Computational Geometry-Theory and Applications最新文献

英文中文

Engineering an algorithm for constructing low-stretch geometric graphs with near-greedy average degrees 构造具有近贪婪平均度的低拉伸几何图的算法

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2026-01-01 Epub Date: 2025-06-04 DOI: 10.1016/j.comgeo.2025.102201

FNU Shariful , Justin Weathers , Anirban Ghosh , Giri Narasimhan

We design and engineer Fast-Sparse-Spanner, a simple and practical (fast and memory-efficient) algorithm for constructing sparse low stretch factor geometric graphs on large pointsets in the plane. To our knowledge, this is the first practical algorithm to construct fast low stretch factor graphs on large pointsets with average degrees (hence, the number of edges) competitive with that of greedy spanners, the sparsest known class of Euclidean geometric spanners. Although theoretically not guaranteed to produce t-spanners, we always found in our rigorous experiments that Fast-Sparse-Spanner generated near-greedy size t-spanners.

To evaluate our implementation in terms of computation speed, memory usage, and quality of output, we performed extensive experiments with synthetic and real-world pointsets, and by comparing it to our closest competitor Bucketing, the fastest known greedy spanner algorithm for pointsets in the plane, devised by Alewijnse et al. (2017) [5]. Our experiment with constructing a 1.1-spanner on a large synthetic pointset with 128K points uniformly distributed within a square shows more than a 41-fold speedup with roughly a third of the memory usage of that of Bucketing, but with only a 3% increase in the average degree of the resulting graph. When ran on a pointset with a million points drawn from the same distribution, we observed a 130-fold speedup, with roughly a fourth of the memory usage of that of Bucketing, and just a 6% increase in the average degree. In terms of diameter, the graphs generated by Fast-Sparse-Spanner beat greedy spanners in most cases (have substantially lower diameter) while maintaining near-greedy average degree. Further, our algorithm can be easily parallelized to take advantage of parallel environments.

We share the implementations via GitHub for broader uses and future research.

GitHub repository. https://github.com/ghoshanirban/FSS.

我们设计和工程fast - sparse - spanner，一个简单实用（快速和内存高效）的算法，用于在平面上的大点集上构造稀疏的低拉伸因子几何图形。据我们所知，这是第一个在大型点集上构建快速低拉伸因子图的实用算法，其平均度（因此，边的数量）与贪婪扳手（已知最稀疏的欧几里德几何扳手类）竞争。虽然理论上不能保证产生t型扳手，但我们总是在严格的实验中发现Fast-Sparse-Spanner生成了接近贪婪尺寸的t型扳手。为了评估我们在计算速度、内存使用和输出质量方面的实现，我们对合成和现实世界的点集进行了广泛的实验，并将其与我们最接近的竞争对手bucket进行比较，bucket是由Alewijnse等人（2017）设计的平面上点集已知最快的贪婪扳手算法。我们的实验是在一个大的合成点集上构造一个1.1扳手，其中128K个点均匀分布在一个正方形内，结果显示，在内存使用大约是bucket的三分之一的情况下，加速速度提高了41倍以上，但结果图的平均程度只增加了3%。当在从相同分布中提取一百万个点的点集上运行时，我们观察到速度提高了130倍，内存使用量大约是bucket的四分之一，平均程度仅提高了6%。在直径方面，Fast-Sparse-Spanner生成的图在大多数情况下（具有更低的直径）都优于贪婪扳手，同时保持接近贪婪的平均度。此外，我们的算法可以很容易地并行化，以利用并行环境。我们通过GitHub分享实现，以供更广泛的使用和未来的研究。GitHub库。https://github.com/ghoshanirban/FSS。

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引用次数: 0

Realizable dimension of periodic frameworks 周期框架的可实现维度

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2026-01-01 Epub Date: 2025-04-29 DOI: 10.1016/j.comgeo.2025.102200

Ryoshun Oba, Shin-ichi Tanigawa

Belk and Connelly introduced the realizable dimension

rd (G)

of a finite graph G, which is the minimum nonnegative integer d such that every framework

(G, p)

in any dimension admits a framework in

R^{d}

with the same edge lengths. They characterized finite graphs with realizable dimension at most 1, 2, or 3 in terms of forbidden minors. In this paper, we consider periodic frameworks and extend the notion to

Z

-symmetric graphs. We give a forbidden minor characterization of

Z

-symmetric graphs with realizable dimension at most 1 or 2, and show that the characterization can be checked in linear time when a graph is given as a quotient

Z

-labeled graph.

