Computational Geometry-Theory and Applications最新文献

{"title":"Straight-line drawings of 1-planar graphs","authors":"Franz J. Brandenburg","doi":"10.1016/j.comgeo.2023.102036","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102036","url":null,"abstract":"<div><p>A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. However, there are 1-planar graphs that do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges such that edges of the same color do not cross. Thus 1-planar graphs have geometric thickness two. In addition, the drawing is nearly 1-planar, that is, it is 1-planar if all fan-crossed edges are removed. An edge is fan-crossed if it is crossed by edges with a common vertex if it is crossed more than twice. The drawing algorithm uses high precision arithmetic with numbers with <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> digits and computes the straight-line drawing from a 1-planar drawing in linear time on a real RAM.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"116 ","pages":"Article 102036"},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49846967","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":"Approximating Gromov-Hausdorff distance in Euclidean space","authors":"Sushovan Majhi ,&nbsp;Jeffrey Vitter ,&nbsp;Carola Wenk","doi":"10.1016/j.comgeo.2023.102034","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102034","url":null,"abstract":"<div><p>The Gromov-Hausdorff distance <span><math><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi><mi>H</mi></mrow></msub><mo>)</mo></math></span> proves to be a useful distance measure between shapes. In order to approximate <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi><mi>H</mi></mrow></msub></math></span> for <span><math><mi>X</mi><mo>,</mo><mi>Y</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, we look into its relationship with <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi><mi>s</mi><mi>o</mi></mrow></msub></math></span>, the infimum Hausdorff distance under Euclidean isometries. As already known for dimension <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi><mi>s</mi><mi>o</mi></mrow></msub></math></span> cannot be bounded above by a constant factor times <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi><mi>H</mi></mrow></msub></math></span>. For <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, however, we prove that <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>i</mi><mi>s</mi><mi>o</mi></mrow></msub><mo>≤</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi><mi>H</mi></mrow></msub></math></span>. We also show that the bound is tight. In effect, for <span><math><mi>X</mi><mo>,</mo><mi>Y</mi><mo>⊂</mo><mi>R</mi></math></span> with at most <em>n</em> points, this gives rise to 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 to approximate <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi><mi>H</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> with an approximation factor of <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></math></span>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"116 ","pages":"Article 102034"},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49804595","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":"Guest editorial: Special issue on the 33rd Canadian Conference on Computational Geometry (CCCG)","authors":"Meng He,&nbsp;Don Sheehy","doi":"10.1016/j.comgeo.2023.102035","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102035","url":null,"abstract":"","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"116 ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49804597","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":"From trees to barcodes and back again II: Combinatorial and probabilistic aspects of a topological inverse problem","authors":"Justin Curry,&nbsp;Jordan DeSha,&nbsp;Adélie Garin,&nbsp;Kathryn Hess,&nbsp;Lida Kanari,&nbsp;Brendan Mallery","doi":"10.1016/j.comgeo.2023.102031","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102031","url":null,"abstract":"<div><p>In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. We show how to lift this combinatorial characterization of barcodes to an analogous combinatorialization of merge trees. As result of this study, we provide the first clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees: generic phylogenetic trees on <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> leaf nodes fall into <span><math><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>!</mo><mo>!</mo></math></span> distinct equivalence classes, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>!</mo><mi>n</mi><mo>!</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup></math></span>. The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation of our distribution in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data, which opens the door to doing more precise statistics and science.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"116 ","pages":"Article 102031"},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49846968","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":"Density of triangulated ternary disc packings","authors":"Thomas Fernique ,&nbsp;Daria Pchelina","doi":"10.1016/j.comgeo.2023.102032","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102032","url":null,"abstract":"<div><p>We consider <em>ternary</em> disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each “hole” is bounded by three pairwise tangent discs are called <em>triangulated</em>. There are 164 pairs <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span>, <span><math><mn>1</mn><mo>&gt;</mo><mi>r</mi><mo>&gt;</mo><mi>s</mi></math></span>, allowing triangulated packings by discs of radii 1, <em>r</em> and <em>s</em>. In this paper, we enhance existing methods of dealing with maximal-density packings in order to find ternary triangulated packings which maximize the density among all the packings with the same disc radii. We showed for 16 pairs that the density is maximized by a triangulated ternary packing; for 16 other pairs, we proved the density to be maximized by a triangulated packing using only two sizes of discs; for 45 pairs, we found non-triangulated packings strictly denser than any triangulated one; finally, we classified the remaining cases where our methods are not applicable.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"115 ","pages":"Article 102032"},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49791317","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":"Range updates and range sum queries on multidimensional points with monoid weights","authors":"Shangqi Lu,&nbsp;Yufei Tao","doi":"10.1016/j.comgeo.2023.102030","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102030","url":null,"abstract":"<div><p>Let <em>P</em> be a set of <em>n</em> points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> where each point <span><math><mi>p</mi><mo>∈</mo><mi>P</mi></math></span> carries a <em>weight</em><span> drawn from a commutative monoid </span><span><math><mo>(</mo><mi>M</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. Given a <em>d</em>-rectangle <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>upd</mi></mrow></msub></math></span> (i.e., an orthogonal rectangle in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>) and a value <span><math><mi>Δ</mi><mo>∈</mo><mi>M</mi></math></span>, a <em>range update</em> adds Δ to the weight of every point <span><math><mi>p</mi><mo>∈</mo><mi>P</mi><mo>∩</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>upd</mi></mrow></msub></math></span>; given a <em>d</em>-rectangle <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>qry</mi></mrow></msub></math></span>, a <em>range sum query</em> returns the total weight of the points in <span><math><mi>P</mi><mo>∩</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>qry</mi></mrow></msub></math></span>. The goal is to store <em>P</em> in a structure to support updates and queries with attractive performance guarantees. We describe a structure of <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>n</mi><mo>)</mo></math></span> space that handles an update in <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>upd</mi></mrow></msub><mo>)</mo></math></span> time and a query in <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>qry</mi></mrow></msub><mo>)</mo></math></span> time for arbitrary functions <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>upd</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>qry</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> satisfying <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>upd</mi></mrow></msub><mo>⋅</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>qry</mi></mrow></msub><mo>=</mo><mi>n</mi></math></span>. The result holds for any fixed dimensionality <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span><span>. