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Algorithms for radius-optimally augmenting trees in a metric space 度量空间中半径最优扩充树的算法
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-10-01 DOI: 10.1016/j.comgeo.2023.102018
Joachim Gudmundsson , Yuan Sha

Let T be a tree with n vertices in a metric space. We consider the problem of adding one shortcut edge to T to minimize the radius of the resulting graph.

For the continuous version of the problem where a center may be a point in the interior of an edge of the graph we give a linear time algorithm. In the case when the center is restricted to lie on a vertex, the discrete version, we give an O(nlogn) expected time algorithm.

Previously linear-time algorithms were known for the special case when the input graph is a path.

设T是一个在度量空间中有n个顶点的树。我们考虑向T添加一条快捷边以最小化生成图的半径的问题。对于问题的连续版本,其中中心可能是图边缘内部的一个点,我们给出了一个线性时间算法。在中心被限制在一个顶点上的情况下,离散形式,我们给出了一个O(nlog⁡n) 预期时间算法。以前的线性时间算法对于输入图是路径的特殊情况是已知的。
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引用次数: 0
The constant of point–line incidence constructions 点-线关联构造的常数
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-10-01 DOI: 10.1016/j.comgeo.2023.102009
Martin Balko , Adam Sheffer , Ruiwen Tang

We study a lower bound for the constant of the Szemerédi–Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies I(P,L)(c+o(1))|P|2/3|L|2/3, with c1.27. Our technique is based on studying a variety of properties of Euler's totient function. We also improve the current best constant for Elekes's construction from 1 to about 1.27. From an expository perspective, this is the first full analysis of the constant of Erdős's construction.

我们研究了Szemerédi–Trotter定理常数的一个下界。特别地,我们证明了最近的无穷一族点线配置满足I(P,L)≥(c+o(1))|P|2/3|L|2/3,其中c≈1.27。我们的技术是基于对欧拉瞬变函数的各种性质的研究。我们还将Elekes结构的当前最佳常数从1提高到约1.27。从阐释的角度来看,这是第一次全面分析埃尔德斯结构的常数。
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引用次数: 1
Simple linear time algorithms for piercing pairwise intersecting disks 穿透成对相交圆盘的简单线性时间算法
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-10-01 DOI: 10.1016/j.comgeo.2023.102011
Ahmad Biniaz , Prosenjit Bose , Yunkai Wang

A set D of disks in the plane is said to be pierced by a point set P if each disk in D contains a point of P. Any set of pairwise intersecting unit disks can be pierced by 3 points (Hadwiger and Debrunner (1955) [7]). Stachó and independently Danzer established that any set of pairwise intersecting arbitrary disks can be pierced by 4 points (Stachó (1981–1984) [16]. Danzer (1986) [4]). Existing linear-time algorithms for finding a set of 4 or 5 points that pierce pairwise intersecting disks of arbitrary radius use the LP-type problem as a subroutine. We present simple linear-time algorithms for finding 3 points for piercing pairwise intersecting unit disks, and 5 points for piercing pairwise intersecting disks of arbitrary radius. Our algorithms use simple geometric transformations and avoid heavy machinery. We also show that 3 points are sometimes necessary for piercing pairwise intersecting unit disks.

如果平面中的一组圆盘D包含P的一个点,则称平面中的圆盘D被点集P刺穿。任何一组成对相交的单位圆盘都可以被3个点刺穿(Hadwiger和Debrunner(1955)[7])。Stachó和Danzer独立地建立了任何一组成对相交的任意圆盘都可以被4个点刺穿(Stachó(1981–1984)[16]。Danzer(1986)[4])。现有的线性时间算法用于寻找穿透任意半径的成对相交圆盘的4或5个点的集合,使用LP型问题作为子程序。我们提出了简单的线性时间算法,用于寻找穿透成对相交单位圆盘的3个点,以及穿透任意半径的成对相交圆盘的5个点。我们的算法使用简单的几何变换,避免使用重型机械。我们还证明,有时需要3个点来穿透成对相交的单位圆盘。
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引用次数: 1
Cut locus realizations on convex polyhedra 凸多面体上切割轨迹的实现
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-10-01 DOI: 10.1016/j.comgeo.2023.102010
Joseph O'Rourke , Costin Vîlcu

We prove that every positively weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T edge weights matching C(x) edge lengths. If T has n leaves, P has (in general) n+1 vertices. We show there is in fact a continuum of polyhedra P each realizing T for some xP. Three main tools in the proof are properties of the star unfolding of P, Alexandrov's gluing theorem, and a new cut-locus partition lemma. The construction of P from T is surprisingly simple.

