Computational Geometry-Theory and Applications最新文献

{"title":"Discrete Fréchet distance for closed curves","authors":"Evgeniy Vodolazskiy","doi":"10.1016/j.comgeo.2022.101967","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101967","url":null,"abstract":"<div><p>The paper presents a discrete variation of the Fréchet distance between closed curves, which can be seen as an approximation of the continuous measure. A rather straightforward approach to compute the discrete Fréchet distance between two closed sequences of <em>m</em> and <em>n</em> points using binary search takes <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>m</mi><mi>n</mi><mo>)</mo></math></span> time. We present an algorithm that takes <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>n</mi><msup><mrow><mi>log</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>⁡</mo><mi>m</mi><mi>n</mi><mo>)</mo></math></span> time, where <span><math><msup><mrow><mi>log</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is the iterated logarithm.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"111 ","pages":"Article 101967"},"PeriodicalIF":0.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49810275","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":"Complexity results on untangling red-blue matchings","authors":"Arun Kumar Das ,&nbsp;Sandip Das ,&nbsp;Guilherme D. da Fonseca ,&nbsp;Yan Gerard ,&nbsp;Bastien Rivier","doi":"10.1016/j.comgeo.2022.101974","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101974","url":null,"abstract":"<div><p>Given a matching between <em>n</em> red points and <em>n</em> blue points by line segments in the plane, we consider the problem of obtaining a crossing-free matching through flip operations that replace two crossing segments by two non-crossing ones. We first show that (i) it is NP-hard to <em>α</em>-approximate the shortest flip sequence, for any constant <em>α</em><span>. Second, we show that when the red points are collinear, (ii) given a matching, a flip sequence of length at most </span><span><math><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></math></span> always exists, and (iii) the number of flips in any sequence never exceeds <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>6</mn></mrow></mfrac></math></span>. Finally, we present (iv) a lower bounding flip sequence with roughly <span><math><mn>1.5</mn><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span> flips, which shows that the <span><math><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></math></span> flips attained in the convex case are not the maximum, and (v) a convex matching from which any flip sequence has roughly <span><math><mn>1.5</mn><mspace></mspace><mi>n</mi></math></span> flips. The last four results, based on novel analyses, improve the constants of state-of-the-art bounds.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"111 ","pages":"Article 101974"},"PeriodicalIF":0.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49851372","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":"An algorithmic framework for the single source shortest path problem with applications to disk graphs","authors":"Katharina Klost","doi":"10.1016/j.comgeo.2022.101979","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101979","url":null,"abstract":"&lt;div&gt;&lt;p&gt;&lt;span&gt;Shortest path problems are among the fundamental problems in graph theory. It is folklore that the unweighted single source shortest path (&lt;/span&gt;&lt;span&gt;SSSP&lt;/span&gt;) problem in general graphs can be solved optimally with breadth first search (&lt;span&gt;BFS&lt;/span&gt;) in &lt;span&gt;&lt;math&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;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; time. In this paper, we develop an algorithmic framework that generalizes a batched &lt;span&gt;BFS&lt;/span&gt; approach to give efficient &lt;span&gt;SSSP&lt;/span&gt;&lt;span&gt; algorithms for several graph classes. The running time of these algorithms depends on the running time of three main ingredients. The first is a preprocessing step, to define a &lt;/span&gt;&lt;em&gt;shortcut graph&lt;/em&gt;&lt;span&gt; that maintains some distance information. Then during one run of the algorithm repeatably there are the steps of efficiently finding a set of candidate vertices adjacent in the shortcut graph to a given set of vertices and finally finding the subset of the candidate vertices that actually form an edge in the original graph.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;A &lt;em&gt;disk graph&lt;/em&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a graph that is defined on a set &lt;em&gt;S&lt;/em&gt; of point sites 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;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, where each site &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; has an associated radius &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. The vertex set of &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is &lt;em&gt;S&lt;/em&gt; and two sites &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;/math&gt;&lt;/span&gt; are connected by an edge &lt;em&gt;st&lt;/em&gt; in &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; if and only if the disks induced by &lt;em&gt;s&lt;/em&gt; and &lt;em&gt;t&lt;/em&gt;&lt;span&gt; intersect. These graphs are also called the intersection graph of disks. Our results are algorithms that use the framework to efficiently solve the &lt;/span&gt;&lt;span&gt;SSSP&lt;/span&gt; problem in intersection graphs. For disk graphs in the &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-metric, we can show that after &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;log&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;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt; preprocessing time we can solve the &lt;/span&gt;&lt;span&gt;SSSP&lt;/span&gt; problem in &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;log&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; time. This significantly improves the previous best bound of &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&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; &lt;span&gt;[1]&lt;/span&gt;, &lt;span&gt;[2]&lt;/span&gt;. In the case of intersection graphs of axis-parallel squares, we are even able to reduce the preprocessing time to an optimal &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;log&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;. As intersection grap","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"111 ","pages":"Article 101979"},"PeriodicalIF":0.