Computational Geometry-Theory and Applications最新文献

{"title":"Navigating planar topologies in near-optimal space and time","authors":"José Fuentes-Sepúlveda ,&nbsp;Gonzalo Navarro ,&nbsp;Diego Seco","doi":"10.1016/j.comgeo.2022.101922","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101922","url":null,"abstract":"<div><p><span>We show that any embedding of a planar graph can be encoded succinctly while efficiently answering a number of topological queries near-optimally. More precisely, we build on a succinct representation that encodes an embedding of </span><em>m</em> edges within 4<em>m</em> bits, which is close to the information-theoretic lower bound of about 3.58<em>m</em>. With <span><math><mn>4</mn><mi>m</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>m</mi><mo>)</mo></math></span> bits of space, we show how to answer a number of topological queries relating nodes, edges, and faces, most of them in any time in <span><math><mi>ω</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. Indeed, <span><math><mn>3.58</mn><mi>m</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>m</mi><mo>)</mo></math></span> bits suffice if the graph has no self-loops and no nodes of degree one. Further, we show that with <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mo>)</mo></math></span> bits of space we can solve all those operations in <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> time.</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":"49790779","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":"Simplex closing probabilities in directed graphs","authors":"Florian Unger ,&nbsp;Jonathan Krebs ,&nbsp;Michael G. Müller","doi":"10.1016/j.comgeo.2022.101941","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101941","url":null,"abstract":"<div><p>Recent work in mathematical neuroscience has calculated the directed graph homology of the directed simplicial complex given by the brain's sparse adjacency graph, the so called connectome. These biological connectomes show an abundance of both high-dimensional directed simplices and Betti-numbers in all viable dimensions – in contrast to Erdős–Rényi-graphs of comparable size and density. An analysis of synthetically trained connectomes reveals similar findings, raising questions about the graphs comparability and the nature of origin of the simplices.</p><p>We present a new method capable of delivering insight into the emergence of simplices and thus simplicial abundance. Our approach allows to easily distinguish simplex-rich connectomes of different origin. The method relies on the novel concept of an almost-d-simplex, that is, a simplex missing exactly one edge, and consequently the almost-d-simplex closing probability by dimension. We also describe a fast algorithm to identify almost-d-simplices in a given graph. Applying this method to biological and artificial data allows us to identify a mechanism responsible for simplex emergence, and suggests this mechanism is responsible for the simplex signature of the excitatory subnetwork of a statistical reconstruction of the mouse primary visual cortex. Our highly optimized code for this new method is publicly available.</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":"49790781","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":"1-planarity testing and embedding: An experimental study","authors":"Carla Binucci,&nbsp;Walter Didimo,&nbsp;Fabrizio Montecchiani","doi":"10.1016/j.comgeo.2022.101900","DOIUrl":"10.1016/j.comgeo.2022.101900","url":null,"abstract":"<div><p>Many papers study the natural problem of drawing nonplanar graphs with few crossings per edge. In particular, a graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity<span> is solvable in linear time and several efficient algorithms have been described in the literature, deciding whether a graph is 1-planar is NP-complete, even for restricted classes of graphs. Despite some polynomial-time algorithms are known for recognizing specific subfamilies of 1-planar graphs, there is still a lack of practical 1-planarity testing algorithms and no implementation is available for general graphs. This paper investigates the feasibility of a 1-planarity testing and embedding algorithm based on a backtracking strategy. Our contribution provides initial indications that have the potential to stimulate further research on the design of practical approaches for the 1-planarity testing problem. On the one hand, our experiments show that a backtracking strategy can be successfully applied to graphs with up to 30 vertices. On the other hand, our study suggests that alternative techniques are needed to attack larger graphs.</span></p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48906491","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":"Approximating the packedness of polygonal curves","authors":"Joachim Gudmundsson ,&nbsp;Yuan Sha,&nbsp;Sampson Wong","doi":"10.1016/j.comgeo.2022.101920","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101920","url":null,"abstract":"<div><p>In 2012 Driemel et al. introduced the concept of <em>c</em>-packed curves as a realistic input model. In the case when <em>c</em> is a constant they gave a near linear time <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximation algorithm for computing the Fréchet distance between two <em>c</em><span>-packed polygonal curves. Since then a number of papers have used the model.