[{"oa":1,"publisher":"Springer Nature","scopus_import":"1","doi":"10.1007/s41468-020-00063-x","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"date_updated":"2025-04-14T07:43:51Z","intvolume":"         5","citation":{"apa":"Brown, A., Bobrowski, O., Munch, E., &#38; Wang, B. (2021). Probabilistic convergence and stability of random mapper graphs. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-020-00063-x\">https://doi.org/10.1007/s41468-020-00063-x</a>","ieee":"A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence and stability of random mapper graphs,” <i>Journal of Applied and Computational Topology</i>, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021.","ama":"Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability of random mapper graphs. <i>Journal of Applied and Computational Topology</i>. 2021;5(1):99-140. doi:<a href=\"https://doi.org/10.1007/s41468-020-00063-x\">10.1007/s41468-020-00063-x</a>","mla":"Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper Graphs.” <i>Journal of Applied and Computational Topology</i>, vol. 5, no. 1, Springer Nature, 2021, pp. 99–140, doi:<a href=\"https://doi.org/10.1007/s41468-020-00063-x\">10.1007/s41468-020-00063-x</a>.","short":"A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational Topology 5 (2021) 99–140.","ista":"Brown A, Bobrowski O, Munch E, Wang B. 2021. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. 5(1), 99–140.","chicago":"Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic Convergence and Stability of Random Mapper Graphs.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s41468-020-00063-x\">https://doi.org/10.1007/s41468-020-00063-x</a>."},"issue":"1","publication":"Journal of Applied and Computational Topology","volume":5,"type":"journal_article","arxiv":1,"day":"01","publication_status":"published","abstract":[{"lang":"eng","text":"We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data."}],"fulldoi":"https://doi.org/10.1007/s41468-020-00063-x","file_date_updated":"2021-02-11T14:43:59Z","article_type":"original","author":[{"full_name":"Brown, Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam","last_name":"Brown"},{"last_name":"Bobrowski","full_name":"Bobrowski, Omer","first_name":"Omer"},{"last_name":"Munch","full_name":"Munch, Elizabeth","first_name":"Elizabeth"},{"last_name":"Wang","full_name":"Wang, Bei","first_name":"Bei"}],"has_accepted_license":"1","month":"03","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"language":[{"iso":"eng"}],"_id":"9111","ec_funded":1,"year":"2021","status":"public","quality_controlled":"1","project":[{"name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425"}],"page":"99-140","external_id":{"arxiv":["1909.03488"]},"oa_version":"Published Version","acknowledgement":"AB was supported in part by the European Union’s Horizon 2020 research and innovation\r\nprogramme under the Marie Sklodowska-Curie GrantAgreement No. 754411 and NSF IIS-1513616. OB was supported in part by the Israel Science Foundation, Grant 1965/19. BW was supported in part by NSF IIS-1513616 and DBI-1661375. EM was supported in part by NSF CMMI-1800466, DMS-1800446, and CCF-1907591.We would like to thank the Institute for Mathematics and its Applications for hosting a workshop titled Bridging Statistics and Sheaves in May 2018, where this work was conceived.\r\nOpen Access funding provided by Institute of Science and Technology (IST Austria).","title":"Probabilistic convergence and stability of random mapper graphs","article_processing_charge":"Yes (via OA deal)","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_published":"2021-03-01T00:00:00Z","date_created":"2021-02-11T14:41:02Z","file":[{"creator":"dernst","file_name":"2020_JourApplCompTopology_Brown.pdf","checksum":"3f02e9d47c428484733da0f588a3c069","date_created":"2021-02-11T14:43:59Z","success":1,"access_level":"open_access","content_type":"application/pdf","date_updated":"2021-02-11T14:43:59Z","file_id":"9112","relation":"main_file","file_size":2090265}],"ddc":["510"],"department":[{"_id":"HeEd"}]},{"oa":1,"publisher":"IEEE","scopus_import":"1","doi":"10.1109/bigdata50022.2020.9378374","date_updated":"2023-08-07T14:00:13Z","citation":{"apa":"Heiler, G., Reisch, T., Hurt, J., Forghani, M., Omani, A., Hanbury, A., &#38; Karimipour, F. (2021). Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic. In <i>2020 IEEE International Conference on Big Data</i> (pp. 3123–3132). Atlanta, GA, United States: IEEE. <a href=\"https://doi.org/10.1109/bigdata50022.2020.9378374\">https://doi.org/10.1109/bigdata50022.2020.9378374</a>","ieee":"G. Heiler <i>et al.</i>, “Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic,” in <i>2020 IEEE International Conference on Big Data</i>, Atlanta, GA, United States, 2021, pp. 3123–3132.","ista":"Heiler G, Reisch T, Hurt J, Forghani M, Omani A, Hanbury A, Karimipour F. 2021. Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic. 2020 IEEE International Conference on Big Data. Big Data: International Conference on Big Data, 3123–3132.","short":"G. Heiler, T. Reisch, J. Hurt, M. Forghani, A. Omani, A. Hanbury, F. Karimipour, in:, 2020 IEEE International Conference on Big Data, IEEE, 2021, pp. 3123–3132.","ama":"Heiler G, Reisch T, Hurt J, et al. Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic. In: <i>2020 IEEE International Conference on Big Data</i>. IEEE; 2021:3123-3132. doi:<a href=\"https://doi.org/10.1109/bigdata50022.2020.9378374\">10.1109/bigdata50022.2020.9378374</a>","mla":"Heiler, Georg, et al. “Country-Wide Mobility Changes Observed Using Mobile Phone Data during COVID-19 Pandemic.” <i>2020 IEEE International Conference on Big Data</i>, IEEE, 2021, pp. 3123–32, doi:<a href=\"https://doi.org/10.1109/bigdata50022.2020.9378374\">10.1109/bigdata50022.2020.9378374</a>.","chicago":"Heiler, Georg, Tobias Reisch, Jan Hurt, Mohammad Forghani, Aida Omani, Allan Hanbury, and Farid Karimipour. “Country-Wide Mobility Changes Observed Using Mobile Phone Data during COVID-19 Pandemic.” In <i>2020 IEEE International Conference on Big Data</i>, 3123–32. IEEE, 2021. <a href=\"https://doi.org/10.1109/bigdata50022.2020.9378374\">https://doi.org/10.1109/bigdata50022.2020.9378374</a>."},"publication":"2020 IEEE International Conference on Big Data","type":"conference","day":"19","arxiv":1,"conference":{"name":"Big Data: International Conference on Big Data","location":"Atlanta, GA, United States","start_date":"2020-12-10","end_date":"2020-12-13"},"publication_status":"published","abstract":[{"lang":"eng","text":"In March 2020, the Austrian government introduced a widespread lock-down in response to the COVID-19 pandemic. Based on subjective impressions and anecdotal evidence, Austrian public and private life came to a sudden halt. Here we assess the effect of the lock-down quantitatively for all regions in Austria and present an analysis of daily changes of human mobility throughout Austria using near-real-time anonymized mobile phone data. We describe an efficient data aggregation pipeline and analyze the mobility by quantifying mobile-phone traffic at specific point of interests (POIs), analyzing individual trajectories and investigating the cluster structure of the origin-destination graph. We found a reduction of commuters at Viennese metro stations of over 80% and the number of devices with a radius of gyration of less than 500 m almost doubled. The results of studying crowd-movement behavior highlight considerable changes in the structure of mobility networks, revealed by a higher modularity and an increase from 12 to 20 detected communities. We demonstrate the relevance of mobility data for epidemiological studies by showing a significant correlation of the outflow from the town of Ischgl (an early COVID-19 hotspot) and the reported COVID-19 cases with an 8-day time lag. This research indicates that mobile phone usage data permits the moment-by-moment quantification of mobility behavior for a whole country. We emphasize the need to improve the availability of such data in anonymized form to empower rapid response to combat COVID-19 and future pandemics."