我们证明了每一个正加权树T都可以实现为凸多面体P上点x的切割轨迹C(x),其中T的边权重与C(x)的边长度相匹配。如果T有n个叶子,则P(通常)有n+1个顶点。我们证明了事实上存在一个多面体P的连续体,每个多面体P对一些x∈P实现T。证明中的三个主要工具是P的星展开性质、Alexandrov的粘合定理和一个新的割轨迹配分引理。从T构造P的过程非常简单。
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引用次数: 3
Multi-robot motion planning for unit discs with revolving areas 具有旋转区域的单元圆盘的多机器人运动规划
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-10-01 DOI: 10.1016/j.comgeo.2023.102019
Pankaj K. Agarwal , Tzvika Geft , Dan Halperin , Erin Taylor

We study the problem of motion planning for a collection of n labeled unit disc robots in a polygonal environment. We assume that the robots have revolving areas around their start and final positions: that each start and each final is contained in a radius 2 disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a weakly-monotone motion plan, in which robots move according to an ordering as follows: during the turn of a robot R in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As R passes through a revolving area, a robot R that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time (1+ε)-approximation algorithm.

On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an O(1) factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time O(lognloglogn)-approximation algorithm for this problem.

我们研究了多边形环境中n个标记单元圆盘机器人的运动规划问题。我们假设机器人在其起始位置和最终位置周围有旋转区域:每个起始位置和每个最终位置都包含在自由空间中的半径为2的圆盘中,不一定与起始位置或最终位置同心,该圆盘与其他起始位置或终末位置无关。这一假设允许一个弱单调运动计划,其中机器人根据如下顺序移动:在机器人R的顺序中,它从开始位置完全移动到最终位置,而其他机器人不会离开它们的旋转区域。当R穿过旋转区域时,位于该区域内的机器人R′可以在旋转区域内移动以避免碰撞。尽管存在运动计划,但我们证明,在这种设置下,特别是当运动计划被限制为弱单调时,最小化总行进距离是APX困难的,排除了任何多项式时间(1+ε)近似算法。在积极的方面,我们提出了计算可行的弱单调运动计划的第一个常因子近似算法。机器人行进的总距离在最优运动计划的O(1)因子内,该最优运动计划不必是弱单调的。我们的算法扩展到在线设置,其中多边形环境是固定的,但机器人的初始和最终位置是以在线方式指定的。最后,我们观察到,在编辑路径以避免机器人与机器人碰撞时,我们添加的总成本开销可能会因我们选择的顺序而发生显著变化。已知在这方面找到最佳排序是NP困难的,并且我们提供了多项式时间O(log⁡nlog⁡日志⁡n) -这个问题的近似算法。
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引用次数: 0
On reverse shortest paths in geometric proximity graphs 关于几何邻近图中的逆最短路径
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-09-11 DOI: 10.1016/j.comgeo.2023.102053
Pankaj K. Agarwal , Matthew J. Katz , Micha Sharir
<div><p>Let <em>S</em> be a set of <em>n</em><span> geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in </span><span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and let <span><math><mi>ϱ</mi><mo>:</mo><mi>S</mi><mo>×</mo><mi>S</mi><mo>→</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span> be a <em>distance function</em> on <em>S</em>. For a parameter <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, we define the <em>proximity graph</em> <span><math><mi>G</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>S</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> where <span><math><mi>E</mi><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>∈</mo><mi>S</mi><mo>×</mo><mi>S</mi><mo>|</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mspace></mspace><mi>ϱ</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>≤</mo><mi>r</mi><mo>}</mo></math></span>. Given <em>S</em>, <span><math><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>S</mi></math></span>, and an integer <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, the <em>reverse-shortest-path</em> (RSP) problem asks for computing the smallest value <span><math><msup><mrow><mi>r</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>≥</mo><mn>0</mn></math></span> such that <span><math><mi>G</mi><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span> contains a path from <em>s</em> to <em>t</em> of length at most <em>k</em>.