</span></p><p>In this paper we consider the problem of computing the smallest <em>c</em> for which a given polygonal curve in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is <em>c</em><span>-packed. We present two approximation algorithms. The first algorithm is a 2-approximation algorithm and runs in </span><span><math><mi>O</mi><mo>(</mo><mi>d</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time. In the case <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> we develop a faster algorithm that returns a <span><math><mo>(</mo><mn>6</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximation and runs in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>n</mi><mo>/</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mi>polylog</mi><mo>(</mo><mi>n</mi><mo>/</mo><mi>ε</mi><mo>)</mo><mo>)</mo><mo>)</mo></math></span> time.</p><p>We also implemented the first algorithm and computed the approximate packedness-value for 16 sets of real-world trajectories. The experiments indicate that the notion of <em>c</em>-packedness is a useful realistic input model for many curves and trajectories.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49800577","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 enumeration of integer tetrahedra","authors":"James East ,&nbsp;Michael Hendriksen ,&nbsp;Laurence Park","doi":"10.1016/j.comgeo.2022.101915","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101915","url":null,"abstract":"<div><p><span>We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding problem for triangles, which has long been solved. We expect there are no closed-form solutions to the tetrahedron enumeration problems, but we explore the extent to which they can be approached via classical methods, such as orbit enumeration. We also discuss algorithms for computing the numbers, and present several tables and figures that can be used to visualise the data. Several intriguing patterns seem to emerge, leading to a number of natural conjectures. The central conjecture is that the number of integer tetrahedra of perimeter </span><em>n</em>, up to congruence, is asymptotic to <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>/</mo><mi>C</mi></math></span> for some constant <span><math><mi>C</mi><mo>≈</mo><mn>229000</mn></math></span>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49895834","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 4-approximation of the 2π3-MST","authors":"Stav Ashur,&nbsp;Matthew J. Katz","doi":"10.1016/j.comgeo.2022.101914","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101914","url":null,"abstract":"<div><p><span>Bounded-angle (minimum) spanning trees were first introduced in the context of wireless networks with directional antennas. They are reminiscent of bounded-degree (minimum) spanning trees, which have received significant attention. Let </span><em>P</em> be a set of <em>n</em> points in the plane, and let <span><math><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mn>2</mn><mi>π</mi></math></span> be an angle. An <em>α</em>-spanning tree (<em>α</em>-ST) of <em>P</em> is a spanning tree of the complete Euclidean graph over <em>P</em>, with the following property: For each vertex <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>P</mi></math></span>, the (smallest) angle that is spanned by all the edges incident to <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is at most <em>α</em>. An <em>α</em>-minimum spanning tree (<em>α</em>-MST) is an <em>α</em>-ST of <em>P</em> of minimum weight, where the weight of an <em>α</em>-ST is the sum of the lengths of its edges. In this paper, we consider the problem of computing an <em>α</em>-MST for the case where <span><math><mi>α</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>. We present a 4-approximation algorithm, thus improving upon the previous results of Aschner and Katz and Biniaz et al., who presented algorithms with approximation ratios 6 and <span><math><mfrac><mrow><mn>16</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>, respectively.</p><p>To obtain this result, we devise an <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span><span>-time algorithm that, given any Hamiltonian path Π of </span><em>P</em>, constructs a <span><math><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>-ST <span><math><mi>T</mi></math></span> of <em>P</em>, such that <span><math><mi>T</mi></math></span>'s weight is at most twice that of Π and, moreover, <span><math><mi>T</mi></math></span> is a 3-hop spanner of Π. This latter result is optimal (with respect to <span><math><mi>T</mi></math></span>'s weight), since for any <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> there exists a polygonal path for which every <span><math><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>-ST (of the corresponding set of points) has weight greater than <span><math><mn>2</mn><mo>−</mo><mi>ε</mi></math></span> times the weight of the path.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49800574","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":"Folding polyiamonds into octahedra","authors":"Eva Stehr,&nbsp;Linda Kleist","doi":"10.1016/j.comgeo.2022.101917","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101917","url":null,"abstract":"<div><p><span>We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid – the </span>octahedron<span>. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of positive area, namely each but one polyiamond is foldable. Secondly, we show that a convex polyiamond folds into the octahedron if and only if it contains one of five polyiamonds. We thirdly present a sharp size bound: While there exist unfoldable polyiamonds of size 14, every polyiamond of size at least 15 folds into the octahedron. This clearly implies that one can test in polynomial time whether a given polyiamond folds into the octahedron. Lastly, we show that for any assignment of positive integers to the faces, there exists a polyiamond that folds into the octahedron such that the number of triangles covering a face is equal to the assigned number.