Computational Geometry-Theory and Applications最新文献

{"title":"On exact covering with unit disks","authors":"Ji Hoon Chun,&nbsp;Christian Kipp,&nbsp;Sandro Roch","doi":"10.1016/j.comgeo.2025.102193","DOIUrl":"10.1016/j.comgeo.2025.102193","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102193"},"PeriodicalIF":0.4,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximation algorithms for 1-Wasserstein distance between persistence diagrams","authors":"Samantha Chen,&nbsp;Yusu Wang","doi":"10.1016/j.comgeo.2025.102190","DOIUrl":"10.1016/j.comgeo.2025.102190","url":null,"abstract":"<div><div>Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations. A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in near-linear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102190"},"PeriodicalIF":0.4,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738783","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}

{"title":"Pattern formation for fat robots with memory","authors":"Rusul J. Alsaedi,&nbsp;Joachim Gudmundsson,&nbsp;André van Renssen","doi":"10.1016/j.comgeo.2025.102189","DOIUrl":"10.1016/j.comgeo.2025.102189","url":null,"abstract":"<div><div>Given a set of <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> 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 <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>-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 <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> rounds with probability at least <span><math><mn>1</mn><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>q</mi></mrow></msup></math></span>. The algorithms are collision-free and do not require the knowledge of the number of robots.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102189"},"PeriodicalIF":0.4,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parallel line centers with guaranteed separation","authors":"Chaeyoon Chung ,&nbsp;Taehoon Ahn ,&nbsp;Sang Won Bae ,&nbsp;Hee-Kap Ahn","doi":"10.1016/j.comgeo.2025.102185","DOIUrl":"10.1016/j.comgeo.2025.102185","url":null,"abstract":"<div><div>Given a set <em>P</em> of <em>n</em> points in the plane and an integer <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, the <em>k</em>-line-center problem asks <em>k</em> slabs whose union covers <em>P</em> that minimizes the maximum width of the <em>k</em> slabs. In this paper, we introduce a new variant of the <em>k</em>-line-center problem for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, in which the resulting <em>k</em> 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 <em>k</em> parallel slabs, to be the minimum distance between any two slabs, divided by the width of the smallest slab enclosing the <em>k</em> slabs. We present efficient algorithms for the following problems: (1) Given a real <span><math><mn>0</mn><mo>&lt;</mo><mi>ρ</mi><mo>≤</mo><mn>1</mn></math></span>, compute <em>k</em> parallel slabs of minimum width that cover <em>P</em> with gap-ratio at least <em>ρ</em>. (2) Compute <em>k</em> parallel slabs that cover <em>P</em> with maximum possible gap-ratio. Our algorithms run in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup><mo>⋅</mo><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mi>k</mi><mi>n</mi><mo>)</mo><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msubsup><mo>⋅</mo><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mi>k</mi><mi>n</mi><mo>)</mo><mo>)</mo></math></span> time, respectively, using <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>k</mi><mo>)</mo></math></span> space, where <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow></msub></math></span> denotes the maximum possible gap-ratio of any <em>k</em> parallel slabs that cover <em>P</em>. Using linear space, the running times only slightly increase to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup><mo>⋅</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msubsup><mo>⋅</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102185"},"PeriodicalIF":0.4,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681459","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}

{"title":"On line-separable weighted unit-disk coverage and related problems","authors":"Gang Liu,&nbsp;Haitao Wang","doi":"10.1016/j.comgeo.2025.102188","DOIUrl":"10.1016/j.comgeo.2025.102188","url":null,"abstract":"<div><div>Given a set <em>P</em> of <em>n</em> points and a set <em>S</em> of <em>n</em> weighted disks in the plane, the disk coverage problem is to compute a subset of disks of smallest total weight such that the union of the disks in the subset covers all points of <em>P</em>. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of <em>P</em> by a line <em>ℓ</em>. We present an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time algorithm for the problem. This improves the previously best work of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time. Our result leads to an algorithm of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>7</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time for the halfplane coverage problem (i.e., using <em>n</em> weighted halfplanes to cover <em>n</em> points), an improvement over the previous <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time solution. If all halfplanes are lower ones, our algorithm runs in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time, while the previous best algorithm takes <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102188"},"PeriodicalIF":0.4,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681460","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}

{"title":"Decomposition of geometric graphs into star-forests","authors":"János Pach ,&nbsp;Morteza Saghafian ,&nbsp;Patrick Schnider","doi":"10.1016/j.comgeo.2025.102186","DOIUrl":"10.1016/j.comgeo.2025.102186","url":null,"abstract":"<div><div>We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on <em>n</em> vertices cannot be decomposed into fewer than <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102186"},"PeriodicalIF":0.4,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759060","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}

