Computational Geometry-Theory and Applications最新文献

Realizable dimension of periodic frameworks 周期框架的可实现维度

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-04-29 DOI: 10.1016/j.comgeo.2025.102200

Ryoshun Oba, Shin-ichi Tanigawa

引用次数: 0

On the rectilinear crossing number of complete balanced multipartite graphs and balanced layered graphs 完全平衡多部图与平衡层图的直线交叉数

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-04-28 DOI: 10.1016/j.comgeo.2025.102199

Ruy Fabila-Monroy , Rosna Paul , Jenifer Viafara-Chanchi , Alexandra Weinberger

{"title":"On the rectilinear crossing number of complete balanced multipartite graphs and balanced layered graphs","authors":"Ruy Fabila-Monroy , Rosna Paul , Jenifer Viafara-Chanchi , 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}

引用次数: 0

Algorithms for computing closest points for segments 计算线段最近点的算法

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-04-17 DOI: 10.1016/j.comgeo.2025.102196

Haitao Wang

{"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}

引用次数: 0

Growth rates of the number of empty triangles and simplices 空三角形和单形的数量增长率

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-04-17 DOI: 10.1016/j.comgeo.2025.102197

Bhaswar B. Bhattacharya , Sandip Das , Sk Samim Islam , Saumya Sen

{"title":"Growth rates of the number of empty triangles and simplices","authors":"Bhaswar B. Bhattacharya , Sandip Das , Sk Samim Islam , 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}

引用次数: 0

CGTA CGTA

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-04-15 DOI: 10.1016/j.comgeo.2025.102195

Michael A. Bekos, Charis Papadopoulos

引用次数: 0

Revisiting the Fréchet distance between piecewise smooth curves 重新审视片断光滑曲线之间的弗雷谢特距离

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-04-09 DOI: 10.1016/j.comgeo.2025.102194

Jacobus Conradi , Anne Driemel , Benedikt Kolbe

{"title":"Revisiting the Fréchet distance between piecewise smooth curves","authors":"Jacobus Conradi , Anne Driemel , 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}

引用次数: 0

Bounds on the edge-length ratio of 2-outerplanar graphs 2-外平面图边长比的界

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-27 DOI: 10.1016/j.comgeo.2025.102192

Emilio Di Giacomo , Walter Didimo , Giuseppe Liotta , Henk Meijer , Fabrizio Montecchiani , Stephen Wismath

引用次数: 0

Constrained boundary labeling 约束边界标注

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-26 DOI: 10.1016/j.comgeo.2025.102191

Thomas Depian , Martin Nöllenburg , Soeren Terziadis , Markus Wallinger

{"title":"Constrained boundary labeling","authors":"Thomas Depian , Martin Nöllenburg , Soeren Terziadis , Markus Wallinger","doi":"10.1016/j.comgeo.2025.102191","DOIUrl":"10.1016/j.comgeo.2025.102191","url":null,"abstract":"<div><div>Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce <em>grouping</em> and <em>ordering constraints</em> in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is <span>NP</span>-hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is <span>NP</span>-complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102191"},"PeriodicalIF":0.4,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On exact covering with unit disks 用单位圆盘精确覆盖

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-26 DOI: 10.1016/j.comgeo.2025.102193

Ji Hoon Chun, Christian Kipp, Sandro Roch

引用次数: 0

Approximation algorithms for 1-Wasserstein distance between persistence diagrams 持久图之间1-Wasserstein距离的近似算法

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-25 DOI: 10.1016/j.comgeo.2025.102190

Samantha Chen, Yusu Wang

{"title":"Approximation algorithms for 1-Wasserstein distance between persistence diagrams","authors":"Samantha Chen, 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}

引用次数: 0