Computational Geometry-Theory and Applications最新文献

{"title":"Guarding points on a terrain by watchtowers","authors":"Byeonguk Kang ,&nbsp;Junhyeok Choi ,&nbsp;Jeesun Han ,&nbsp;Hee-Kap Ahn","doi":"10.1016/j.comgeo.2025.102210","DOIUrl":"10.1016/j.comgeo.2025.102210","url":null,"abstract":"<div><div>We study the problem of guarding points on an <em>x</em>-monotone polygonal chain, called a terrain, using <em>k</em> watchtowers. A watchtower is a vertical segment whose bottom endpoint lies on the terrain. A point on the terrain is visible from a watchtower if the line segment connecting the point and the top endpoint of the watchtower does not cross the terrain. Given a sequence of point sites lying on a terrain, we aim to partition the sequence into <em>k</em> contiguous subsequences and place <em>k</em> watchtowers on the terrain such that every point site in a subsequence is visible from the same watchtower and the maximum length of the watchtowers is minimized. We present efficient algorithms for two variants of the problem.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"131 ","pages":"Article 102210"},"PeriodicalIF":0.4,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523131","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":"Piercing unit geodesic disks","authors":"Ahmad Biniaz ,&nbsp;Prosenjit Bose ,&nbsp;Thomas Shermer","doi":"10.1016/j.comgeo.2025.102209","DOIUrl":"10.1016/j.comgeo.2025.102209","url":null,"abstract":"<div><div>We prove that at most 3 points are always sufficient to pierce a set of <em>m</em> pairwise intersecting unit geodesic disks inside a simple polygon <em>P</em> with <em>n</em> vertices of which <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> are reflex. We provide an <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo></math></span>-time algorithm to compute these at most 3 piercing points. Our bound is tight since it is known that in certain cases 3 points are necessary.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"131 ","pages":"Article 102209"},"PeriodicalIF":0.4,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322380","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":"Engineering an algorithm for constructing low-stretch geometric graphs with near-greedy average degrees","authors":"FNU Shariful ,&nbsp;Justin Weathers ,&nbsp;Anirban Ghosh ,&nbsp;Giri Narasimhan","doi":"10.1016/j.comgeo.2025.102201","DOIUrl":"10.1016/j.comgeo.2025.102201","url":null,"abstract":"<div><div>We design and engineer <span>Fast-Sparse-Spanner</span>, a simple and practical (fast and memory-efficient) algorithm for constructing sparse low stretch factor geometric graphs on large pointsets in the plane. To our knowledge, this is the first practical algorithm to construct fast low stretch factor graphs on large pointsets with average degrees (hence, the number of edges) competitive with that of greedy spanners, the sparsest known class of Euclidean geometric spanners. Although theoretically not guaranteed to produce <em>t</em>-spanners, we always found in our rigorous experiments that <span>Fast-Sparse-Spanner</span> generated near-greedy size <em>t</em>-spanners.</div><div>To evaluate our implementation in terms of computation speed, memory usage, and quality of output, we performed extensive experiments with synthetic and real-world pointsets, and by comparing it to our closest competitor <span>Bucketing</span>, the fastest known greedy spanner algorithm for pointsets in the plane, devised by Alewijnse et al. (2017) <span><span>[5]</span></span>. Our experiment with constructing a 1.1-spanner on a large synthetic pointset with 128<em>K</em> points uniformly distributed within a square shows more than a 41-fold speedup with roughly a third of the memory usage of that of <span>Bucketing</span>, but with only a 3% increase in the average degree of the resulting graph. When ran on a pointset with a million points drawn from the same distribution, we observed a 130-fold speedup, with roughly a fourth of the memory usage of that of <span>Bucketing</span>, and just a 6% increase in the average degree. In terms of diameter, the graphs generated by <span>Fast-Sparse-Spanner</span> beat greedy spanners in most cases (have substantially lower diameter) while maintaining near-greedy average degree. Further, our algorithm can be easily parallelized to take advantage of parallel environments.</div><div>We share the implementations via <span>GitHub</span> for broader uses and future research.</div><div><strong>GitHub repository.