcs.SoCG.2026.103,
author = {Fekete, S\'{a}ndor P. and Keldenich, Phillip and Perk, Michael and Wallner, Tobias},
title = {{Scalable Algorithmic Methods for Simulating Heavy-Rain Events}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {103:1--103:6},
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.2026.103},
URN = {urn:nbn:de:0030-drops-259094},
doi = {10.4230/LIPIcs.SoCG.2026.103},
annote = {Keywords: Heavy-rain events, hydrodynamic flows, scalable simulation, applied computational geometry, sensitivity analysis}
}
Document
Media Exposition
Interactive Visualization and Verification Tools for Tesseract Path Unfoldings (Media Exposition)
Abstract
This paper introduces interactive software tools for studying 2-face path unfoldings of the tesseract (4D hypercube). We present: (1) an algorithm to verify whether a given 24-omino is a valid path unfolding of the tesseract, (2) a web-based visualization tool for exploring and animating unfolding sequences with smooth 3D interpolation, and (3) a design interface integrated with SVG Painter, with a similar design to Demaine’s SVG Painter, to create custom unfoldings. We demonstrate these tools by designing a geometric font of 36 path unfoldings resembling Latin letters and digits, illustrating the rich diversity and accessibility of tesseract geometry.
Cite as
Soham Samanta, Hugo A. Akitaya, Erik Demaine, and Martin Demaine. Interactive Visualization and Verification Tools for Tesseract Path Unfoldings (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 104:1-104:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{samanta_et_al:LIPIcs.SoCG.2026.104,
author = {Samanta, Soham and A. Akitaya, Hugo and Demaine, Erik and Demaine, Martin},
title = {{Interactive Visualization and Verification Tools for Tesseract Path Unfoldings}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {104:1--104:5},
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.2026.104},
URN = {urn:nbn:de:0030-drops-259104},
doi = {10.4230/LIPIcs.SoCG.2026.104},
annote = {Keywords: unfolding, polyominoes, tesseract, path unfolding, visualization}
}
Document
CG Challenge
ETH Flippers Approach to Parallel Reconfiguration of Triangulations: SAT Formulation and Heuristics (CG Challenge)
Abstract
We describe the algorithms used by the ETH Flippers team in the CG:SHOP 2026 Challenge. Each instance consists of a set of triangulations on a common point set, and the objective is to find a central triangulation that minimizes the total parallel flip distance to the input set. Our strategy combines an exact solver for small and medium-sized instances with a suite of heuristics for larger instances. For the exact approach, we formulate the problem as a SAT instance with XOR clauses to model edge transitions across multiple rounds, further optimized by lower bounds derived from exact pairwise distances. For larger instances, we use a greedy local search and edge-coloring techniques to identify maximal sets of independent flips. Our approach ranked second overall and first in the junior category, computing provably optimal solutions for 186 out of 250 instances.
Cite as
Lorenzo Battini and Marko Milenković. ETH Flippers Approach to Parallel Reconfiguration of Triangulations: SAT Formulation and Heuristics (CG Challenge). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 105:1-105:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{battini_et_al:LIPIcs.SoCG.2026.105,
author = {Battini, Lorenzo and Milenkovi\'{c}, Marko},
title = {{ETH Flippers Approach to Parallel Reconfiguration of Triangulations: SAT Formulation and Heuristics}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {105:1--105:6},
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.2026.105},
URN = {urn:nbn:de:0030-drops-259115},
doi = {10.4230/LIPIcs.SoCG.2026.105},
annote = {Keywords: exact solution, heuristic, SAT solver, XOR clauses, computational geometry}
}
Document
CG Challenge
Engineering Greedy Heuristics and Simulated Annealing Methods for the Median Triangulation Under the Parallel Flip Distance (CG Challenge)
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
Shadoks Approach to Parallel Reconfiguration of Triangulations (CG Challenge)
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
CG Challenge
CG#Hunters Approach to Central Triangulation Under Parallel Flip Operations (CG Challenge)
Abstract
In the CG:SHOP 2026 Challenge, the goal is to compute a central triangulation for a given set of triangulations on the same point set while minimizing the sum of parallel flip distances. To address the problem, our team (CG#Hunters) constructs an initial solution by iteratively applying parallel flips to reduce the total number of crossings between the triangulations until none remain. To optimize these solutions, we shorten the paths by using a score-based greedy edge selection and refine the central triangulation via a large scale neighborhood search. Additionally, a representative-set-based approach is utilized to efficiently handle large instances. With these combined approaches, we achieved third place by successfully computing central triangulations with sufficiently short parallel flip paths for all 250 instances.
Cite as
Jaegun Lee, Seokyun Kang, Hyeonseok Lee, Hyeyun Yang, and Taehoon Ahn. CG#Hunters Approach to Central Triangulation Under Parallel Flip Operations (CG Challenge). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 108:1-108:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lee_et_al:LIPIcs.SoCG.2026.108,
author = {Lee, Jaegun and Kang, Seokyun and Lee, Hyeonseok and Yang, Hyeyun and Ahn, Taehoon},
title = {{CG#Hunters Approach to Central Triangulation Under Parallel Flip Operations}},
booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)},
pages = {108:1--108:8},
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.2026.108},
URN = {urn:nbn:de:0030-drops-259147},
doi = {10.4230/LIPIcs.SoCG.2026.108},
annote = {Keywords: Central triangulation, Parallel flip operations, Crossing number, Large scale neighborhood search, Representative set}
}