Computational Geometry-Theory and Applications最新文献

英文中文

Geometric TSP on sets 集合上的几何TSP

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-03-18 DOI: 10.1016/j.comgeo.2025.102187

Henk Alkema, Mark de Berg

<div><div>In One-of-a-Set TSP, also known as the Generalised TSP, the input is a collection <math><mi>P</mi><mo>:</mo><mo>=</mo><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>}</mo></math> of sets in a metric space and the goal is to compute a minimum-length tour that visits one element from each set.</div><div>In the Euclidean variant of this problem, each <math><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub></math> is a set of points in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math>. Let <math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math> be a hypercube that contains <math><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub></math>, for <math><mn>1</mn><mo>⩽</mo><mi>i</mi><mo>⩽</mo><mi>r</mi></math>. We investigate how the complexity of Euclidean One-of-a-Set TSP depends on λ, the ply of the set <math><mi>H</mi><mo>:</mo><mo>=</mo><mo>{</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>}</mo></math> of hypercubes. (The ply is the smallest λ such that every point in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math> is contained in at most λ of the hypercubes). We show that the problem can be solved in <math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup><mo>)</mo></mrow></msup></math> time, where <math><mi>n</mi><mo>:</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></math> is the total number of points, and that the problem cannot be solved in <math><msup><mrow><mn>2</mn></mrow><mrow><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math> time when <math><mi>λ</mi><mo>=</mo><mi>Θ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math>, unless the Exponential Time Hypothesis (ETH) fails.</div><div>In Rectilinear One-of-a-Cube TSP, the input is a set <math><mi>H</mi></math> of hypercubes in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math> and the goal is to compute a minimum-length rectilinear tour that visits every hypercube. We show that the problem can be solved in <math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>d</mi></

在One-of-a-Set TSP中，也称为广义TSP，输入是一个集合P:={P1，…，Pr}是度量空间中的集合，目标是计算访问每个集合中的一个元素的最小长度巡回。在这个问题的欧几里得变体中，每个Pi是Rd中点的集合。设Hi是一个包含Pi的超立方体，对于1≤i≤r。我们研究了欧几里得单集TSP的复杂度如何依赖于λ，集合H的层数：={H1，…，超立方体的Hr}。（铺层是最小的λ，使得Rd中的每个点最多包含在超立方体的λ中）。我们证明了问题可以在2O（λ1/dn1−1/d）时间内解决，其中n:=∑i=1r|Pi|是总点数，并且当λ=Θ(n)时，除非指数时间假设（ETH）失效，否则问题不能在2O(n)时间内解决。在直角1 -of-a- cube TSP中，输入是Rd中的超立方体集合H，目标是计算访问每个超立方体的最小长度的直线巡回。我们证明了这个问题可以在2O（λ1/dn1−1/dlog (n)）时间内解决，其中n是超立方体的个数。

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 : 2025-03-13 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

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 : 2025-02-26 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

Minimum-width double-slabs and widest empty slabs in high dimensions 最小宽度的双层板和最宽的高尺寸空板

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications

Pub Date : 2025-02-25 DOI: 10.1016/j.comgeo.2025.102173

Taehoon Ahn , Chaeyoon Chung , Hee-Kap Ahn , Sang Won Bae , Otfried Cheong , Sang Duk Yoon

A slab in d-dimensional space

R^{d}

is the set of points enclosed by two parallel hyperplanes. We consider the problem of finding an optimal pair of parallel slabs, called a double-slab, that covers a given set P of n points in

R^{d}

. We address two optimization problems in

R^{d}

for any fixed dimension

d ⩾ 3

: the minimum-width double-slab problem, in which one wants to minimize the maximum width of the two slabs of the resulting double-slab, and the widest empty slab problem, in which one wants to maximize the gap between the two slabs. Our results include the first nontrivial exact algorithms that solve the former problem for

d ⩾ 3

and the latter problem for

d ⩾ 4

d维空间Rd中的平板是由两个平行超平面包围的点的集合。我们考虑寻找最优的平行平板对的问题，称为双平板，覆盖Rd中给定的n个点的集合P。对于任何固定维度d大于或等于3，我们在Rd中解决两个优化问题：最小宽度双平板问题，其中人们希望最小化所得到的双平板的两个平板的最大宽度，以及最宽的空平板问题，其中人们希望最大化两个平板之间的间隙。我们的结果包括解决d大于或等于3的前一个问题和d大于或等于4的后一个问题的第一个非平凡精确算法。

