Computational Geometry-Theory and Applications最新文献

{"title":"Many order types on integer grids of polynomial size","authors":"Manfred Scheucher","doi":"10.1016/j.comgeo.2022.101924","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101924","url":null,"abstract":"<div><p>Two sets of labeled points <span><math><mo>{</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> and <span><math><mo>{</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> are of the same <span><em>labeled </em><em>order type</em></span> if, for every <span><math><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></math></span>, the triples <span><math><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> have the same orientation. In the 1980's, Goodman, Pollack, and Sturmfels showed that (i) the number of labeled order types on <em>n</em> points is of order <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn><mi>n</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span>, (ii) all order types can be realized with double-exponential integer coordinates, and that (iii) certain order types indeed require double-exponential integer coordinates. In 2018, Caraballo, Díaz-Báñez, Fabila-Monroy, Hidalgo-Toscano, Leaños, and Montejano showed that at least <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mi>n</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> labeled <em>n</em>-point order types can be realized on an integer grid of polynomial size. In this article, we improve their result by showing that at least <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn><mi>n</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> labeled <em>n</em>-point order types can be realized on an integer grid of polynomial size, which is asymptotically tight in the exponent. Finally we conclude that there are <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mi>n</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> order types in the <em>unlabeled</em> setting and that <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mi>n</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> of them can be realized on an integer grid of polynomial size.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49790783","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 discrete harmonic differential forms in a given cohomology class using finite element exterior calculus","authors":"Anil N. Hirani ,&nbsp;Kaushik Kalyanaraman ,&nbsp;Han Wang ,&nbsp;Seth Watts","doi":"10.1016/j.comgeo.2022.101937","DOIUrl":"10.1016/j.comgeo.2022.101937","url":null,"abstract":"<div><p><span><span>Computational topology research of the past two decades has emphasized combinatorial techniques while numerical methods such as numerical linear algebra remain underutilized. While the combinatorial techniques have been very successful in diverse areas, for some applications, it is worth considering the numerical counterparts. We discuss one such application. </span>Harmonic forms<span><span> are elements of the kernel of the Hodge Laplacian operator and contain information about the topology of the manifold. If a particular </span>cohomology class<span> is chosen, the closed differential form with the smallest norm in that class is a harmonic form. We use these well-known facts to give an algorithm for solving the following problem: given a piecewise flat manifold simplicial complex (with or without boundary) and a closed </span></span></span>piecewise polynomial<span><span><span> differential form representing a cohomology class, find the discrete harmonic form in that cohomology class. We give a least squares<span> based algorithm to solve this problem and show that the computed form satisfies the finite element exterior calculus (FEEC) equations for being a harmonic form. The piecewise polynomial spaces used are the spaces of trimmed </span></span>polynomial forms, that is, arbitrary </span>degree polynomial generalizations of Whitney forms which are used in FEEC. We also survey other methods for finding harmonic forms.</span></p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43300293","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":"Unfolding 3-separated polycube graphs of arbitrary genus","authors":"Mirela Damian ,&nbsp;Robin Flatland","doi":"10.1016/j.comgeo.2022.101944","DOIUrl":"10.1016/j.comgeo.2022.101944","url":null,"abstract":"<div><p>A <em>polycube graph</em><span> is a polyhedron composed of cubes glued together along whole faces, whose surface is a 2-manifold. A polycube graph is 3</span><em>-separated</em> if no two boxes of degree 3 or higher are adjacent, and no grid edge is entirely surrounded by boxes (i.e., there is no cycle of length 4). We show that every 3-separated polycube graph can be unfolded with a <span><math><mn>7</mn><mo>×</mo><mn>7</mn></math></span> refinement of the grid faces. This result extends the class of well-separated polycube graphs known to have an unfolding by allowing boxes of degree 2 to be adjacent to each other and to higher degree boxes.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41758132","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 pairwise intersecting geodesic disks by five points","authors":"A. Karim Abu-Affash ,&nbsp;Paz Carmi ,&nbsp;Meytal Maman","doi":"10.1016/j.comgeo.2022.101947","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101947","url":null,"abstract":"<div><p><span>Given a simple polygon </span><em>P</em> on <em>n</em> vertices and a set <span><math><mi>D</mi></math></span> of <em>m</em> pairwise intersecting geodesic disks in <em>P</em>, we show that five points in <em>P</em> are always sufficient to pierce all the disks in <span><math><mi>D</mi></math></span>. The points can be computed in <span><math><mi>O</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mi>log</mi><mo>⁡</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo></math></span> time, where <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> is the number of the reflex vertices of <em>P</em>. This improves the previous bound of 14, obtained by Bose, Carmi, and Shermer <span>[1]</span>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49790769","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":"Decomposing filtered chain complexes: Geometry behind barcoding algorithms","authors":"Wojciech Chachólski,&nbsp;Barbara Giunti,&nbsp;Alvin Jin,&nbsp;Claudia Landi","doi":"10.1016/j.comgeo.2022.101938","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101938","url":null,"abstract":"<div><p>In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called interval spheres. In this paper, we provide an algorithm to decompose filtered chain complexes into such interval spheres. This algorithm provides geometric insights into various aspects of the standard persistence algorithm and two of its runtime optimizations. Moreover, since it works for any filtered chain complexes, our algorithm can be applied in more general cases. As an application, we show how to decompose filtered kernels with it.