{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:53:46Z","timestamp":1760241226817,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,30]],"date-time":"2019-12-30T00:00:00Z","timestamp":1577664000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we present an algorithm to compute the filtered generalized \u010cech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris\u2013Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized \u010cech complex of a finite collection of d-dimensional disks in     R d    , and we show the performance of our algorithm.<\/jats:p>","DOI":"10.3390\/a13010011","type":"journal-article","created":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T03:28:53Z","timestamp":1578022133000},"page":"11","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Numerical Approach for the Filtered Generalized \u010cech Complex"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5601-3634","authenticated-orcid":false,"given":"Jes\u00fas F.","family":"Espinoza","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Sonora, C.P. 83000, Hermosillo, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rosal\u00eda","family":"Hern\u00e1ndez-Amador","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Sonora, C.P. 83000, Hermosillo, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"H\u00e9ctor A.","family":"Hern\u00e1ndez-Hern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Sonora, C.P. 83000, Hermosillo, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Beatriz","family":"Ramonetti-Valencia","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Sonora, C.P. 83000, Hermosillo, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1090\/S0273-0979-09-01249-X","article-title":"Topology and data","volume":"46","author":"Carlsson","year":"2009","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1017\/S0962492914000051","article-title":"Topological pattern recognition for point cloud data","volume":"23","author":"Carlsson","year":"2014","journal-title":"Acta Numer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1090\/S0273-0979-07-01191-3","article-title":"Barcodes: The persistent topology of data","volume":"45","author":"Ghrist","year":"2008","journal-title":"Bull. Am. Math. Soc."},{"doi-asserted-by":"crossref","unstructured":"Lum, P.Y., Singh, G., Lehman, A., Ishkanov, T., Vejdemo-Johansson, M., Alagappan, M., Carlsson, J., and Carlsson, G. (2013). Extracting insights from the shape of complex data using topology. Sci. Rep., 3.","key":"ref_4","DOI":"10.1038\/srep01236"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s00454-004-1146-y","article-title":"Computing persistent homology","volume":"33","author":"Zomorodian","year":"2005","journal-title":"Discret. Comput. Geom."},{"doi-asserted-by":"crossref","unstructured":"Edelsbrunner, H., and Harer, J. (2010). Computational Topology: An Introduction, American Mathematical Society.","key":"ref_6","DOI":"10.1090\/mbk\/069"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/j.cag.2010.03.007","article-title":"Fast construction of the Vietoris-Rips complex","volume":"34","author":"Zomorodian","year":"2010","journal-title":"Comput. Graph."},{"doi-asserted-by":"crossref","unstructured":"Le, N.K., Martins, P., Decreusefond, L., and Vergne, A. (2015, January 11\u201314). Construction of the Generalized \u010cech Complex. Proceedings of the 2015 IEEE 81st Vehicular Technology Conference (VTC Spring), Glasgow, UK.","key":"ref_8","DOI":"10.1109\/VTCSpring.2015.7145759"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"708","DOI":"10.1016\/j.cag.2012.02.016","article-title":"Efficient construction of the \u010cech complex","volume":"36","author":"Dantchev","year":"2012","journal-title":"Comput. Graph."},{"unstructured":"Morozov, D. (2019, October 23). Dionysus 2. Available online: http:\/\/mrzv.org\/software\/dionysus2\/.","key":"ref_10"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1140\/epjds\/s13688-017-0109-5","article-title":"A roadmap for the computation of persistent homology","volume":"6","author":"Otter","year":"2017","journal-title":"EPJ Data Sci."},{"doi-asserted-by":"crossref","unstructured":"Cai, L., Cheng, S.W., and Lam, T.W. (2013). Approximate \u010cech complex in low and high dimensions. Algorithms and Computation, Springer.","key":"ref_12","DOI":"10.1007\/978-3-642-45030-3"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1214\/15-AOAS886","article-title":"Persistent homology analysis of brain artery trees","volume":"10","author":"Bendich","year":"2016","journal-title":"Ann. Appl. Stat."},{"unstructured":"Goldfarb, D. (2014). An application of topological data analysis to hockey analytics. arXiv.","key":"ref_14"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1177\/0278364915586713","article-title":"Topological trajectory classification with filtrations of simplicial complexes and persistent homology","volume":"35","author":"Pokorny","year":"2016","journal-title":"Int. J. Robot. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.physd.2016.03.007","article-title":"Principal component analysis of persistent homology rank functions with case studies of spatial point patterns, sphere packing and colloids","volume":"334","author":"Robins","year":"2016","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"339","DOI":"10.2140\/agt.2007.7.339","article-title":"Coverage in sensor networks via persistent homology","volume":"7","author":"deSilva","year":"2007","journal-title":"Algebr. Geom. Topol."},{"unstructured":"Bell, G., Lawson, A., Martin, J., Rudzinski, J., and Smyth, C. (2017). Weighted persistent homology. arXiv.","key":"ref_18"},{"key":"ref_19","first-page":"241","article-title":"Ueber die kleinste Kugel, die eine r\u00e4umliche Figur einschliesst","volume":"123","author":"Jung","year":"1901","journal-title":"J. Reine Angew. Math."},{"doi-asserted-by":"crossref","unstructured":"Bollobas, B., and Bollob\u00e1s, B. (2006). The Art of Mathematics: Coffee Time in Memphis, Cambridge University Press.","key":"ref_20","DOI":"10.1017\/CBO9780511816574"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"630","DOI":"10.1007\/978-3-540-39658-1_57","article-title":"Fast smallest-enclosing-ball computation in high dimensions","volume":"Volume 2832","author":"Zwick","year":"2003","journal-title":"Algorithms\u2014ESA 2003"},{"doi-asserted-by":"crossref","unstructured":"Maurer, H. (1991). Smallest enclosing disks (balls and ellipsoids). New Results and New Trends in Computer Science, Springer.","key":"ref_22","DOI":"10.1007\/BFb0038178"},{"unstructured":"Fischer, K., G\u00e4rtner, B., and Kutz, M. (2019, October 23). Miniball (GitHub Repository). Available online: https:\/\/github.com\/hbf\/miniball\/.","key":"ref_23"},{"unstructured":"Research Group in Geometric and Combinatorial Structures (2019, October 23). Mathematics Department, University of Sonora\u2014M\u00e9xico. Available online: www.gcs.mat.uson.mx.","key":"ref_24"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/1\/11\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:46:53Z","timestamp":1760190413000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/1\/11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,30]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["a13010011"],"URL":"https:\/\/doi.org\/10.3390\/a13010011","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2019,12,30]]}}}