</strong> <span><span>https://github.com/ghoshanirban/FSS</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102201"},"PeriodicalIF":0.4,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270838","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":"VHS: A package for homological simplification of voxelized plant root data for skeletonization","authors":"Erin W. Chambers ,&nbsp;Tao Ju ,&nbsp;David Letscher ,&nbsp;Hannah Schreiber ,&nbsp;Dan Zeng","doi":"10.1016/j.comgeo.2025.102198","DOIUrl":"10.1016/j.comgeo.2025.102198","url":null,"abstract":"<div><div>In this work, we present VHS (<strong>V</strong>oxelized <strong>H</strong>omological <strong>S</strong>implification), a C++ package whose purpose is to de-noise voxelized data and output a topologically accurate simplified shape. In contrast to previous work on voxelized homological simplification tools, our main goal is offering a better starting point for computing curve skeletons for shape analysis. This goal necessitates additional simplification beyond what other packages provide, although our approach extends and improves prior work on heuristic methods which compute approximate solutions for the homological simplification problem. Our tool is designed for and tested on voxelized plant roots, although it is potentially useful beyond this data set. While the homological simplification problem is NP-hard in general, our package is able to simplify almost all of the topological noise when used on data from plant root systems. Compared with existing simplification tools, our method strikes a better balance between topological simplicity and geometric accuracy, resulting in higher usability of the resulting skeletons. Our code is publicly available at <span><span>https://github.com/davidletscher/VHS/</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102198"},"PeriodicalIF":0.4,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144194543","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":"Realizable dimension of periodic frameworks","authors":"Ryoshun Oba,&nbsp;Shin-ichi Tanigawa","doi":"10.1016/j.comgeo.2025.102200","DOIUrl":"10.1016/j.comgeo.2025.102200","url":null,"abstract":"<div><div>Belk and Connelly introduced the realizable dimension <span><math><mi>rd</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a finite graph <em>G</em>, which is the minimum nonnegative integer <em>d</em> such that every framework <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> in any dimension admits a framework in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with the same edge lengths. They characterized finite graphs with realizable dimension at most 1, 2, or 3 in terms of forbidden minors. In this paper, we consider periodic frameworks and extend the notion to <span><math><mi>Z</mi></math></span>-symmetric graphs. We give a forbidden minor characterization of <span><math><mi>Z</mi></math></span>-symmetric graphs with realizable dimension at most 1 or 2, and show that the characterization can be checked in linear time when a graph is given as a quotient <span><math><mi>Z</mi></math></span>-labeled graph.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102200"},"PeriodicalIF":0.4,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144071810","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 the rectilinear crossing number of complete balanced multipartite graphs and balanced layered graphs","authors":"Ruy Fabila-Monroy ,&nbsp;Rosna Paul ,&nbsp;Jenifer Viafara-Chanchi ,&nbsp;Alexandra Weinberger","doi":"10.1016/j.comgeo.2025.102199","DOIUrl":"10.1016/j.comgeo.2025.102199","url":null,"abstract":"<div><div>A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments and its vertices are points in general position. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all rectilinear drawings of the graph. Let <span><math><mi>n</mi><mo>≥</mo><mi>r</mi></math></span> be positive integers. The graph <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, is the complete balanced <em>r</em>-partite graph on <em>n</em> vertices, in which every set of the partition has at least <span><math><mo>⌊</mo><mi>n</mi><mo>/</mo><mi>r</mi><mo>⌋</mo></math></span> vertices. The balanced layered graph, <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, is an <em>r</em>-partite graph on <em>n</em> vertices, where <em>n</em> is multiple of <em>r</em>. Every partition of <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> contains <span><math><mi>n</mi><mo>/</mo><mi>r</mi></math></span> vertices; for every <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>r</mi><mo>−</mo><mn>1</mn></math></span>, all the vertices in the <em>i</em>-th partition are adjacent to all the vertices in the <span><math><mo>(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-th partition, and these are the only edges of <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>. In this paper, we give upper bounds on the rectilinear crossing numbers of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102199"},"PeriodicalIF":0.4,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143904056","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 computing closest points for segments","authors":"Haitao