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Discrete and Computational Geometry最新文献

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The Complexity of the Hausdorff Distance 豪斯多夫距离的复杂性
Discrete and Computational Geometry Pub Date : 2023-09-27 DOI: 10.1007/s00454-023-00562-5
Paul Jungeblut, Linda Kleist, Tillmann Miltzow
{"title":"The Complexity of the Hausdorff Distance","authors":"Paul Jungeblut, Linda Kleist, Tillmann Miltzow","doi":"10.1007/s00454-023-00562-5","DOIUrl":"https://doi.org/10.1007/s00454-023-00562-5","url":null,"abstract":"Abstract We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class $${ forall exists _{<}mathbb {R}} $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>∀</mml:mo> <mml:msub> <mml:mo>∃</mml:mo> <mml:mo><</mml:mo> </mml:msub> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . This implies that the problem is -, -, $$exists mathbb {R} $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>∃</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> -, and $$forall mathbb {R} $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>∀</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> -hard.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135535321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Geometry of Random Tournaments 随机比赛的几何学
Discrete and Computational Geometry Pub Date : 2023-09-21 DOI: 10.1007/s00454-023-00571-4
Mario Sanchez, Brett Kolesnik
{"title":"The Geometry of Random Tournaments","authors":"Mario Sanchez, Brett Kolesnik","doi":"10.1007/s00454-023-00571-4","DOIUrl":"https://doi.org/10.1007/s00454-023-00571-4","url":null,"abstract":"Abstract A tournament is an orientation of a graph. Each edge is a match, directed towards the winner. The score sequence lists the number of wins by each team. In this article, by interpreting score sequences geometrically, we generalize and extend classical theorems of Landau (Bull. Math. Biophys. 15 , 143–148 (1953)) and Moon (Pac. J. Math. 13 , 1343–1345 (1963)), via the theory of zonotopal tilings.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136102132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Distributing Persistent Homology via Spectral Sequences 利用谱序列分布持久同源性
Discrete and Computational Geometry Pub Date : 2023-09-20 DOI: 10.1007/s00454-023-00549-2
Álvaro Torras-Casas
{"title":"Distributing Persistent Homology via Spectral Sequences","authors":"Álvaro Torras-Casas","doi":"10.1007/s00454-023-00549-2","DOIUrl":"https://doi.org/10.1007/s00454-023-00549-2","url":null,"abstract":"We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of using them. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self contained presentation of the Mayer-Vietoris spectral sequence. Then we study the Persistent Mayer-Vietoris spectral sequence and present a solution to the extension problem. Finally, we review PerMaViss, a method implementing these ideas. This procedure distributes simplicial data, while focusing on merging homological information.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136307117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
Covering the Edges of a Complete Geometric Graph with Convex Polygons 用凸多边形覆盖完整几何图的边缘
Discrete and Computational Geometry Pub Date : 2023-09-16 DOI: 10.1007/s00454-023-00548-3
Rom Pinchasi, Oren Yerushalmi
{"title":"Covering the Edges of a Complete Geometric Graph with Convex Polygons","authors":"Rom Pinchasi, Oren Yerushalmi","doi":"10.1007/s00454-023-00548-3","DOIUrl":"https://doi.org/10.1007/s00454-023-00548-3","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135305810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Labelings vs. Embeddings: On Distributed and Prioritized Representations of Distances 标签与嵌入:关于距离的分布和优先表示
Discrete and Computational Geometry Pub Date : 2023-09-15 DOI: 10.1007/s00454-023-00565-2
Arnold Filtser, Lee-Ad Gottlieb, Robert Krauthgamer
{"title":"Labelings vs. Embeddings: On Distributed and Prioritized Representations of Distances","authors":"Arnold Filtser, Lee-Ad Gottlieb, Robert Krauthgamer","doi":"10.1007/s00454-023-00565-2","DOIUrl":"https://doi.org/10.1007/s00454-023-00565-2","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135395710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Discrete Isothermic Nets Based on Checkerboard Patterns 基于棋盘图案的离散等温网
Discrete and Computational Geometry Pub Date : 2023-09-14 DOI: 10.1007/s00454-023-00558-1