Belk和Connelly引入了有限图G的可实现维数rd(G)，它是最小非负整数d，使得任意维上的每一个框架（G,p）在rd中都有一个具有相同边长的框架。他们用禁忌次元描述了可实现维数最多为1、2或3的有限图。本文考虑周期框架，并将其推广到z对称图。我们给出了可实现维数最多为1或2的z对称图的一个禁忌小特征，并证明了当图被给定为商z标记图时，该特征可以在线性时间内得到检验。

引用次数: 0

Parallel line centers with guaranteed separation 平行线中心，保证分离

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-12-01 Epub Date: 2025-03-18 DOI: 10.1016/j.comgeo.2025.102185

Chaeyoon Chung , Taehoon Ahn , Sang Won Bae , Hee-Kap Ahn

Given a set P of n points in the plane and an integer

k \geq 1

, the k-line-center problem asks k slabs whose union covers P that minimizes the maximum width of the k slabs. In this paper, we introduce a new variant of the k-line-center problem for

k \geq 2

, in which the resulting k lines are parallel and a prescribed separation between two line centers is guaranteed. More precisely, we define a measure of separation, namely the gap-ratio of k parallel slabs, to be the minimum distance between any two slabs, divided by the width of the smallest slab enclosing the k slabs. We present efficient algorithms for the following problems: (1) Given a real

0 < ρ \leq 1

, compute k parallel slabs of minimum width that cover P with gap-ratio at least ρ. (2) Compute k parallel slabs that cover P with maximum possible gap-ratio. Our algorithms run in

O (ρ^{- k} \cdot (n \log n + k n))

and

O (ρ_{\max}^{- k} \cdot (n \log n + k n))

time, respectively, using

O (k n \log k)

space, where

ρ_{\max}

denotes the maximum possible gap-ratio of any k parallel slabs that cover P. Using linear space, the running times only slightly increase to

O (ρ^{- k} \cdot k n \log n)

and

O (ρ_{\max}^{- k} \cdot k n \log n)

给定平面上有n个点的集合P，且整数k≥1，k线中心问题要求k个板，其并集覆盖P，使k个板的最大宽度最小。本文引入了k≥2时k-线中心问题的一个新变体，其中k条线是平行的，并且保证两条线中心之间有规定的间隔。更准确地说，我们定义了一种分离度量，即k块平行板的间隙比，是任意两块板之间的最小距离，除以围住k块板的最小板的宽度。我们给出了以下问题的有效算法：(1)给定一个实数0<；ρ≤1，计算k个覆盖P的最小宽度平行板，其间隙比至少为ρ。(2)以最大可能的间隙比计算覆盖P的k个平行板。我们的算法运行时间分别为O（ρ−k⋅(nlog (n) +kn)）和O(ρmax - k⋅(nlog (n) +kn))，使用O（knlog (k)）空间，其中ρmax表示覆盖p的任意k个平行平板的最大可能间隙比，使用线性空间，运行时间仅略微增加到O（ρ−k⋅klog (n)）和O（ρmax - k⋅klog (n)）。

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引用次数: 0

Constrained boundary labeling 约束边界标注

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-12-01 Epub Date: 2025-03-26 DOI: 10.1016/j.comgeo.2025.102191

Thomas Depian , Martin Nöllenburg , Soeren Terziadis , Markus Wallinger

Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce grouping and ordering constraints in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is NP-hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is NP-complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.

边界标记是计算几何中的一种技术，用于标记插图中的特征集。它包括沿轴平行的边界框放置标签，并使用不交叉的引线将每个标签与其相应的特征连接起来。尽管边界标注已经得到了很好的研究，但对标注的语义约束还没有进行深入的研究。本文在边界标注中引入了分组约束和排序约束：分组约束要求一组中的所有标签在边界上连续放置，排序约束要求标签在边界上部分有序。我们证明了它是np困难找到一个标记的任意大小的标签，无限制的位置沿边界的一侧。然而，如果我们将这个问题限制为等高标签或有限候选位置集，我们将获得多项式时间算法。此外，我们证明了在边界的两个相对侧找到标记是np完全的，即使对于等高度标记和有限的标记位置也是如此。最后，我们通过实验证实了我们的方法也具有实际意义。

