Computational Geometry-Theory and Applications最新文献

{"title":"Maximum inscribed and minimum enclosing tropical balls of tropical polytopes and applications to volume estimation and uniform sampling","authors":"David Barnhill ,&nbsp;Ruriko Yoshida ,&nbsp;Keiji Miura","doi":"10.1016/j.comgeo.2025.102163","DOIUrl":"10.1016/j.comgeo.2025.102163","url":null,"abstract":"<div><div>We consider a minimum enclosing and maximum inscribed tropical balls for any given tropical polytope over the tropical projective torus in terms of the tropical metric with the max-plus algebra. We show that we can obtain such tropical balls via linear programming. Then we apply minimum enclosing and maximum inscribed tropical balls of any given tropical polytope to estimate the volume of and sample uniformly from the tropical polytope.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"128 ","pages":"Article 102163"},"PeriodicalIF":0.4,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147717","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":"A geometric condition for uniqueness of Fréchet means of persistence diagrams","authors":"Yueqi Cao,&nbsp;Anthea Monod","doi":"10.1016/j.comgeo.2024.102162","DOIUrl":"10.1016/j.comgeo.2024.102162","url":null,"abstract":"<div><div>The Fréchet mean is an important statistical summary and measure of centrality of data; it has been defined and studied for persistent homology captured by persistence diagrams. However, the complicated geometry of the space of persistence diagrams implies that the Fréchet mean for a given set of persistence diagrams is not necessarily unique, which prohibits theoretical guarantees for empirical means with respect to population means. In this paper, we derive a variance expression for a set of persistence diagrams exhibiting a multi-matching between the persistence points known as a grouping. Moreover, we propose a condition for groupings, which we refer to as flatness; we prove that sets of persistence diagrams that exhibit flat groupings give rise to unique Fréchet means. We derive a finite sample convergence result for general groupings, which results in convergence for Fréchet means if the groupings are flat. We then interpret flat groupings in a recently-proposed general framework of Fréchet means in Alexandrov geometry. Finally, we show that for manifold-valued data, the persistence diagrams can be truncated to construct flat groupings.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"128 ","pages":"Article 102162"},"PeriodicalIF":0.4,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pattern formation for fat robots with lights","authors":"Rusul J. Alsaedi,&nbsp;Joachim Gudmundsson,&nbsp;André van Renssen","doi":"10.1016/j.comgeo.2024.102161","DOIUrl":"10.1016/j.comgeo.2024.102161","url":null,"abstract":"<div><div>Given a set of <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> unit disk robots in the Euclidean plane, we consider the Pattern Formation problem, i.e., the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. Recently, this problem was solved in the asynchonous model for fat robots that agree on at least one axis in the robots with lights model where each robot is equipped with an externally visible persistent light that can assume colors from a fixed set of colors <span><span>[1]</span></span>. In this work, we reduce the number of colors needed and remove the axis-agreement requirement in the fully synchronous model. In particular, we present an algorithm requiring 7 colors when scaling the target pattern is allowed and an 8-color algorithm if scaling is not allowed. Our algorithms run in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> rounds with probability at least <span><math><mn>1</mn><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>q</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"128 ","pages":"Article 102161"},"PeriodicalIF":0.4,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143147720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Infinite circle packings on surfaces with conical singularities","authors":"Philip L. Bowers ,&nbsp;Lorenzo Ruffoni","doi":"10.1016/j.comgeo.2024.102160","DOIUrl":"10.1016/j.comgeo.2024.102160","url":null,"abstract":"<div><div>We show that given an infinite triangulation <em>K</em> of a surface with punctures (i.e., with no vertices at the punctures) and a set of target cone angles smaller than <em>π</em> at the punctures that satisfy a Gauss-Bonnet inequality, there exists a hyperbolic metric that has the prescribed angles and supports a circle packing in the combinatorics of <em>K</em>. Moreover, if <em>K</em> is very symmetric, then we can identify the underlying Riemann surface and show that it does not depend on the angles. In particular, this provides examples of a triangulation <em>K</em> and a conformal class <em>X</em> such that there are infinitely many conical hyperbolic structures in the conformal class <em>X</em> with a circle packing in the combinatorics of <em>K</em>. This is in sharp contrast with a conjecture of Kojima-Mizushima-Tan in the closed case.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"127 ","pages":"Article 102160"},"PeriodicalIF":0.4,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165469","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":"Geometric and algorithmic solutions to the generalised alibi query","authors":"Arthur Jansen,&nbsp;Bart Kuijpers","doi":"10.1016/j.comgeo.2024.102159","DOIUrl":"10.1016/j.comgeo.2024.102159","url":null,"abstract":"<div><div>Space-time prisms provide a framework to model the uncertainty on the space-time points that a moving object may have visited between measured space-time locations, provided that a bound on the speed of the moving object is given. In this model, the <em>alibi query</em> asks whether two moving objects, given by their respective measured space-time locations and speed bound, may have met. An analytical solution to this problem was first given by Othman <span><span>[15]</span></span>. In this paper, we address the <em>generalised alibi query</em> that asks the same question for an arbitrary number <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> of moving objects. We provide several solutions (mainly via the spatial and temporal projection) to this query with varying time complexities. These algorithmic solutions rely on techniques from convex and semi-algebraic geometry. We also address variants of the generalised alibi query where the question is asked for a given spatial location or a given moment in time.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"127 ","pages":"Article 102159"},"PeriodicalIF":0.4,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165468","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":"Realizability of free spaces of curves","authors":"Hugo A. Akitaya ,&nbsp;Maike Buchin ,&nbsp;Majid Mirzanezhad ,&nbsp;Leonie Ryvkin ,&nbsp;Carola Wenk","doi":"10.1016/j.comgeo.2024.102151","DOIUrl":"10.1016/j.comgeo.2024.102151","url":null,"abstract":"<div><div>The free space diagram is a popular tool to compute the well-known Fréchet distance. As the Fréchet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often the question arises whether a certain pattern in the free space diagram is “<em>realizable</em>”, i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram?</div><div>Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>&gt;</mo><mn>2</mn></mrow></msup></math></span>, showing <span><math><mo>∃</mo><mi>R</mi></math></span>-hardness. We use this to show that for curves in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>≥</mo><mn>2</mn></mrow></msup></math></span> the realizability problem is <span><math><mo>∃</mo><mi>R</mi></math></span>-complete, both for continuous and discrete Fréchet distances. We prove that the continuous case in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> we show that the problem becomes solvable in polynomial time.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"127 ","pages":"Article 102151"},"PeriodicalIF":0.4,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748635","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":"Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces","authors":"Eunku Park,&nbsp;Antoine Vigneron","doi":"10.1016/j.comgeo.2024.102150","DOIUrl":"10.1016/j.comgeo.2024.102150","url":null,"abstract":"<div><div>We give an embedding of the Poincaré halfspace <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>D</mi></mrow></msup></math></span> into a discrete metric space based on a binary tiling of <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>D</mi></mrow></msup></math></span>, with additive distortion <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>D</mi><mo>)</mo></math></span>. It yields the following results. We show that any subset <em>P</em> of <em>n</em> points in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>D</mi></mrow></msup></math></span> can be embedded into a graph-metric with <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi></math></span> vertices and edges, and with additive distortion <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>D</mi><mo>)</mo></math></span>. We also show how to construct, for any <em>k</em>, an <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>log</mi><mo>⁡</mo><mi>D</mi><mo>)</mo></math></span>-purely additive spanner of <em>P</em> with <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi></math></span> Steiner vertices and <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi><mo>⋅</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> edges, where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the <em>k</em>th-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for <em>P</em> of size <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi></math></span>. It allows us to answer approximate near-neighbor queries in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mo>+</mo><mi>O</mi><mo>(</mo><mi>D</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time, with additive error <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>D</mi><mo>)</mo></math></span>. These constructions can be done in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span> time.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"126 ","pages":"Article 102150"},"PeriodicalIF":0.4,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652454","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":"Parameterized inapproximability of Morse matching","authors":"Ulrich Bauer ,&nbsp;Abhishek Rathod","doi":"10.1016/j.comgeo.2024.102148","DOIUrl":"10.1016/j.comgeo.2024.102148","url":null,"abstract":"<div><div>We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of <span>Min-Morse Matching</span> within a factor of <span><math><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>log</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo></mrow></msup><mo>⁡</mo><mi>n</mi></mrow></msup></math></span>. Our second result shows that <span>Min-Morse Matching</span> is <span><math><mi>W</mi><mo>[</mo><mi>P</mi><mo>]</mo></math></span>-hard with respect to the standard parameter. Next, we show that <span>Min-Morse Matching</span> with standard parameterization has no FPT approximation algorithm for <em>any</em> approximation factor <em>ρ</em>. The above hardness results are applicable to complexes of dimension ≥2.</div><div>On the positive side, we provide a factor <span><math><mi>O</mi><mo>(</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></mfrac><mo>)</mo></math></span> approximation algorithm for <span>Min-Morse Matching</span> on 2-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa–Farber model of random complexes.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"126 ","pages":"Article 102148"},"PeriodicalIF":0.4,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652500","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 orthogonal Grünbaum partition problem in dimension three","authors":"Gerardo L. Maldonado,&nbsp;Edgardo Roldán-Pensado","doi":"10.1016/j.comgeo.2024.102149","DOIUrl":"10.1016/j.comgeo.2024.102149","url":null,"abstract":"<div><div>Grünbaum's equipartition problem asked if for any measure <em>μ</em> on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> there are always <em>d</em> hyperplanes which divide <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> into <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup></math></span> <em>μ</em>-equal parts. This problem is known to have a positive answer for <span><math><mi>d</mi><mo>≤</mo><mn>3</mn></math></span> and a negative one for <span><math><mi>d</mi><mo>≥</mo><mn>5</mn></math></span>. A variant of this question is to require the hyperplanes to be mutually orthogonal. This variant is known to have a positive answer for <span><math><mi>d</mi><mo>≤</mo><mn>2</mn></math></span> and there is reason to expect it to have a negative answer for <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>. In this note we exhibit measures that prove this. Additionally, we describe an algorithm that checks if a set of 8<em>n</em> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> can be split evenly by 3 mutually orthogonal planes. To our surprise, it seems the probability that a random set of 8 points chosen uniformly and independently in the unit cube does not admit such a partition is less than 0.001.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"126 ","pages":"Article 102149"},"PeriodicalIF":0.4,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593413","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":"Computing Euclidean distance and maximum likelihood retraction maps for constrained optimization","authors":"Alexander Heaton ,&nbsp;Matthias Himmelmann","doi":"10.1016/j.comgeo.2024.102147","DOIUrl":"10.1016/j.comgeo.2024.102147","url":null,"abstract":"<div><div>Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence <em>retraction maps</em> are used to approximate the exponential map and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For implicitly-defined manifolds, suitable retraction maps are difficult to compute. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and prove convergence results.</div><div>We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. Replacing Euclidean distance with maximum likelihood results in a map which we prove is a retraction. In fact, we prove the retraction is second-order; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"126 ","pages":"Article 102147"},"PeriodicalIF":0.4,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432072","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}