</p><p>In this paper we present a general randomized technique that solves the RSP problem efficiently for a large family of geometric objects and distance functions. Using standard, and sometimes more involved, semi-algebraic range-searching techniques, we first give an efficient algorithm for the decision problem, namely, given a value <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, determine whether <span><math><mi>G</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> contains a path from <em>s</em> to <em>t</em> of length at most <em>k</em>. Next, we adapt our decision algorithm and combine it with a random-sampling method to compute <span><math><msup><mrow><mi>r</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, by efficiently performing a binary search over an implicit set of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> candidate ‘critical’ values that contains <span><math><msup><mrow><mi>r</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>.</p><p>We illustrate the versatility of our general technique by applying it to a variety of g
设S是R2中n个恒定复杂度的几何对象(例如,点、线段、圆盘、椭圆)的集合,并且设ϱ:S×S→R≥0是S上的距离函数。对于参数R≥0,我们定义了邻近图G(R)=(S,E),其中E={(e1,e2)∈S×S|e1≠e2,ϱ(e1、e2)≤R}。给定S,S,t∈S,且整数k≥1,反最短路径(RSP)问题要求计算最小值r≥0,使得G(r)包含从S到t的最大长度为k的路径。使用标准的,有时更复杂的半代数范围搜索技术,我们首先给出了决策问题的一个有效算法,即,给定值r≥0,确定G(r)是否包含从s到t的路径,长度至多为k。接下来,我们调整我们的决策算法,并将其与随机抽样方法相结合来计算r,通过在包含r的O(n2)个候选“临界”值的隐式集合上有效地执行二进制搜索。我们通过将其应用于各种几何邻近图来说明我们的通用技术的多功能性。例如,我们得到了(i)一个O(n4/3)期望时间随机化算法(其中O(·)隐藏了polylog(n)因子),其中S是R2中的一组(可能相交)线段,并且ϱ(e1,e2)=minx∈e1,y∈e2⁡‖x−y‖(其中‖是欧几里得距离),以及(ii)当S是位于具有n个顶点的x单调多边形链T上的m个点的集合时的O(n+m4/3)期望时间随机化算法,并且对于p,q∈S,ϱ(p,q)是最小值h,使得点p′:=p+(0,h)和q′:=q+(0、h)彼此可见,即。,线段p′q′上的所有点都位于多边形链T之上或之上。
{"title":"On reverse shortest paths in geometric proximity graphs","authors":"Pankaj K. Agarwal ,&nbsp;Matthew J. Katz ,&nbsp;Micha Sharir","doi":"10.1016/j.comgeo.2023.102053","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102053","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;em&gt;S&lt;/em&gt; be a set of &lt;em&gt;n&lt;/em&gt;&lt;span&gt; geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, and let &lt;span&gt;&lt;math&gt;&lt;mi&gt;ϱ&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be a &lt;em&gt;distance function&lt;/em&gt; on &lt;em&gt;S&lt;/em&gt;. For a parameter &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we define the &lt;em&gt;proximity graph&lt;/em&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;ϱ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Given &lt;em&gt;S&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and an integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, the &lt;em&gt;reverse-shortest-path&lt;/em&gt; (RSP) problem asks for computing the smallest value &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; contains a path from &lt;em&gt;s&lt;/em&gt; to &lt;em&gt;t&lt;/em&gt; of length at most &lt;em&gt;k&lt;/em&gt;.&lt;/p&gt;&lt;p&gt;In this paper we present a general randomized technique that solves the RSP problem efficiently for a large family of geometric objects and distance functions. Using standard, and sometimes more involved, semi-algebraic range-searching techniques, we first give an efficient algorithm for the decision problem, namely, given a value &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, determine whether &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; contains a path from &lt;em&gt;s&lt;/em&gt; to &lt;em&gt;t&lt;/em&gt; of length at most &lt;em&gt;k&lt;/em&gt;. Next, we adapt our decision algorithm and combine it with a random-sampling method to compute &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, by efficiently performing a binary search over an implicit set of &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; candidate ‘critical’ values that contains &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/p&gt;&lt;p&gt;We illustrate the versatility of our general technique by applying it to a variety of g","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"117 ","pages":"Article 102053"},"PeriodicalIF":0.6,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49799333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Clustering with faulty centers 具有故障中心的聚类
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-08-21 DOI: 10.1016/j.comgeo.2023.102052
Emily Fox , Hongyao Huang , Benjamin Raichel