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49810276","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":"Lions and contamination: Monotone clearings","authors":"Daniel Bertschinger,&nbsp;Meghana M. Reddy ,&nbsp;Enrico Mann","doi":"10.1016/j.comgeo.2022.101961","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101961","url":null,"abstract":"<div><p>We consider a special variant of a pursuit-evasion game called lions and contamination. In a graph whose vertices are originally contaminated, a set of lions walks around the graph and each lion clears the contamination from every vertex it visits. The contamination, however, simultaneously spreads to any adjacent vertex not occupied by a lion. We study the relationship between different types of clearings of graphs, such as clearings which do not allow recontamination, clearings where at most one lion moves at each time step and clearings where lions are forbidden to be stacked on the same vertex. We answer several questions raised by Adams et al. <span>[1]</span>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"110 ","pages":"Article 101961"},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49812089","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":"Rectangular partitions of a rectilinear polygon","authors":"Hwi Kim ,&nbsp;Jaegun Lee ,&nbsp;Hee-Kap Ahn","doi":"10.1016/j.comgeo.2022.101965","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101965","url":null,"abstract":"<div><p><span>We investigate the problem of partitioning a rectilinear polygon </span><em>P</em> with <em>n</em> vertices and no holes into rectangles using disjoint line segments drawn inside <em>P</em><span> under two optimality criteria. In the minimum ink partition, the total length of the line segments drawn inside </span><em>P</em> is minimized. We present an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>-time algorithm using <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> space that returns a minimum ink partition of <em>P</em>. In the thick partition, the minimum side length over all resulting rectangles is maximized. We present an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</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 using <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> space that returns a thick partition using line segments incident to vertices of <em>P</em>, and an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>6</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 using <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span> space that returns a thick partition using line segments incident to the boundary of <em>P</em>. We also show that if the input rectilinear polygon has holes, the corresponding decision problem for the thick partition problem using line segments incident to vertices of the polygon is NP-complete. We also present an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>-time 3-approximation algorithm for the minimum ink partition for a rectangle containing <em>m</em> point holes.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"110 ","pages":"Article 101965"},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49854411","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":"Rectangular Spiral Galaxies are still hard","authors":"Erik D. Demaine ,&nbsp;Maarten Löffler ,&nbsp;Christiane Schmidt","doi":"10.1016/j.comgeo.2022.101949","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101949","url":null,"abstract":"<div><p>Spiral Galaxies is a pencil-and-paper puzzle played on a grid of unit squares: given a set of points called <em>centers</em>, the goal is to partition the grid into polyominoes such that each polyomino contains exactly one center and is <span><math><msup><mrow><mn>180</mn></mrow><mrow><mo>∘</mo></mrow></msup></math></span> rotationally symmetric about its center. We show that this puzzle is NP-complete, ASP-complete, and #P-complete even if (a) all solutions to the puzzle have rectangles for polyominoes; or (b) the polyominoes are required to be rectangles and all solutions to the puzzle have just <span><math><mn>1</mn><mo>×</mo><mn>1</mn></math></span>, <span><math><mn>1</mn><mo>×</mo><mn>3</mn></math></span>, and <span><math><mn>3</mn><mo>×</mo><mn>1</mn></math></span> rectangles. The proof for the latter variant also implies NP/ASP/#P-completeness of finding a noncrossing perfect matching in distance-2 grid graphs where edges connect vertices of Euclidean distance 2. Moreover, we prove NP-completeness of the design problem of minimizing the number of centers such that there exists a set of galaxies that exactly cover a given shape.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"110 ","pages":"Article 101949"},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49812086","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":"Intersecting disks using two congruent disks","authors":"Byeonguk Kang ,&nbsp;Jongmin Choi ,&nbsp;Hee-Kap Ahn","doi":"10.1016/j.comgeo.2022.101966","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101966","url":null,"abstract":"<div><p>We consider the following Euclidean 2-center problem. Given <em>n</em> disks in the plane, find two smallest congruent disks such that every input disk intersects at least one of the two congruent disks. We present a deterministic algorithm for the problem that returns an optimal pair of congruent disks in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span><span> time. We also present a randomized algorithm with </span><span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> expected time. These results improve upon the previously best deterministic and randomized algorithms, making a step closer to the optimal algorithms for the problem. We show that the same algorithms also work for two variants of the problem, the 2-piercing problem and the restricted 2-cover problem on disks. We also consider the 2-center problem and its two variants on <em>n</em><span> convex polygons, each with </span><span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> vertices in the plane and present efficient algorithms for them.