</span></p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49800576","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":"The impact of geometry on monochrome regions in the flip Schelling process","authors":"Thomas Bläsius ,&nbsp;Tobias Friedrich ,&nbsp;Martin S. Krejca ,&nbsp;Louise Molitor","doi":"10.1016/j.comgeo.2022.101902","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101902","url":null,"abstract":"<div><p>Schelling's classical segregation model gives a coherent explanation for the wide-spread phenomenon of residential segregation. We introduce an agent-based saturated open-city variant, the Flip Schelling Process (FSP), in which agents, placed on a graph, have one out of two types and, based on the predominant type in their neighborhood, decide whether to change their types; similar to a new agent arriving as soon as another agent leaves the vertex.</p><p><span>We investigate the probability that an edge </span><span><math><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo></math></span> is monochrome, i.e., that both vertices <em>u</em> and <em>v</em><span><span> have the same type in the FSP, and we provide a general framework for analyzing the influence of the underlying graph topology on residential segregation. In particular, for two </span>adjacent vertices, we show that a highly decisive common neighborhood, i.e., a common neighborhood where the absolute value of the difference between the number of vertices with different types is high, supports segregation and, moreover, that large common neighborhoods are more decisive.</span></p><p><span><span>As an application, we study the expected behavior of the FSP on two common random graph models with and without geometry: (1) For random </span>geometric graphs, we show that the existence of an edge </span><span><math><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo></math></span> makes a highly decisive common neighborhood for <em>u</em> and <em>v</em> more likely. Based on this, we prove the existence of a constant <span><math><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> such that the expected fraction of monochrome edges after the FSP is at least <span><math><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>c</mi></math></span>. (2) For Erdős–Rényi graphs we show that large common neighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP is at most <span><math><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math></span>. Our results indicate that the cluster structure of the underlying graph has a significant impact on the obtained segregation strength.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49800575","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 dominating sets - a minimum version of the No-Three-In-Line Problem","authors":"Oswin Aichholzer ,&nbsp;David Eppstein ,&nbsp;Eva-Maria Hainzl","doi":"10.1016/j.comgeo.2022.101913","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101913","url":null,"abstract":"<div><p>We consider a minimizing variant of the well-known <em>No-Three-In-Line Problem</em>, the <span><em>Geometric </em><em>Dominating Set</em><em> Problem</em></span>: What is the smallest number of points in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> grid such that every grid point lies on a common line with two of the points in the set? We show a lower bound of <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span> points and provide a constructive upper bound of size <span><math><mn>2</mn><mo>⌈</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌉</mo></math></span>. If the points of the dominating sets are required to be in general position we provide optimal solutions for grids of size up to <span><math><mn>12</mn><mo>×</mo><mn>12</mn></math></span>. For arbitrary <em>n</em> the currently best upper bound for points in general position remains the obvious 2<em>n</em>. Finally, we discuss the problem on the discrete torus where we prove an upper bound of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>. For <em>n</em> even or a multiple of 3, we can even show a constant upper bound of 4. We also mention a number of open questions and some further variations of the problem.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49800644","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":"Crossing lemma for the odd-crossing number","authors":"János Karl ,&nbsp;Géza Tóth","doi":"10.1016/j.comgeo.2022.101901","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101901","url":null,"abstract":"<div><p>A graph is 1-planar, if it can be drawn in the plane such that there is at most one crossing on every edge. It is known, that 1-planar graphs have at most <span><math><mn>4</mn><mi>n</mi><mo>−</mo><mn>8</mn></math></span> edges.</p><p>We prove the following odd-even generalization. If a graph can be drawn in the plane such that every edge is crossed by at most one other edge <em>an odd number of times</em>, then it is called 1-odd-planar and it has at most <span><math><mn>5</mn><mi>n</mi><mo>−</mo><mn>9</mn></math></span> edges. As a consequence, we improve the constant in the Crossing Lemma for the odd-crossing number, if adjacent edges cross an even number of times. We also give upper bound for the number of edges of <em>k</em>-odd-planar graphs.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49800578","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}