Wang","doi":"10.1016/j.comgeo.2025.102196","DOIUrl":"10.1016/j.comgeo.2025.102196","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> segments in the plane, we consider the problem of computing for each segment of <em>S</em> its closest point in <em>P</em>. The previously best algorithm solves the problem in <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mi>log</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> time [Bespamyatnikh, 2003] and a lower bound (under a somewhat restricted model) <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span> has also been proved. In this paper, we present an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span> time algorithm, which matches the above lower bound. In addition, we also present data structures for solving the online version of the problem, i.e., given a query segment (or a line as a special case), find its closest point in <em>P</em>. Our new results improve the previous work.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102196"},"PeriodicalIF":0.4,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859372","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":"Growth rates of the number of empty triangles and simplices","authors":"Bhaswar B. Bhattacharya ,&nbsp;Sandip Das ,&nbsp;Sk Samim Islam ,&nbsp;Saumya Sen","doi":"10.1016/j.comgeo.2025.102197","DOIUrl":"10.1016/j.comgeo.2025.102197","url":null,"abstract":"<div><div>Given a set <em>P</em> of <em>n</em> points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, in general position, denote by <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo></math></span> the number of empty triangles with vertices in <em>P</em>. In this paper we investigate by how much <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo></math></span> changes if a point <em>x</em> is removed from <em>P</em>. By constructing a graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> based on the arrangement of the empty triangles incident on <em>x</em>, we transform this geometric problem to the problem of counting triangles in the graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span>. We study properties of the graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> and, in particular, show that it is diamond-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa–Szemerédi problem. We also derive similar bounds for the growth rate of the number of empty simplices for point sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"130 ","pages":"Article 102197"},"PeriodicalIF":0.4,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863742","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":"CGTA","authors":"Michael A. Bekos,&nbsp;Charis Papadopoulos","doi":"10.1016/j.comgeo.2025.102195","DOIUrl":"10.1016/j.comgeo.2025.102195","url":null,"abstract":"","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102195"},"PeriodicalIF":0.4,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143842584","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":"Revisiting the Fréchet distance between piecewise smooth curves","authors":"Jacobus Conradi ,&nbsp;Anne Driemel ,&nbsp;Benedikt Kolbe","doi":"10.1016/j.comgeo.2025.102194","DOIUrl":"10.1016/j.comgeo.2025.102194","url":null,"abstract":"<div><div>Since its introduction to computational geometry by Alt and Godau in 1992, the Fréchet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision problem for piecewise smooth curves that is both conceptually simpler and naturally extends to the first algorithm for the problem for piecewise smooth curves in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</div><div>We assume that the algorithm is given two continuous curves, each consisting of a sequence of <em>m</em>, resp. <em>n</em>, smooth pieces, where each piece belongs to a sufficiently well-behaved class of curves, such as the set of algebraic curves of bounded degree. We introduce a decomposition of the free space diagram into a controlled number of pieces that can be used to solve the decision problem similarly to the polygonal case, in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>n</mi><mo>)</mo></math></span> time, leading to a computation of the Fréchet distance that runs in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>m</mi><mi>n</mi><mo>)</mo><mo>)</mo></math></span> time.</div><div>Furthermore, we study approximation algorithms for piecewise smooth curves that are also <em>c</em>-packed for some fixed value <em>c</em>. We adapt the existing framework for polygonal curves that leads to a near-linear <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximation to the Fréchet distance to the setting of piecewise smooth curves.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102194"},"PeriodicalIF":0.4,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826153","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}