}],"fulldoi":"https://doi.org/10.1109/bigdata50022.2020.9378374","author":[{"first_name":"Georg","full_name":"Heiler, Georg","last_name":"Heiler"},{"last_name":"Reisch","full_name":"Reisch, Tobias","first_name":"Tobias"},{"last_name":"Hurt","first_name":"Jan","full_name":"Hurt, Jan"},{"last_name":"Forghani","first_name":"Mohammad","full_name":"Forghani, Mohammad"},{"full_name":"Omani, Aida","first_name":"Aida","last_name":"Omani"},{"last_name":"Hanbury","full_name":"Hanbury, Allan","first_name":"Allan"},{"last_name":"Karimipour","orcid":"0000-0001-6746-4174","id":"2A2BCDC4-CF62-11E9-BE5E-3B1EE6697425","full_name":"Karimipour, Farid","first_name":"Farid"}],"month":"03","publication_identifier":{"isbn":["9781728162515"]},"language":[{"iso":"eng"}],"_id":"9253","year":"2021","quality_controlled":"1","status":"public","main_file_link":[{"url":"https://arxiv.org/abs/2008.10064","open_access":"1"}],"page":"3123-3132","isi":1,"external_id":{"arxiv":["2008.10064"],"isi":["000662554703032"]},"oa_version":"Preprint","title":"Country-wide mobility changes observed using mobile phone data during COVID-19 pandemic","article_processing_charge":"No","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_published":"2021-03-19T00:00:00Z","date_created":"2021-03-21T11:34:07Z","department":[{"_id":"HeEd"}]},{"author":[{"last_name":"Aichholzer","first_name":"Oswin","full_name":"Aichholzer, Oswin"},{"full_name":"Arroyo Guevara, Alan M","first_name":"Alan M","id":"3207FDC6-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-2401-8670","last_name":"Arroyo Guevara"},{"last_name":"Masárová","orcid":"0000-0002-6660-1322","full_name":"Masárová, Zuzana","first_name":"Zuzana","id":"45CFE238-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Parada, Irene","first_name":"Irene","last_name":"Parada"},{"full_name":"Perz, Daniel","first_name":"Daniel","last_name":"Perz"},{"last_name":"Pilz","first_name":"Alexander","full_name":"Pilz, Alexander"},{"last_name":"Tkadlec","orcid":"0000-0002-1097-9684","full_name":"Tkadlec, Josef","first_name":"Josef","id":"3F24CCC8-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Vogtenhuber","first_name":"Birgit","full_name":"Vogtenhuber, Birgit"}],"arxiv":1,"conference":{"location":"Yangon, Myanmar","name":"WALCOM: Algorithms and Computation","end_date":"2021-03-02","start_date":"2021-02-28"},"day":"16","publication_status":"published","abstract":[{"lang":"eng","text":" matching is compatible to two or more labeled point sets of size n with labels   {1,…,n}  if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with   ⌊2n−−√⌋  edges. More generally, for any   ℓ  labeled point sets we construct compatible matchings of size   Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any   ℓ  given sets of n points there exists a labeling of each set such that the largest compatible matching has   O(n2/(ℓ+1))  edges. Finally, we show that   Θ(logn)  copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge."}],"fulldoi":"https://doi.org/10.1007/978-3-030-68211-8_18","alternative_title":["LNCS"],"citation":{"ieee":"O. Aichholzer <i>et al.</i>, “On compatible matchings,” in <i>15th International Conference on Algorithms and Computation</i>, Yangon, Myanmar, 2021, vol. 12635, pp. 221–233.","apa":"Aichholzer, O., Arroyo Guevara, A. M., Masárová, Z., Parada, I., Perz, D., Pilz, A., … Vogtenhuber, B. (2021). On compatible matchings. In <i>15th International Conference on Algorithms and Computation</i> (Vol. 12635, pp. 221–233). Yangon, Myanmar: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-030-68211-8_18\">https://doi.org/10.1007/978-3-030-68211-8_18</a>","chicago":"Aichholzer, Oswin, Alan M Arroyo Guevara, Zuzana Masárová, Irene Parada, Daniel Perz, Alexander Pilz, Josef Tkadlec, and Birgit Vogtenhuber. “On Compatible Matchings.” In <i>15th International Conference on Algorithms and Computation</i>, 12635:221–33. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/978-3-030-68211-8_18\">https://doi.org/10.1007/978-3-030-68211-8_18</a>.","ama":"Aichholzer O, Arroyo Guevara AM, Masárová Z, et al. On compatible matchings. In: <i>15th International Conference on Algorithms and Computation</i>. Vol 12635. Springer Nature; 2021:221-233. doi:<a href=\"https://doi.org/10.1007/978-3-030-68211-8_18\">10.1007/978-3-030-68211-8_18</a>","mla":"Aichholzer, Oswin, et al. “On Compatible Matchings.” <i>15th International Conference on Algorithms and Computation</i>, vol. 12635, Springer Nature, 2021, pp. 221–33, doi:<a href=\"https://doi.org/10.1007/978-3-030-68211-8_18\">10.1007/978-3-030-68211-8_18</a>.","short":"O. Aichholzer, A.M. Arroyo Guevara, Z. Masárová, I. Parada, D. Perz, A. Pilz, J. Tkadlec, B. Vogtenhuber, in:, 15th International Conference on Algorithms and Computation, Springer Nature, 2021, pp. 221–233.","ista":"Aichholzer O, Arroyo Guevara AM, Masárová Z, Parada I, Perz D, Pilz A, Tkadlec J, Vogtenhuber B. 2021. On compatible matchings. 