Computational Geometry-Theory and Applications最新文献

英文中文

Approximating average bounded-angle minimum spanning trees

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-02-18 DOI: 10.1016/j.comgeo.2025.102172

Ahmad Biniaz , Prosenjit Bose , Patrick Devaney

Motivated by the problem of orienting directional antennas in wireless communication networks, we study average bounded-angle minimum spanning trees. Let P be a set of points in the plane and let α be an angle. An α-spanning tree (α-ST) of P is a spanning tree of the complete Euclidean graph induced by P such that all edges incident to each point

p \in P

lie in a fixed wedge of angle α with apex p. An α-minimum spanning tree (α-MST) of P is an α-ST with minimum total edge length.

An average-α-spanning tree (denoted by

\overline{α}

-ST) is a spanning tree with the relaxed condition that incident edges to all points lie in wedges with average angle α. An average-α-minimum spanning tree (

\overline{α}

-MST) is an

\overline{α}

-ST with minimum total edge length.

Let

A (α)

be the smallest ratio of the length of the

\overline{α}

-MST to the length of the standard MST, over all sets of points in the plane. We investigate bounds for

A (α)

. For

α = \frac{2 π}{3}

, Biniaz, Bose, Lubiw, and Maheshwari (Algorithmica 2022) showed that

\frac{4}{3} \leq A (\frac{2 π}{3}) \leq \frac{3}{2}

. We improve the upper bound and show that

A (\frac{2 π}{3}) \leq \frac{13}{9}

. We also study this for

α = \frac{π}{2}

and prove that

\frac{3}{2} \leq A (\frac{π}{2}) \leq 4

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引用次数: 0

Finding a largest-area triangle in a terrain in near-linear time

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-02-17 DOI: 10.1016/j.comgeo.2025.102171

Sergio Cabello , Arun Kumar Das , Sandip Das , Joydeep Mukherjee

A terrain is an x-monotone polygon whose lower boundary is a single line segment. We present an algorithm to find in a terrain a triangle of largest area in

O (n \log n)

time, where n is the number of vertices defining the terrain. The best previous algorithm for this problem has a running time of

O (n^{2})

引用次数: 0

Maximum inscribed and minimum enclosing tropical balls of tropical polytopes and applications to volume estimation and uniform sampling

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-01-27 DOI: 10.1016/j.comgeo.2025.102163

David Barnhill , Ruriko Yoshida , Keiji Miura

We consider a minimum enclosing and maximum inscribed tropical balls for any given tropical polytope over the tropical projective torus in terms of the tropical metric with the max-plus algebra. We show that we can obtain such tropical balls via linear programming. Then we apply minimum enclosing and maximum inscribed tropical balls of any given tropical polytope to estimate the volume of and sample uniformly from the tropical polytope.

引用次数: 0

A geometric condition for uniqueness of Fréchet means of persistence diagrams

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2024-12-30 DOI: 10.1016/j.comgeo.2024.102162

Yueqi Cao, Anthea Monod

The Fréchet mean is an important statistical summary and measure of centrality of data; it has been defined and studied for persistent homology captured by persistence diagrams. However, the complicated geometry of the space of persistence diagrams implies that the Fréchet mean for a given set of persistence diagrams is not necessarily unique, which prohibits theoretical guarantees for empirical means with respect to population means. In this paper, we derive a variance expression for a set of persistence diagrams exhibiting a multi-matching between the persistence points known as a grouping. Moreover, we propose a condition for groupings, which we refer to as flatness; we prove that sets of persistence diagrams that exhibit flat groupings give rise to unique Fréchet means. We derive a finite sample convergence result for general groupings, which results in convergence for Fréchet means if the groupings are flat. We then interpret flat groupings in a recently-proposed general framework of Fréchet means in Alexandrov geometry. Finally, we show that for manifold-valued data, the persistence diagrams can be truncated to construct flat groupings.

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引用次数: 0

Pattern formation for fat robots with lights

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2024-12-27 DOI: 10.1016/j.comgeo.2024.102161

Rusul J. Alsaedi, Joachim Gudmundsson, André van Renssen

Given a set of

n \geq 1

unit disk robots in the Euclidean plane, we consider the Pattern Formation problem, i.e., 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. Recently, this problem was solved in the asynchonous model for fat robots that agree on at least one axis in the robots with lights model where each robot is equipped with an externally visible persistent light that can assume colors from a fixed set of colors [1]. In this work, we reduce the number of colors needed and remove the axis-agreement requirement in the fully synchronous model. In particular, we present an algorithm requiring 7 colors when scaling the target pattern is allowed and an 8-color algorithm if scaling is not allowed. Our algorithms run in

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

rounds with probability at least

1 - n^{- q}

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引用次数: 0

Infinite circle packings on surfaces with conical singularities

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2024-12-20 DOI: 10.1016/j.comgeo.2024.102160

Philip L. Bowers , Lorenzo Ruffoni

We show that given an infinite triangulation K of a surface with punctures (i.e., with no vertices at the punctures) and a set of target cone angles smaller than π at the punctures that satisfy a Gauss-Bonnet inequality, there exists a hyperbolic metric that has the prescribed angles and supports a circle packing in the combinatorics of K. Moreover, if K is very symmetric, then we can identify the underlying Riemann surface and show that it does not depend on the angles. In particular, this provides examples of a triangulation K and a conformal class X such that there are infinitely many conical hyperbolic structures in the conformal class X with a circle packing in the combinatorics of K. This is in sharp contrast with a conjecture of Kojima-Mizushima-Tan in the closed case.