{"title":"Geometric TSP on sets","authors":"Henk Alkema,&nbsp;Mark de Berg","doi":"10.1016/j.comgeo.2025.102187","DOIUrl":"10.1016/j.comgeo.2025.102187","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In &lt;span&gt;One-of-a-Set TSP&lt;/span&gt;, also known as the &lt;span&gt;Generalised TSP&lt;/span&gt;, the input is a collection &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&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;P&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;mo&gt;.&lt;/mo&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;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of sets in a metric space and the goal is to compute a minimum-length tour that visits one element from each set.&lt;/div&gt;&lt;div&gt;In the Euclidean variant of this problem, each &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is a set of points in &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;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be a hypercube that contains &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, for &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;⩽&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;⩽&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We investigate how the complexity of &lt;span&gt;Euclidean One-of-a-Set TSP&lt;/span&gt; depends on &lt;em&gt;λ&lt;/em&gt;, the ply of the set &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&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;H&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;mo&gt;.&lt;/mo&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;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of hypercubes. (The ply is the smallest &lt;em&gt;λ&lt;/em&gt; such that every point in &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;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; is contained in at most &lt;em&gt;λ&lt;/em&gt; of the hypercubes). We show that the problem can be solved in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; time, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the total number of points, and that the problem cannot be solved in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; time when &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, unless the Exponential Time Hypothesis (ETH) fails.&lt;/div&gt;&lt;div&gt;In &lt;span&gt;Rectilinear One-of-a-Cube TSP&lt;/span&gt;, the input is a set &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of hypercubes in &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;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and the goal is to compute a minimum-length rectilinear tour that visits every hypercube. We show that the problem can be solved in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102187"},"PeriodicalIF":0.4,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Flips in odd matchings","authors":"Oswin Aichholzer ,&nbsp;Anna Brötzner ,&nbsp;Daniel Perz ,&nbsp;Patrick Schnider","doi":"10.1016/j.comgeo.2025.102184","DOIUrl":"10.1016/j.comgeo.2025.102184","url":null,"abstract":"<div><div>Let <span><math><mi>P</mi></math></span> be a set of <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></math></span> points in the plane in general position. We define the graph <span><math><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> whose vertex set is the set of all plane matchings on <span><math><mi>P</mi></math></span> with exactly <em>m</em> edges. Two vertices in <span><math><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> are connected if the two corresponding matchings have <span><math><mi>m</mi><mo>−</mo><mn>1</mn></math></span> edges in common. In this work we show that <span><math><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> is connected and give an upper bound of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> on its diameter. Moreover, we present a lower bound of <span><math><mi>n</mi><mo>−</mo><mn>2</mn></math></span> and an upper bound of <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> for the diameter of <span><math><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> for <span><math><mi>P</mi></math></span> in convex position.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102184"},"PeriodicalIF":0.4,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143637278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Connected matchings","authors":"Oswin Aichholzer ,&nbsp;Sergio Cabello ,&nbsp;Viola Mészáros ,&nbsp;Patrick Schnider ,&nbsp;Jan Soukup","doi":"10.1016/j.comgeo.2025.102174","DOIUrl":"10.1016/j.comgeo.2025.102174","url":null,"abstract":"<div><div>We show that each set of <span><math><mi>n</mi><mo>⩾</mo><mn>2</mn></math></span> points in the plane in general position has a straight-line matching with at least <span><math><mo>(</mo><mn>5</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>27</mn></math></span> edges whose segments form a connected set, and such a matching can be computed in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> 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 <span><math><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>⌉</mo></math></span> edges. We also consider a colored version, where each edge of the matching should connect points with different colors.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102174"},"PeriodicalIF":0.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143550937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Minimum-width double-slabs and widest empty slabs in high dimensions","authors":"Taehoon Ahn ,&nbsp;Chaeyoon Chung ,&nbsp;Hee-Kap Ahn ,&nbsp;Sang Won Bae ,&nbsp;Otfried Cheong ,&nbsp;Sang Duk Yoon","doi":"10.1016/j.comgeo.2025.102173","DOIUrl":"10.1016/j.comgeo.2025.102173","url":null,"abstract":"<div><div>A <em>slab</em> in <em>d</em>-dimensional space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is the set of points enclosed by two parallel hyperplanes. We consider the problem of finding an optimal pair of parallel slabs, called a <em>double-slab</em>, that covers a given set <em>P</em> of <em>n</em> points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We address two optimization problems in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for any fixed dimension <span><math><mi>d</mi><mo>⩾</mo><mn>3</mn></math></span>: the <em>minimum-width double-slab</em> problem, in which one wants to minimize the maximum width of the two slabs of the resulting double-slab, and the <em>widest empty slab</em> 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 <span><math><mi>d</mi><mo>⩾</mo><mn>3</mn></math></span> and the latter problem for <span><math><mi>d</mi><mo>⩾</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102173"},"PeriodicalIF":0.4,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143550936","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}