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49790773","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":"New formulas for cup-i products and fast computation of Steenrod squares","authors":"Anibal M. Medina-Mardones","doi":"10.1016/j.comgeo.2022.101921","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101921","url":null,"abstract":"<div><p><span>Operations on the cohomology<span> of spaces are important tools enhancing the descriptive power of this computable invariant. For cohomology with mod 2 coefficients, Steenrod squares are the most significant of these operations. Their effective computation relies on formulas defining a cup-</span></span><em>i</em><span> construction, a structure on (co)chains which is important in its own right, having connections to lattice<span> field theory, convex geometry and higher category theory among others. In this article we present new formulas defining a cup-</span></span><em>i</em><span> construction, and use them to introduce a fast algorithm for the computation of Steenrod squares on the cohomology of finite simplicial complexes. In forthcoming work we use these formulas to axiomatically characterize the cup-</span><em>i</em> construction they define, showing additionally that all other formulas in the literature define the same cup-<em>i</em> construction up to isomorphism.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49831341","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 the Fréchet distance between uncertain curves in one dimension","authors":"Kevin Buchin ,&nbsp;Maarten Löffler ,&nbsp;Tim Ophelders ,&nbsp;Aleksandr Popov ,&nbsp;Jérôme Urhausen ,&nbsp;Kevin Verbeek","doi":"10.1016/j.comgeo.2022.101923","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101923","url":null,"abstract":"<div><p>We consider the problem of computing the Fréchet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given <em>uncertainty region</em> for that vertex, and the objective is to place vertices so as to minimise the Fréchet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all.</p><p>We present the first general algorithmic framework for this problem. We prove that it results in a polynomial-time algorithm for curves in 1D with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance.</p><p>We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49831313","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":"Connectivity of spaces of directed paths in geometric models for concurrent computation","authors":"Martin Raussen","doi":"10.1016/j.comgeo.2022.101942","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101942","url":null,"abstract":"<div><p>Higher Dimensional Automata (HDA) are higher dimensional relatives to transition systems in concurrency theory taking into account to which degree various actions commute. Mathematically, they take the form of labelled cubical complexes. It is important to know, and challenging from a geometric/topological perspective, whether the space of directed paths (executions in the model) between two vertices (states) is connected; more generally, to estimate higher connectivity of these path spaces.</p><p>This paper presents an approach for such an estimation for particularly simple HDA arising from PV programs and modelling the access of a number of processors to a number of resources with given limited capacity each. It defines the spare capacity of a concurrent program with prescribed periods of access of the processors to the resources using only the syntax of individual programs and the capacities of shared resources. It shows that the connectivity of spaces of directed paths can be estimated (from above) by spare capacities. Moreover, spare capacities can also be used to detect deadlocks and critical states in such a simple HDA.</p><p>The key theoretical ingredient is a transition from the calculation of local connectivity bounds (of the upper links of vertices of an HDA) to global ones by applying a version of the nerve lemma due to Anders Björner.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49790770","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":"Practical volume approximation of high-dimensional convex bodies, applied to modeling portfolio dependencies and financial crises","authors":"Ludovic Calès ,&nbsp;Apostolos Chalkis ,&nbsp;Ioannis Z. Emiris ,&nbsp;Vissarion Fisikopoulos","doi":"10.1016/j.comgeo.2022.101916","DOIUrl":"10.1016/j.comgeo.2022.101916","url":null,"abstract":"<div><p><span><span>We examine volume computation of general-dimensional polytopes and more general </span>convex bodies, defined by the intersection of a simplex by a family of parallel </span>hyperplanes<span>, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest for the public in general and for policy makers in particular. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula<span>; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be estimated by computing volumes of convex bodies.</span></span></p><p><span>We design and implement practical algorithms in the exact and approximate setting, and experimentally juxtapose them in order to study the trade-off of exactness and accuracy for speed. We also experimentally find an efficient parameter-tuning to achieve a sufficiently good estimation of the probability density of each copula. Our C++ software, based on Eigen and available on </span><span>github</span>, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47182787","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":"Acrophobic guard watchtower problem","authors":"Ritesh Seth ,&nbsp;Anil Maheshwari ,&nbsp;Subhas C. Nandy","doi":"10.1016/j.comgeo.2022.101918","DOIUrl":"10.1016/j.comgeo.2022.101918","url":null,"abstract":"<div><p>In the <em>acrophobic guard watchtower problem</em> for a polyhedral terrain, a square axis-aligned platform is placed on the top of a tower whose bottom end-point lies on the surface of the terrain. As in the standard watchtower problem, the objective is to minimize the height (i.e., the length) of the watchtower such that every point on the surface of the terrain is weakly visible from the platform placed on the top of the tower. In this paper, we show that in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> the problem can be solved in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> time, and in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> it takes <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time, where <em>n</em> is the total number of vertices of the terrain.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47553882","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}