arXiv subjects

Geert van Wordragen

Publications and source records attributed to Geert van Wordragen.

9 recordsLinked to original sources

arXiv2026

Fast Thick-Thin Decomposition for Sparse Spanners on Hyperbolic Surfaces

We consider spanners for point sets lying in the hyperbolic plane or on a closed hyperbolic surface with the restriction that spanner edges are not allowed to cross. This is a natural generalization of non-crossing Euclidean spanners. Thus, the resulting spanner graphs are embedded in the hyperbolic plane or on the hyperbolic surface. As our main contribution, we show that there are sparse $(1+\varepsilon)$-spanners for these problems when we are allowed to use Steiner points: - on the hyperbolic plane we get a non-crossing Steiner $(1+\varepsilon)$-spanner with $\mathcal{O}(n / \varepsilon^2)$ edges, - on hyperbolic surfaces of genus $g$ we get a Steiner $(1+\varepsilon)$-spanner with $\mathcal{O}(n / \varepsilon^{3/2} + g/\varepsilon^2)$ non-crossing edges, or with $\mathcal{O}(n / \sqrt{\varepsilon} + g/\varepsilon)$ edges that are allowed to cross. In particular, our spanners on surfaces have sparsity with linear dependence on $g$, rather than the easier-to-attain exponential dependence, and the terms $n/\varepsilon^{3/2}$ and $n/\sqrt{\varepsilon}$ match the current best Euclidean results for plane and crossing Steiner spanners, respectively. As a corollary of our non-crossing spanner and techniques from the existing literature on light spanners and minor-free TSP, we get an EPTAS for TSP on hyperbolic surfaces. Our surface constructions rely on the thick-thin decomposition, a standard tool for studying hyperbolic surfaces. For convex hyperbolic polygons, we introduce an analogous neck decomposition. We give algorithms that compute the thick-thin decomposition of a genus-$g$ surface in $\mathcal{O}(g^4\log g)$ time and the neck decomposition of an $n$-vertex polygon in $\mathcal{O}(n)$ time.

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arXiv2026

Fine-Grained Complexity of Continuous Euclidean k-Center

In the (continuous) Euclidean $k$-center problem, given $n$ points in $\mathbb{R}^d$ and an integer $k$, the goal is to find $k$ center points in $\mathbb{R}^d$ that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. $\bullet$ Parameterized by $k$: Assuming the Exponential Time Hypothesis (ETH), we show that there is no $f(k)n^{o(k^{1-1/d})}$-time algorithm for the Euclidean $k$-center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any $(1+\varepsilon)$-approximation algorithm running in time $(k/\varepsilon)^{o(k^{1-1/d})}n^{O(1)}$, thereby establishing near-optimality of the corresponding approximation scheme by the same authors. $\bullet$ Small $k$: Assuming the 3-SUM hypothesis, we prove that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the Euclidean $2$-center problem in $\mathbb{R}^3$. This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any $\varepsilon > 0$, the Euclidean $6$-center problem in $\mathbb{R}^2$ also admits no $O(n^{2-\varepsilon})$-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

cs.CG↗

arXiv2026

Gap-ETH-Tight Algorithms for Hyperbolic TSP and Steiner Tree

We give an approximation scheme for the TSP in $d$-dimensional hyperbolic space that has optimal dependence on $\varepsilon$ under Gap-ETH. For any fixed dimension $d\geq 2$ and for any $\varepsilon>0$ our randomized algorithm gives a $(1+\varepsilon)$-approximation in time $2^{O(1/\varepsilon^{d-1})}n^{1+o(1)}$. 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.

cs.CG↗

arXiv2026

Improved Bounds for Discrete Voronoi Games

In the planar one-round discrete Voronoi game, two players $\mathcal{P}$ and $\mathcal{Q}$ compete over a set $V$ of $n$ voters represented by points in $\mathbb{R}^2$. First, $\mathcal{P}$ places a set $P$ of $k$ points, then $\mathcal{Q}$ places a set $Q$ of $\ell$ points, and then each voter $v\in V$ is won by the player who has placed a point closest to $v$. It is well known that if $k=\ell=1$, then $\mathcal{P}$ can always win $n/3$ voters and that this is worst-case optimal. We study the setting where $k>1$ and $\ell=1$. We present lower bounds on the number of voters that $\mathcal{P}$ can always win, which improve the existing bounds for all $k\geq 4$. As a by-product, we obtain improved bounds on small $\varepsilon$-nets for convex ranges. These results are for the $L_2$ metric. We also obtain lower bounds on the number of voters that $\mathcal{P}$ can always win when distances are measured in the $L_1$ metric.

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arXiv2025

Fine-Grained Complexity of Earth Mover's Distance under Translation

The Earth Mover's Distance is a popular similarity measure in several branches of computer science. It measures the minimum total edge length of a perfect matching between two point sets. The Earth Mover's Distance under Translation ($\mathrm{EMDuT}$) is a translation-invariant version thereof. It minimizes the Earth Mover's Distance over all translations of one point set. For $\mathrm{EMDuT}$ in $\mathbb{R}^1$, we present an $\tilde{\mathcal{O}}(n^2)$-time algorithm. We also show that this algorithm is nearly optimal by presenting a matching conditional lower bound based on the Orthogonal Vectors Hypothesis. For $\mathrm{EMDuT}$ in $\mathbb{R}^d$, we present an $\tilde{\mathcal{O}}(n^{2d+2})$-time algorithm for the $L_1$ and $L_\infty$ metric. We show that this dependence on $d$ is asymptotically tight, as an $n^{o(d)}$-time algorithm for $L_1$ or $L_\infty$ would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.