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引用次数: 0

Approximating average bounded-angle minimum spanning trees 近似平均有界角最小生成树

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

针对无线通信网络中定向天线的定位问题，研究了平均有界角最小生成树。设P是平面上点的集合，设α是一个角。P的α-生成树（α- st）是由P生成的完全欧氏图的生成树，使得与每个点P∈P相关的所有边都位于以顶点P为角α的固定楔中。P的α-最小生成树（α- mst）是总边长度最小的α- st。平均-α-生成树（用α -ST表示）是一种松弛条件，即所有点的入射边都在平均角为α的楔形中。平均-α-最小生成树（α -MST）是具有最小总边长度的α -ST。设A（α）是在平面上所有点的集合上，α -MST的长度与标准MST的长度之比最小。我们研究了A（α）的界。对于α=2π3, Biniaz, Bose， Lubiw和Maheshwari （Algorithmica 2022）证明43≤A(2π3)≤32。我们改进了上界，证明了A(2π3)≤139。我们也对α=π2进行了研究，证明了32≤A(π2)≤4。

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引用次数: 0

Finding a largest-area triangle in a terrain in near-linear time 在近线性时间内找到地形上面积最大的三角形

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})

地形是一个x单调多边形，它的下界是一条线段。我们提出了一种算法，在O（nlog (n)）时间内在地形中找到面积最大的三角形，其中n是定义地形的顶点数。对于这个问题，之前最好的算法的运行时间是O（n2）。

引用次数: 0

Maximum inscribed and minimum enclosing tropical balls of tropical polytopes and applications to volume estimation and uniform sampling 热带多面体的最大内切和最小封闭热带球及其在体积估计和均匀抽样中的应用

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.

我们考虑了在热带射影环面上任意给定热带多面体的最小外接和最大内切的热带球，并给出了具有max-plus代数的热带度量。我们证明了用线性规划可以得到这样的热带球。然后利用任意给定热带多面体的最小外接球和最大内接球来均匀估计热带多面体的体积和样本。

引用次数: 0

A geometric condition for uniqueness of Fréchet means of persistence diagrams 持久性图的一个几何条件

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.

法氏平均是一种重要的统计汇总和数据中心性度量；它已经被定义并研究了持久化图所捕获的持久化同构。然而，持久性图空间的复杂几何结构意味着给定的持久性图集合的fr平均值不一定是唯一的，这就妨碍了对总体平均值的经验平均值的理论保证。在本文中，我们推导了一组持久性图的方差表达式，这些持久性图显示了称为分组的持久性点之间的多重匹配。此外，我们提出了一个条件，我们称之为平坦性；我们证明了显示平面分组的持久性图集产生了唯一的fr切法。我们得到了一般群的有限样本收敛结果，该结果表明当群是平的时，对于frimchet means是收敛的。然后，我们在最近提出的亚历山德罗夫几何中fr均值的一般框架中解释平群。最后，我们证明了对于流形值数据，可以截断持久性图以构造平面分组。

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引用次数: 0

Pattern formation for fat robots with lights 有光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 : 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}

给定欧几里得平面上的一组n≥1个单元的圆盘机器人，我们考虑模式形成问题，即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。最近，这个问题在胖机器人的异步模型中得到了解决，在机器人带灯模型中，每个机器人都配备了一个外部可见的持久光，可以从一组固定的颜色中选择颜色[1]。在这项工作中，我们减少了所需的颜色数量，并消除了完全同步模型中的轴一致要求。特别是，我们提出了一种算法，当允许缩放目标图案时需要7种颜色，如果不允许缩放则需要8种颜色。我们的算法运行在O(n)+O（qlog (n)）轮中，概率至少为1−n−q。

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引用次数: 0

Infinite circle packings on surfaces with conical singularities 具有圆锥奇点的曲面上的无限圆填充

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.

我们表明,给定一个无限表面的三角K穿刺(也就是说,没有顶点穿刺)和一组目标锥角小于π的穿刺满足Gauss-Bonnet不平等,存在一个双曲规规定的角度和支持一个圆的组合包装K .此外,如果K是非常对称的,然后我们可以识别潜在的黎曼曲面,表明它并不依赖于角度。特别地，这提供了一个三角剖分K和一个共形类X的例子，使得在共形类X中有无穷多个圆弧双曲结构，在K的组合中有一个圆填充。这与Kojima-Mizushima-Tan在封闭情况下的一个猜想形成鲜明对比。

引用次数: 0

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