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引用次数: 0

Bounds on the edge-length ratio of 2-outerplanar graphs 2-外平面图边长比的界

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-12-01 Epub Date: 2025-03-27 DOI: 10.1016/j.comgeo.2025.102192

Emilio Di Giacomo , Walter Didimo , Giuseppe Liotta , Henk Meijer , Fabrizio Montecchiani , Stephen Wismath

The edge-length ratio of a planar straight-line drawing Γ of a graph G is the largest ratio between the lengths of every pair of edges of Γ. If the ratio is measured by considering only pairs of edges that are incident to a common vertex, we talk about local edge-length ratio. The (local) edge-length ratio of a planar graph is the infimum over all (local) edge-length ratios of its planar straight-line drawings. It is known that the edge-length ratio of outerplanar graphs is upper bounded by a constant, while there exist graph families with non-constant outerplanarity that have non-constant lower bounds on their edge-length ratios. In this paper we prove an

Ω (\sqrt{n})

lower bound on the local edge-length ratio (and hence on the edge-length ratio) of the n-vertex 2-outerplanar graphs. We also prove a constant upper bound on the edge-length ratio of Halin graphs, pseudo-Halin graphs, and their generalizations.

图形G的平面直线图Γ的边长比是Γ的每对边的长度之比的最大值。如果这个比率是通过只考虑与一个公共顶点相关的边对来测量的，我们就讨论局部边长比。平面图的（局部）边长比是其平面直线图的所有（局部）边长比的最小值。已知外平面图的边长比上界有一个常数，而存在外平面度为非常数的图族，其边长比下界为非常数。本文证明了n顶点2-外平面图的局部边长比的Ω(n)下界，从而证明了n顶点2-外平面图的边长比。我们还证明了Halin图、伪Halin图及其推广的边长比的一个常数上界。

引用次数: 0

Connected matchings 连接拼毛

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-12-01 Epub Date: 2025-02-26 DOI: 10.1016/j.comgeo.2025.102174

Oswin Aichholzer , Sergio Cabello , Viola Mészáros , Patrick Schnider , Jan Soukup

We show that each set of

n ⩾ 2

points in the plane in general position has a straight-line matching with at least

(5 n + 1) / 27

edges whose segments form a connected set, and such a matching can be computed in

O (n \log n)

time. As an upper bound, we show that for some planar point sets in general position the largest matching whose segments form a connected set has

⌈ \frac{n - 1}{3} ⌉

edges. We also consider a colored version, where each edge of the matching should connect points with different colors.

我们表明，平面中一般位置上的n个或2个点的每一组与至少(5n+1)/27条边具有直线匹配，其线段形成连接集，并且这种匹配可以在O（nlog ln n）时间内计算。作为上界，我们证明了对于一般位置上的某平面点集，其区段构成连通集的最大匹配具有≤n−13条边。我们还考虑了一个彩色版本，其中匹配的每个边缘都应该连接不同颜色的点。

引用次数: 0

On exact covering with unit disks 用单位圆盘精确覆盖

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-12-01 Epub Date: 2025-03-26 DOI: 10.1016/j.comgeo.2025.102193

Ji Hoon Chun, Christian Kipp, Sandro Roch

We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an exact cover is not possible.

我们研究了用单位圆盘覆盖平面上给定点集的问题，使每个点只覆盖一次。我们证明了17个点总是可以被完全覆盖。另一方面，我们构造一个657个点的集合，其中精确的覆盖是不可能的。

引用次数: 0

Flips in odd matchings 在奇数配对中投掷

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-12-01 Epub Date: 2025-03-13 DOI: 10.1016/j.comgeo.2025.102184

Oswin Aichholzer , Anna Brötzner , Daniel Perz , Patrick Schnider

Let

P

be a set of

n = 2 m + 1

points in the plane in general position. We define the graph

G M_{P}

whose vertex set is the set of all plane matchings on

P

with exactly m edges. Two vertices in

G M_{P}

are connected if the two corresponding matchings have

m - 1

edges in common. In this work we show that

G M_{P}

is connected and give an upper bound of

O (n^{2})

on its diameter. Moreover, we present a lower bound of

n - 2

and an upper bound of

2 n - 2

for the diameter of

G M_{P}

for

P

in convex position.

