{"title":"Chains, Koch Chains, and Point Sets with Many Triangulations","authors":"Daniel Rutschmann, Manuel Wettstein","doi":"https://dl.acm.org/doi/10.1145/3585535","DOIUrl":null,"url":null,"abstract":"<p>We introduce the abstract notion of a chain, which is a sequence of <i>n</i> points in the plane, ordered by <i>x</i>-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations.</p><p>We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have Ω (9.08<sup><i>n</i></sup>) triangulations. This is a significant improvement over the previous and long-standing lower bound of Ω (8.65<sup><i>n</i></sup>) for the maximum number of triangulations of planar point sets.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"27 1","pages":""},"PeriodicalIF":2.3000,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the ACM","FirstCategoryId":"94","ListUrlMain":"https://doi.org/https://dl.acm.org/doi/10.1145/3585535","RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, HARDWARE & ARCHITECTURE","Score":null,"Total":0}
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Abstract
We introduce the abstract notion of a chain, which is a sequence of n points in the plane, ordered by x-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations.
We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have Ω (9.08n) triangulations. This is a significant improvement over the previous and long-standing lower bound of Ω (8.65n) for the maximum number of triangulations of planar point sets.
期刊介绍：
The best indicator of the scope of the journal is provided by the areas covered by its Editorial Board. These areas change from time to time, as the field evolves. The following areas are currently covered by a member of the Editorial Board: Algorithms and Combinatorial Optimization; Algorithms and Data Structures; Algorithms, Combinatorial Optimization, and Games; Artificial Intelligence; Complexity Theory; Computational Biology; Computational Geometry; Computer Graphics and Computer Vision; Computer-Aided Verification; Cryptography and Security; Cyber-Physical, Embedded, and Real-Time Systems; Database Systems and Theory; Distributed Computing; Economics and Computation; Information Theory; Logic and Computation; Logic, Algorithms, and Complexity; Machine Learning and Computational Learning Theory; Networking; Parallel Computing and Architecture; Programming Languages; Quantum Computing; Randomized Algorithms and Probabilistic Analysis of Algorithms; Scientific Computing and High Performance Computing; Software Engineering; Web Algorithms and Data Mining
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