Felix Dellinger
{"title":"Discrete Isothermic Nets Based on Checkerboard Patterns","authors":"Felix Dellinger","doi":"10.1007/s00454-023-00558-1","DOIUrl":"https://doi.org/10.1007/s00454-023-00558-1","url":null,"abstract":"Abstract This paper studies the discrete differential geometry of the checkerboard pattern inscribed in a quadrilateral net by connecting edge midpoints. It turns out to be a versatile tool which allows us to consistently define principal nets, Koenigs nets and eventually isothermic nets as a combination of both. Principal nets are based on the notions of orthogonality and conjugacy and can be identified with sphere congruences that are entities of Möbius geometry. Discrete Koenigs nets are defined via the existence of the so-called conic of Koenigs. We find several interesting properties of Koenigs nets, including their being dualizable and having equal Laplace invariants. Isothermic nets can be defined as Koenigs nets that are also principal nets. We prove that the class of isothermic nets is invariant under both dualization and Möbius transformations. Among other things, this allows a natural construction of discrete minimal surfaces and their Goursat transformations.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134910978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Width Deviation of Convex Polygons 凸多边形的宽度偏差
Discrete and Computational Geometry Pub Date : 2023-09-13 DOI: 10.1007/s00454-023-00545-6
Shigeki Akiyama, Teturo Kamae
{"title":"Width Deviation of Convex Polygons","authors":"Shigeki Akiyama, Teturo Kamae","doi":"10.1007/s00454-023-00545-6","DOIUrl":"https://doi.org/10.1007/s00454-023-00545-6","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135734522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Improved Routing on the Delaunay Triangulation 改进了Delaunay三角定位的路由
Discrete and Computational Geometry Pub Date : 2023-09-07 DOI: 10.1007/s00454-023-00499-9
Nicolas Bonichon, Prosenjit Bose, Jean-Lou De Carufel, Vincent Despré, Darryl Hill, Michiel Smid
{"title":"Improved Routing on the Delaunay Triangulation","authors":"Nicolas Bonichon, Prosenjit Bose, Jean-Lou De Carufel, Vincent Despré, Darryl Hill, Michiel Smid","doi":"10.1007/s00454-023-00499-9","DOIUrl":"https://doi.org/10.1007/s00454-023-00499-9","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135048140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane 论高阶布里渊镶嵌及平面上相关平铺的角度
Discrete and Computational Geometry Pub Date : 2023-09-07 DOI: 10.1007/s00454-023-00566-1
Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian
{"title":"On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane","authors":"Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian","doi":"10.1007/s00454-023-00566-1","DOIUrl":"https://doi.org/10.1007/s00454-023-00566-1","url":null,"abstract":"Abstract For a locally finite set in $${{{mathbb {R}}}}^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , the order- k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k . As an example, a stationary Poisson point process in $${{{mathbb {R}}}}^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6 , 85–127 (1970)).","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135048286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Recovery of Plane Curves from Branch Points 从分支点恢复平面曲线
Discrete and Computational Geometry Pub Date : 2023-09-07 DOI: 10.1007/s00454-023-00538-5
Daniele Agostini, Hannah Markwig, Clemens Nollau, Victoria Schleis, Javier Sendra-Arranz, Bernd Sturmfels
{"title":"Recovery of Plane Curves from Branch Points","authors":"Daniele Agostini, Hannah Markwig, Clemens Nollau, Victoria Schleis, Javier Sendra-Arranz, Bernd Sturmfels","doi":"10.1007/s00454-023-00538-5","DOIUrl":"https://doi.org/10.1007/s00454-023-00538-5","url":null,"abstract":"Abstract We recover plane curves from their branch points under projection onto a line. Our focus lies on cubics and quartics. These have 6 and 12 branch points respectively. The plane Hurwitz numbers 40 and 120 count the orbits of solutions. We determine the numbers of real solutions, and we present exact algorithms for recovery. Our approach relies on 150 years of beautiful algebraic geometry, from Clebsch to Vakil and beyond.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135048490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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