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"110 ","pages":"Article 101966"},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49854402","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":"Advice complexity of online non-crossing matching","authors":"Ali Mohammad Lavasani,&nbsp;Denis Pankratov","doi":"10.1016/j.comgeo.2022.101943","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101943","url":null,"abstract":"&lt;div&gt;&lt;p&gt;We study online matching in the Euclidean 2-dimensional plane with the non-crossing constraint. The offline version was introduced by Atallah in 1985 and the online version was introduced and studied more recently by Bose et al. The input to the monochromatic non-crossing matching (MNM) problem consists of a sequence of points. Upon arrival of a point, an algorithm can decide to match it with a previously unmatched point or leave it unmatched. The line segments corresponding to the edges in the matching should not cross each other, and the goal is to maximize the size of the matching. The decisions are irrevocable, and while an optimal offline solution always matches all the points, an online algorithm cannot match all the points in the worst case, unless it is given some additional information, i.e., advice. In the bichromatic version (BNM), blue points are given in advance and the same number of red points arrive online. The goal is to maximize the number of red points matched to blue points without creating any crossings.&lt;/p&gt;&lt;p&gt;We show that the advice complexity of solving BNM optimally on a circle (or, more generally, on inputs in a convex position) is tightly bounded by the logarithm of the &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mtext&gt;th&lt;/mtext&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; Catalan number from above and below. This result corrects the previous claim of Bose et al. that the advice complexity is &lt;span&gt;&lt;math&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;. At the heart of the result is a connection between the non-crossing constraint in online inputs and the 231-avoiding property of permutations of &lt;/span&gt;&lt;em&gt;n&lt;/em&gt; elements. We also show a lower bound of &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and an upper bound of 3&lt;em&gt;n&lt;/em&gt; on the advice complexity for MNM on a plane. This gives an exponential improvement over the previously best-known lower bound and an improvement in the constant of the leading term in the upper bound. In addition, we establish a lower bound of &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&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;α&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt; on the advice complexity for achieving competitive ratio &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;17&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for MNM on a circle where &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt; is the relative entropy between two Bernoulli random variables with parameters &lt;/span&gt;&lt;em&gt;p&lt;/em&gt; and &lt;em&gt;q&lt;/em&gt;&lt;span&gt;. Standard tools from advice complexity, such as partition trees and reductions from string guessing problems, do not seem to apply to MNM/BNM, so we have to design our lower bounds fr","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"110 ","pages":"Article 101943"},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49854410","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":"Time and space efficient collinearity indexing","authors":"Boris Aronov ,&nbsp;Esther Ezra ,&nbsp;Micha Sharir ,&nbsp;Guy Zigdon","doi":"10.1016/j.comgeo.2022.101963","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101963","url":null,"abstract":"<div><p>The <span><em>collinearity</em><em> testing</em></span><span> problem is a basic problem in computational geometry, in which, given three sets </span><em>A</em>, <em>B</em>, <em>C</em> in the plane, of <em>n</em><span> points each, the task is to detect a collinear triple of points in </span><span><math><mi>A</mi><mo>×</mo><mi>B</mi><mo>×</mo><mi>C</mi></math></span> or report there is no such triple. In this paper we consider a preprocessing variant of this question, namely, the <em>collinearity indexing</em> problem, in which we are given two sets <em>A</em> and <em>B</em>, each of <em>n</em> points in the plane, and our goal is to preprocess <em>A</em> and <em>B</em><span> into a data structure, so that, for any query point </span><span><math><mi>q</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, we can determine whether <em>q</em> is collinear with a pair of points <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>∈</mo><mi>A</mi><mo>×</mo><mi>B</mi></math></span>. We provide a solution to the problem for the case where the points of <em>A</em>, <em>B</em> lie on an integer grid, and the query points lie on a vertical line, with a data structure of subquadratic storage and sublinear query time. We then extend our result to the case where the query points lie on the graph of a polynomial of constant degree. Our solution is based on the function-inversion technique of Fiat and Naor <span>[11]</span>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"110 ","pages":"Article 101963"},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49812088","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":"An optimal algorithm for L1 shortest paths in unit-disk graphs","authors":"Haitao Wang,&nbsp;Yiming Zhao","doi":"10.1016/j.comgeo.2022.101960","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101960","url":null,"abstract":"<div><p>A unit-disk graph <span><math><mi>G</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> of a set <em>P</em> of points in the plane is a graph with <em>P</em> as its vertex set such that two points of <em>P</em> are connected by an edge if the distance between the two points is at most 1 and the weight of the edge is equal to the distance of the two points. Given <em>P</em> and a source point <span><math><mi>s</mi><mo>∈</mo><mi>P</mi></math></span>, we consider the problem of finding shortest paths in <span><math><mi>G</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> from <em>s</em> to all other vertices of <span><math><mi>G</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span>. In the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> case where the distance is measured by the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> metric, the problem has been extensively studied and the current best algorithm runs in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time, with <span><math><mi>n</mi><mo>=</mo><mo>|</mo><mi>P</mi><mo>|</mo></math></span>. In this paper, we study the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> case in which the distance is measured under the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> metric (and each disk becomes a diamond); we present an <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time algorithm, which matches the <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>-time lower bound.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"110 ","pages":"Article 101960"},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49812090","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}