{"scopus_import":"1","article_number":"16","oa_version":"Published Version","publication_status":"published","project":[{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","call_identifier":"FWF","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Discretization in Geometry and Dynamics","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887"}],"file":[{"file_id":"9611","success":1,"content_type":"application/pdf","checksum":"22b11a719018b22ecba2471b51f2eb40","file_size":727817,"file_name":"2021_LIPIcs_Biswas.pdf","relation":"main_file","date_updated":"2021-06-28T13:11:39Z","date_created":"2021-06-28T13:11:39Z","creator":"asandaue","access_level":"open_access"}],"month":"06","date_created":"2021-06-27T22:01:48Z","date_updated":"2023-02-23T14:02:28Z","conference":{"end_date":"2021-06-11","start_date":"2021-06-07","location":"Online","name":"SoCG: International Symposium on Computational Geometry"},"year":"2021","publication_identifier":{"isbn":["9783959771849"],"issn":["18688969"]},"volume":189,"language":[{"iso":"eng"}],"publication":"Leibniz International Proceedings in Informatics","author":[{"first_name":"Ranita","full_name":"Biswas, Ranita","last_name":"Biswas","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-5372-7890"},{"first_name":"Sebastiano","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian"}],"department":[{"_id":"HeEd"}],"status":"public","_id":"9604","quality_controlled":"1","date_published":"2021-06-02T00:00:00Z","ec_funded":1,"alternative_title":["LIPIcs"],"doi":"10.4230/LIPIcs.SoCG.2021.16","citation":{"ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. In: <i>Leibniz International Proceedings in Informatics</i>. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">10.4230/LIPIcs.SoCG.2021.16</a>","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory,” in <i>Leibniz International Proceedings in Informatics</i>, Online, 2021, vol. 189.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Counting Cells of Order-k Voronoi Tessellations in ℝ<sup>3</sup> with Morse Theory.” In <i>Leibniz International Proceedings in Informatics</i>, Vol. 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">https://doi.org/10.4230/LIPIcs.SoCG.2021.16</a>.","mla":"Biswas, Ranita, et al. “Counting Cells of Order-k Voronoi Tessellations in ℝ<sup>3</sup> with Morse Theory.” <i>Leibniz International Proceedings in Informatics</i>, vol. 189, 16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">10.4230/LIPIcs.SoCG.2021.16</a>.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, in:, Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2021. Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. Leibniz International Proceedings in Informatics. SoCG: International Symposium on Computational Geometry, LIPIcs, vol. 189, 16.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2021). Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. In <i>Leibniz International Proceedings in Informatics</i> (Vol. 189). Online: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.16\">https://doi.org/10.4230/LIPIcs.SoCG.2021.16</a>"},"article_processing_charge":"No","intvolume":"       189","title":"Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory","oa":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"ddc":["516"],"day":"02","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","file_date_updated":"2021-06-28T13:11:39Z","type":"conference","abstract":[{"lang":"eng","text":"Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation."}],"has_accepted_license":"1"}