cs.CG↗

arXiv2025

Shortest Paths, Convexity, and Treewidth in Regular Hyperbolic Tilings

Hyperbolic tilings are natural infinite planar graphs where each vertex has degree $q$ and each face has $p$ edges for some $\frac1p+\frac1q<\frac12$. We study the structure of shortest paths in such graphs. We show that given a set of $n$ terminals, we can compute a so-called isometric closure (closely related to the geodesic convex hull) of the terminals in near-linear time, using a classic geometric convex hull algorithm as a black box. We show that the size of the convex hull is $O(N)$ where $N$ is the total length of the paths to the terminals from a fixed origin. Furthermore, we prove that the geodesic convex hull of a set of $n$ terminals has treewidth only $\max(12,O(\log\frac{n}{p + q}))$, a bound independent of the distance of the points involved. As a consequence, we obtain algorithms for subset TSP and Steiner tree with running time $O(N \log N) + \mathrm{poly}(\frac{n}{p + q}) \cdot N$.

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arXiv2025

Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces

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 $(1+\varepsilon)$-spanners of size $O_d(\varepsilon^{(1-d)/2}\log(1/\varepsilon)n)$, nearly matching the lower bounds of Bhore and Tóth [SIDMA'22]. We obtain Steiner $(1+\varepsilon)$-spanners of size $O_d(\varepsilon^{(1-d)/2}\log(1/\varepsilon)n)$ 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 $O(\log(1/\varepsilon))$ 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 $Ω(n\log n)$ edges, and we obtain a $2$-spanner of size $O_d(n\log n)$ in $d$-dimensional hyperbolic space, matching our lower bound for any constant $d$. Finally, we give a Steiner spanner with additive error $\varepsilon$ in hyperbolic space with $O_d(\varepsilon^{(1-d)/2}\log(α(n)/\varepsilon)n)$ edges, where $α(n)$ is the inverse Ackermann function. Our techniques generalize to closed orientable surfaces of constant curvature as well as to some quotient spaces.

cs.CG↗

arXiv2025

Structure and Independence in Hyperbolic Uniform Disk Graphs

We consider intersection graphs of disks of radius $r$ in the hyperbolic plane. Unlike the Euclidean setting, these graph classes are different for different values of $r$, where very small $r$ corresponds to an almost-Euclidean setting and $r \in Ω(\log n)$ corresponds to a firmly hyperbolic setting. We observe that larger values of $r$ create simpler graph classes, at least in terms of separators and the computational complexity of the \textsc{Independent Set} problem. First, we show that intersection graphs of disks of radius $r$ in the hyperbolic plane can be separated with $\mathcal{O}((1+1/r)\log n)$ cliques in a balanced manner. Our second structural insight concerns Delaunay complexes in the hyperbolic plane and may be of independent interest. We show that for any set $S$ of $n$ points with pairwise distance at least $2r$ in the hyperbolic plane the corresponding Delaunay complex has outerplanarity $1+\mathcal{O}(\frac{\log n}{r})$, which implies a similar bound on the balanced separators and treewidth of such Delaunay complexes. Using this outerplanarity (and treewidth) bound we prove that \textsc{Independent Set} can be solved in $n^{\mathcal{O}(1+\frac{\log n}{r})}$ time. The algorithm is based on dynamic programming on some unknown sphere cut decomposition that is based on the solution. The resulting algorithm is a far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is tight under the Exponential Time Hypothesis. In particular, \textsc{Independent Set} is polynomial-time solvable in the firmly hyperbolic setting of $r\in Ω(\log n)$. Finally, in the case when the disks have ply (depth) at most $\ell$, we give a PTAS for \textsc{Maximum Independent Set} that has only quasi-polynomial dependence on $1/\varepsilon$ and $\ell$. Our PTAS is a further generalization of our exact algorithm.

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arXiv2023

A Quadtree, a Steiner Spanner, and Approximate Nearest Neighbours in Hyperbolic Space

We propose a data structure in $d$-dimensional hyperbolic space that can be considered a natural counterpart to quadtrees in Euclidean spaces. Based on this data structure we propose a so-called L-order for hyperbolic point sets, which is an extension of the Z-order defined in Euclidean spaces. Using these quadtrees and the L-order we build geometric spanners. Near-linear size $(1+ε)$-spanners do not exist in hyperbolic spaces, but we are able to create a Steiner spanner that achieves a spanning ratio of $1+ε$ with $\mathcal O_{d,ε}(n)$ edges, using a simple construction that can be maintained dynamically. As a corollary we also get a $(2+ε)$-spanner (in the classical sense) of the same size, where the spanning ratio $2+ε$ is almost optimal among spanners of subquadratic size. Finally, we show that our Steiner spanner directly provides a solution to the approximate nearest neighbour problem: given a point set $P$ in $d$-dimensional hyperbolic space we build the data structure in $\mathcal O_{d,ε}(n\log n)$ time, using $\mathcal O_{d,ε}(n)$ space. Then for any query point $q$ we can find a point $p\in P$ that is at most $1+ε$ times farther from $q$ than its nearest neighbour in $P$ in $\mathcal O_{d,ε}(\log n)$ time. Moreover, the data structure is dynamic and can handle point insertions and deletions with update time $\mathcal O_{d,ε}(\log n)$.

cs.CG↗