设P是平面上一般位置上n=2m+1个点的集合。定义图GMP，其顶点集是P上所有平面匹配的恰好m条边的集合。在GMP中，如果两个对应的匹配有m−1条共同的边，则两个顶点是连通的。在这项工作中，我们证明了GMP是连通的，并给出了其直径的上界O（n2）。此外，我们给出了P在凸位置的GMP直径的下界n−2和上界2n−2。

引用次数: 0

Revisiting the Fréchet distance between piecewise smooth curves 重新审视片断光滑曲线之间的弗雷谢特距离

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-12-01 Epub Date: 2025-04-09 DOI: 10.1016/j.comgeo.2025.102194

Jacobus Conradi , Anne Driemel , Benedikt Kolbe

Since its introduction to computational geometry by Alt and Godau in 1992, the Fréchet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision problem for piecewise smooth curves that is both conceptually simpler and naturally extends to the first algorithm for the problem for piecewise smooth curves in

R^{d}

We assume that the algorithm is given two continuous curves, each consisting of a sequence of m, resp. n, smooth pieces, where each piece belongs to a sufficiently well-behaved class of curves, such as the set of algebraic curves of bounded degree. We introduce a decomposition of the free space diagram into a controlled number of pieces that can be used to solve the decision problem similarly to the polygonal case, in

O (m n)

time, leading to a computation of the Fréchet distance that runs in

O (m n \log (m n))

time.

Furthermore, we study approximation algorithms for piecewise smooth curves that are also c-packed for some fixed value c. We adapt the existing framework for polygonal curves that leads to a near-linear

(1 + ε)

-approximation to the Fréchet distance to the setting of piecewise smooth curves.

自1992年Alt和Godau将其引入计算几何以来，fr切特距离一直是曲线相似度计算算法研究的支柱。研究的重点一直是比较多边形曲线，值得注意的是，Rote（2007）为平面分段光滑曲线的决策问题提供了一个算法例外。我们提出了一种用于分段光滑曲线决策问题的算法，该算法在概念上更简单，并且自然地扩展到rd中分段光滑曲线问题的第一种算法。我们假设该算法给定两条连续曲线，每条曲线由m的序列组成。N个光滑块，其中每个块属于一个足够好的曲线类，例如有界次的代数曲线集。我们将自由空间图分解为可控制数量的块，这些块可用于解决与多边形情况类似的决策问题，在O（mn）时间内，导致在O（mnlog (mn)）时间内运行的fr距离计算。此外，我们研究了分段光滑曲线的近似算法，这些曲线对于某个固定值c也是c填充的。我们采用了现有的多边形曲线框架，使得分段光滑曲线的fr距离近似为近线性（1+ε）。

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引用次数: 0

Pattern formation for fat robots with memory 有记忆的胖机器人的模式形成

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-12-01 Epub Date: 2025-03-19 DOI: 10.1016/j.comgeo.2025.102189

Rusul J. Alsaedi, Joachim Gudmundsson, André van Renssen

Given a set of

n \geq 1

autonomous, anonymous, indistinguishable, silent, and possibly disoriented mobile unit disk (i.e., fat) robots operating following Look-Compute-Move cycles in the Euclidean plane, we consider the Pattern Formation problem: from arbitrary starting positions, the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. We assume that a robot's movement cannot be interrupted by an adversary and that robots have a small

O (1)

-sized memory that they can use to store information, but that cannot be communicated to the other robots. To solve this problem, we present an algorithm that works in three steps. First it establishes mutual visibility, then it elects one robot to be the leader, and finally it forms the required pattern. The whole algorithm runs in

O (n) + O (q \log n)

rounds with probability at least

1 - n^{- q}

. The algorithms are collision-free and do not require the knowledge of the number of robots.

给定一组n≥1个自主的、匿名的、不可区分的、沉默的、可能迷失方向的移动单元磁盘（即脂肪）机器人在欧几里德平面上按照look - comput - move循环操作，我们考虑模式形成问题：从任意起始位置，机器人必须重新定位自己以形成给定的目标模式。这个问题出现在能见度受阻的情况下，如果在两个机器人之间的直线段上有第三个机器人，机器人就看不到另一个机器人。我们假设一个机器人的运动不会被对手打断，并且机器人有一个小的0(1)大小的内存，它们可以用来存储信息，但这些信息不能传递给其他机器人。为了解决这个问题，我们提出了一个分三步工作的算法。首先建立相互可见性，然后选择一个机器人作为领导者，最后形成所需的模式。整个算法运行周期为O(n)+O（qlog (n)）轮，概率至少为1−n−q。该算法是无碰撞的，不需要了解机器人的数量。

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引用次数: 0

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