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture.</span></p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"115 ","pages":"Article 102030"},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49791615","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":"Colouring bottomless rectangles and arborescences","authors":"Jean Cardinal ,&nbsp;Kolja Knauer ,&nbsp;Piotr Micek ,&nbsp;Dömötör Pálvölgyi ,&nbsp;Torsten Ueckerdt ,&nbsp;Narmada Varadarajan","doi":"10.1016/j.comgeo.2023.102020","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102020","url":null,"abstract":"<div><p>We study problems related to colouring families of bottomless rectangles in the plane, in an attempt to improve the <em>polychromatic k-colouring number</em> <span><math><msubsup><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. This number is the smallest <em>m</em> such that any collection of bottomless rectangles can be <em>k</em>-coloured so that any <em>m</em>-fold covered point is covered by all <em>k</em> colours. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, or bottomless rectangles whose left corners lie on a line, <span><math><msubsup><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> is linear in <em>k</em>. We present the lower bound <span><math><msubsup><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>≥</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn></math></span> for general families.</p><p>We also investigate <em>semi-online</em> colouring algorithms, which need not colour each vertex immediately, but must maintain a proper colouring. We prove that for many sweeping orders, for any positive integers <span><math><mi>m</mi><mo>,</mo><mi>k</mi></math></span>, there is no semi-online algorithm that can <em>k</em>-colour bottomless rectangles presented in that order, so that any <em>m</em>-fold covered point is covered by at least two colours. This holds even for translates of quadrants, and is a corollary of a stronger result for arborescence colourings: Any semi-online colouring algorithm that colours an arborescence presented in post-order may produce arbitrarily long monochromatic paths.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"115 ","pages":"Article 102020"},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49791319","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":"Half-plane point retrieval queries with independent and dependent geometric uncertainties","authors":"Rivka Gitik,&nbsp;Leo Joskowicz","doi":"10.1016/j.comgeo.2023.102021","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102021","url":null,"abstract":"&lt;div&gt;&lt;p&gt;This paper addresses a family of geometric half-plane retrieval queries of points in the plane in the presence of geometric uncertainty. The problems include exact and uncertain point sets and half-plane queries defined by an exact or uncertain line whose location uncertainties are independent or dependent and are defined by &lt;em&gt;k&lt;/em&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt; real-valued parameters. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear &lt;/span&gt;approximation of geometric uncertainty that supports parametric dependencies between &lt;/span&gt;point locations. We present an efficient &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&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;/mrow&gt;&lt;/math&gt;&lt;/span&gt; time and space algorithm for computing the envelope of the LPGUM line that defines the half-plane query. For an exact line and an LPGUM &lt;em&gt;n&lt;/em&gt; points set, we present an &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; time query and &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; space algorithm, where &lt;em&gt;m&lt;/em&gt; is the number of LPGUM points on or above the half-plane line. For a LPGUM line and an exact points set, we present a &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&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;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;log&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;log&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;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; time and &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&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;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt; space approximation algorithm, where &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; is the desired approximation error. For a LPGUM line and an LPGUM points set, we present two &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&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;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&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;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; time query and &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;m","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"115 ","pages":"Article 102021"},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49791616","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":"Algorithms for radius-optimally augmenting trees in a metric space","authors":"Joachim Gudmundsson ,&nbsp;Yuan Sha","doi":"10.1016/j.comgeo.2023.102018","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102018","url":null,"abstract":"<div><p>Let <em>T</em> be a tree with <em>n</em> vertices in a metric space. We consider the problem of adding one shortcut edge to <em>T</em> to minimize the radius of the resulting graph.</p><p>For the <em>continuous</em> 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 <em>discrete</em> version, we give an <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> expected time algorithm.</p><p>Previously linear-time algorithms were known for the special case when the input graph is a path.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"114 ","pages":"Article 102018"},"PeriodicalIF":0.6,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49790333","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":"The constant of point–line incidence constructions","authors":"Martin Balko ,&nbsp;Adam Sheffer ,&nbsp;Ruiwen Tang","doi":"10.1016/j.comgeo.2023.102009","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102009","url":null,"abstract":"<div><p>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 <span><math><mi>I</mi><mo>(</mo><mi>P</mi><mo>,</mo><mi>L</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mi>c</mi><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mo>|</mo><mi>P</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>|</mo><mi>L</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span>, with <span><math><mi>c</mi><mo>≈</mo><mn>1.27</mn></math></span>. 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.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"114 ","pages":"Article 102009"},"PeriodicalIF":0.6,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49790332","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}