In this paper we introduce and formally study the problem of k-clustering with faulty centers. Specifically, we study the faulty versions of k-center, k-median, and k-means clustering, where centers have some probability of not existing, as opposed to prior work where clients had some probability of not existing. For all three problems we provide fixed parameter tractable algorithms, in the parameters k, d, and ε, that (1+ε)-approximate the minimum expected cost solutions for points in d dimensional Euclidean space. For Faulty k-center we additionally provide a 5-approximation for general metrics. Significantly, all of our algorithms have only a linear dependence on n.

在本文中,我们引入并正式研究了具有故障中心的k-聚类问题。具体来说,我们研究了k-中心、k-中值和k-均值聚类的错误版本,其中中心有一定的不存在概率,而之前的工作中客户端有一定的可能性不存在。对于这三个问题,我们提供了固定参数的可处理算法,在参数k、d和ε中,(1+ε)-近似于d维欧氏空间中点的最小期望成本解。对于故障k中心,我们还提供了一般度量的5近似值。值得注意的是,我们所有的算法对n只有线性依赖关系。
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引用次数: 0
On algorithmic complexity of imprecise spanners 不精确扳手的算法复杂度
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-08-16 DOI: 10.1016/j.comgeo.2023.102051
Abolfazl Poureidi , Mohammad Farshi

Let t>1 be a real number. A geometric t-spanner is a geometric graph for a point set in Rd with straight line segments between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most t.

An imprecise point set is modeled by a set R of regions in Rd. If one chooses a point inside each region of R, then the resulting point set is called a precise instance from R. An imprecise t-spanner for an imprecise point set R is a graph G=(R,E) such that for each precise instance S from R, graph GS=(S,ES), where ES is the set of edges corresponding to E and S, is a t-spanner.

In this paper, we show an imprecise point set R of n straight-line segments in the plane such that any imprecise t-spanner for R has Ω(n2) edges. Then, we give an algorithm that computes an imprecise t-spanner for a set of n pairwise disjoint d-dimensional balls with arbitrary sizes. This imprecise t-spanner has O(n/(t1)d) edges and can be computed in O(nlogn/(t1)d) time. Finally, we show that given an imprecise spanner, finding a precise instance such that its corresponding precise spanner has minimum dilation between all possible precise instances of the imprecise spanner is NP-hard, no matter if crossing edges are allowed or not.