引用次数: 0

Geometric and algorithmic solutions to the generalised alibi query

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2024-12-16 DOI: 10.1016/j.comgeo.2024.102159

Arthur Jansen, Bart Kuijpers

Space-time prisms provide a framework to model the uncertainty on the space-time points that a moving object may have visited between measured space-time locations, provided that a bound on the speed of the moving object is given. In this model, the alibi query asks whether two moving objects, given by their respective measured space-time locations and speed bound, may have met. An analytical solution to this problem was first given by Othman [15]. In this paper, we address the generalised alibi query that asks the same question for an arbitrary number

n \geq 2

of moving objects. We provide several solutions (mainly via the spatial and temporal projection) to this query with varying time complexities. These algorithmic solutions rely on techniques from convex and semi-algebraic geometry. We also address variants of the generalised alibi query where the question is asked for a given spatial location or a given moment in time.

引用次数: 0

Realizability of free spaces of curves 曲线自由空间的可实现性

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2024-11-22 DOI: 10.1016/j.comgeo.2024.102151

Hugo A. Akitaya , Maike Buchin , Majid Mirzanezhad , Leonie Ryvkin , Carola Wenk

The free space diagram is a popular tool to compute the well-known Fréchet distance. As the Fréchet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often the question arises whether a certain pattern in the free space diagram is “realizable”, i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram?

Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in

R^{> 2}

, showing

\exists R

-hardness. We use this to show that for curves in

R^{\geq 2}

the realizability problem is

\exists R

-complete, both for continuous and discrete Fréchet distances. We prove that the continuous case in

R^{1}

is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in

R^{1}

we show that the problem becomes solvable in polynomial time.

自由空间图是一种流行的工具，用于计算众所周知的fr切距离。由于在许多不同的领域中使用了fr切特距离，因此已经建立了许多变体来满足这些应用程序的特定需求。经常出现的问题是，自由空间图中的某个图案是否“可实现”，即是否存在一对多边形链，其自由空间图与之相对应。这个问题的答案可能有助于确定这些距离度量的计算复杂性，并允许为避免某些自由空间模式的受限输入类设计更有效的算法。因此我们研究反问题：给定一个势自由空间图，是否存在生成这个图的曲线？我们感兴趣的问题与经典的距离几何问题密切相关。我们在R>；2中解决了距离几何的复杂性，并给出了∃r硬度。我们用它来证明，对于R≥2的曲线，无论是对于连续的还是离散的fr距离，可实现问题都是∃R完全的。我们证明了R1中的连续情况是弱np困难的，并给出了一个伪多项式时间算法，证明了它是定参数可处理的。有趣的是，对于R1中的离散情况我们证明了这个问题在多项式时间内是可解的。

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引用次数: 0

Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces 双曲空间的嵌入和近邻搜索与恒定加性误差

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2024-10-30 DOI: 10.1016/j.comgeo.2024.102150

Eunku Park, Antoine Vigneron

We give an embedding of the Poincaré halfspace

H^{D}

into a discrete metric space based on a binary tiling of

H^{D}

, with additive distortion

O (\log D)

. It yields the following results. We show that any subset P of n points in

H^{D}

can be embedded into a graph-metric with

2^{O (D)} n

vertices and edges, and with additive distortion

O (\log D)

. We also show how to construct, for any k, an

O (k \log D)

-purely additive spanner of P with

2^{O (D)} n

Steiner vertices and

2^{O (D)} n \cdot λ_{k} (n)

edges, where

λ_{k} (n)

is the kth-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for P of size

2^{O (D)} n

. It allows us to answer approximate near-neighbor queries in

2^{O (D)} + O (D \log n)

time, with additive error

O (\log D)

. These constructions can be done in

2^{O (D)} n \log n

time.

我们给出了一种基于二元平铺的离散度量空间 HD 的嵌入方法，其附加变形为 O(logD)。它产生了以下结果。我们证明，HD 中任何 n 个点的子集 P 都可以嵌入到一个具有 2O(D)n 个顶点和边的图度量空间中，其附加变形为 O(logD)。我们还展示了如何为任意 k 构建 P 的 O(klogD)-purely additive spanner，该 spanner 具有 2O(D)n 个 Steiner 顶点和 2O(D)n⋅λk(n) 条边，其中 λk(n) 是第 k 行逆阿克曼函数。最后，我们展示了如何为 P 构建大小为 2O(D)n 的近似 Voronoi 图。它允许我们在 2O(D)+O(Dlogn) 时间内回答近似近邻查询，加法误差为 O(logD)。这些构造可以在 2O(D)nlogn 时间内完成。

引用次数: 0

Parameterized inapproximability of Morse matching 莫尔斯匹配的参数化不可逼近性

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2024-10-30 DOI: 10.1016/j.comgeo.2024.102148

Ulrich Bauer , Abhishek Rathod

We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of Min-Morse Matching within a factor of

2^{\log^{(1 - ϵ)} n}

. Our second result shows that Min-Morse Matching is

W [P]

-hard with respect to the standard parameter. Next, we show that Min-Morse Matching with standard parameterization has no FPT approximation algorithm for any approximation factor ρ. The above hardness results are applicable to complexes of dimension ≥2.

On the positive side, we provide a factor

O (\frac{n}{\log n})

approximation algorithm for Min-Morse Matching on 2-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa–Farber model of random complexes.

我们从不可逼近性和参数化复杂性的角度研究了临界简约数最小化问题。我们首先证明了 Min-Morse Matching 在 2log(1-ϵ)n因子范围内的不可逼近性。我们的第二个结果表明，Min-Morse Matching 在标准参数方面是 W[P]-hard 的。从积极的方面看，我们为 2 维复数上的 Min-Morse Matching 提供了系数为 O(nlogn) 的近似算法，而对于更高维的复数，我们还不知道有这样的算法。最后，我们设计了具有极少临界简约的离散梯度，适用于从柯斯达-法伯随机复数模型相当广泛的参数值范围中抽取的典型实例。

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