设t>1为实数。几何t形钳是Rd中点集的几何图,顶点之间有直线段,图中每对顶点之间的最短路径距离(具有欧几里得边长度)与实际欧几里得距离的比率不超过。一个不精确的点集由Rd中的区域集R来建模。如果在R的每个区域内选择一个点,对于不精确点集R的不精确t形扳手是一个图G=(R,E),使得对于来自R的每个精确实例S,图GS=(S,ES),其中ES是对应于E和S的边的集合,是一个t形扳手。本文给出了平面上n个直线段的不精确点集R,使得任意R的不精确t形扳手都有Ω(n2)条边。然后,我们给出了一种算法,用于计算任意大小的n对不相交的d维球的不精确t形扳手。这个不精确的t形扳手有O(n/(t−1)d)条边,可以在O(nlog n/(t−1)d)时间内计算出来。最后,我们证明了给定一个不精确扳手,无论是否允许交叉边,找到一个精确实例,使其对应的精确扳手在所有可能的不精确扳手的精确实例之间具有最小的膨胀是np困难的。
{"title":"On algorithmic complexity of imprecise spanners","authors":"Abolfazl Poureidi ,&nbsp;Mohammad Farshi","doi":"10.1016/j.comgeo.2023.102051","DOIUrl":"10.1016/j.comgeo.2023.102051","url":null,"abstract":"<div><p>Let <span><math><mi>t</mi><mo>&gt;</mo><mn>1</mn></math></span> be a real number. A geometric <em>t</em><span>-spanner is a geometric graph for a point set in </span><span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span><span> with straight line segments between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most </span><em>t</em>.</p><p>An imprecise point set is modeled by a set <em>R</em> of regions in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. If one chooses a point inside each region of <em>R</em>, then the resulting point set is called a precise instance from <em>R</em>. An imprecise <em>t</em>-spanner for an imprecise point set <em>R</em> is a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>R</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> such that for each precise instance <em>S</em> from <em>R</em>, graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>=</mo><mo>(</mo><mi>S</mi><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span> is the set of edges corresponding to <em>E</em> and <em>S</em>, is a <em>t</em>-spanner.</p><p>In this paper, we show an imprecise point set <em>R</em> of <em>n</em> straight-line segments in the plane such that any imprecise <em>t</em>-spanner for <em>R</em> has <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> edges. Then, we give an algorithm that computes an imprecise <em>t</em>-spanner for a set of <em>n</em><span> pairwise disjoint </span><em>d</em>-dimensional balls with arbitrary sizes. This imprecise <em>t</em>-spanner has <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>/</mo><msup><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> edges and can be computed in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>/</mo><msup><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> time. Finally, we show that given an imprecise spanner, finding a precise instance such that its corresponding precise spanner has minimum dilation between all possible precise instances of the imprecise spanner is NP-hard, no matter if crossing edges are allowed or not.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"117 ","pages":"Article 102051"},"PeriodicalIF":0.6,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47292577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Augmenting graphs to minimize the radius 扩充图形以最小化半径
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-08-01 DOI: 10.1016/j.comgeo.2023.101996
Joachim Gudmundsson , Yuan Sha

We study the problem of augmenting a metric graph by adding k edges while minimizing the radius of the augmented graph. We give a simple 3-approximation algorithm and show that there is no polynomial-time (5/3ϵ)-approximation algorithm, for any ϵ>0, unless P=NP.

We also give two exact algorithms for the special case when the input graph is a tree, one of which is generalized to handle metric graphs with bounded treewidth.

我们研究了通过增加k条边来扩充度量图的问题,同时最小化扩充图的半径。我们给出了一个简单的3-近似算法,并证明对于任何一个ε>;0,除非P=NP。对于输入图为树的特殊情况,我们还给出了两个精确的算法,其中一个算法被推广到处理具有有界树宽的度量图。
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引用次数: 0
Keep your distance: Land division with separation 保持距离:土地分割与分离
IF 0.6 4区 计算机科学 Q4 MATHEMATICS Pub Date : 2023-08-01 DOI: 10.1016/j.comgeo.2023.102006
Edith Elkind , Erel Segal-Halevi , Warut Suksompong

This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of proportionality is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the ordinal maximin share approximation, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions.

本文是一项正在进行的努力的一部分,通过处理现实应用中的需求,使公平分配理论更接近实践。我们关注两项源自土地产业划分的要求:(1)每个代理人应获得一块可用几何形状的地块,以及(2)不同代理人的地块必须物理分离。有了这些要求,比例的经典公平概念是不切实际的,因为它可能不可能实现任何乘法近似。相比之下,Budish在2011年引入的序数最大化份额近似提供了有意义的公平保证。当可用形状是正方形、胖矩形或任意轴对齐矩形时,我们证明了可实现的最大共享保证的上界和下界,并探讨了在这种设置下寻找公平分区的算法和查询复杂性。我们的工作利用了计算几何中的工具和概念,如独立的矩形集和剪切分区。
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引用次数: 10
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Computational Geometry-Theory and Applications
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