15th International Conference on Algorithms and Computation. WALCOM: Algorithms and Computation, LNCS, vol. 12635, 221–233."},"publication":"15th International Conference on Algorithms and Computation","volume":12635,"type":"conference","oa":1,"publisher":"Springer Nature","scopus_import":"1","doi":"10.1007/978-3-030-68211-8_18","intvolume":"     12635","date_updated":"2026-04-16T09:18:21Z","date_published":"2021-02-16T00:00:00Z","date_created":"2021-03-28T22:01:41Z","department":[{"_id":"UlWa"},{"_id":"HeEd"},{"_id":"KrCh"}],"isi":1,"external_id":{"isi":["001435069600018"],"arxiv":["2101.03928"]},"oa_version":"Preprint","related_material":{"record":[{"relation":"later_version","status":"public","id":"11938"}]},"title":"On compatible matchings","acknowledgement":"A.A. funded by the Marie Skłodowska-Curie grant agreement No. 754411. Z.M. partially funded by Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31. I.P., D.P., and B.V. partially supported by FWF within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35. A.P. supported by a Schrödinger fellowship of the FWF: J-3847-N35. J.T. partially supported by ERC Start grant no. (279307: Graph Games), FWF grant no. P23499-N23 and S11407-N23 (RiSE).","article_processing_charge":"No","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","quality_controlled":"1","status":"public","year":"2021","project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"grant_number":"279307","call_identifier":"FP7","_id":"2581B60A-B435-11E9-9278-68D0E5697425","name":"Quantitative Graph Games: Theory and Applications"},{"call_identifier":"FWF","_id":"2584A770-B435-11E9-9278-68D0E5697425","grant_number":"P 23499-N23","name":"Modern Graph Algorithmic Techniques in Formal Verification"},{"call_identifier":"FWF","_id":"25863FF4-B435-11E9-9278-68D0E5697425","grant_number":"S11407","name":"Game Theory"}],"main_file_link":[{"url":"https://arxiv.org/abs/2101.03928","open_access":"1"}],"page":"221-233","month":"02","language":[{"iso":"eng"}],"publication_identifier":{"isbn":["9783030682101"],"eisbn":["9783030682118"],"eissn":["1611-3349"],"issn":["0302-9743"]},"_id":"9296","ec_funded":1},{"month":"03","has_accepted_license":"1","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"_id":"9317","ec_funded":1,"project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"year":"2021","quality_controlled":"1","status":"public","page":"1296–1313","external_id":{"isi":["000635460400001"],"pmid":["34720303"]},"isi":1,"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha), and by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF)\r\nOpen Access funding provided by the Institute of Science and Technology (IST Austria).","title":"The multi-cover persistence of Euclidean balls","related_material":{"record":[{"id":"187","status":"public","relation":"earlier_version"}]},"oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2021-03-31T00:00:00Z","date_created":"2021-04-11T22:01:15Z","file":[{"relation":"main_file","file_size":677704,"access_level":"open_access","file_id":"10394","content_type":"application/pdf","date_updated":"2021-12-01T10:56:53Z","checksum":"59b4e1e827e494209bcb4aae22e1d347","success":1,"date_created":"2021-12-01T10:56:53Z","creator":"cchlebak","file_name":"2021_DisCompGeo_Edelsbrunner_Osang.pdf"}],"ddc":["516"],"department":[{"_id":"HeEd"}],"pmid":1,"oa":1,"scopus_import":"1","publisher":"Springer Nature","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"doi":"10.1007/s00454-021-00281-9","date_updated":"2025-06-12T06:36:54Z","intvolume":"        65","citation":{"apa":"Edelsbrunner, H., &#38; Osang, G. F. (2021). The multi-cover persistence of Euclidean balls. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-021-00281-9\">https://doi.org/10.1007/s00454-021-00281-9</a>","ieee":"H. Edelsbrunner and G. F. Osang, “The multi-cover persistence of Euclidean balls,” <i>Discrete and Computational Geometry</i>, vol. 65. Springer Nature, pp. 1296–1313, 2021.","mla":"Edelsbrunner, Herbert, and Georg F. Osang. “The Multi-Cover Persistence of Euclidean Balls.” <i>Discrete and Computational Geometry</i>, vol. 65, Springer Nature, 2021, pp. 1296–1313, doi:<a href=\"https://doi.org/10.1007/s00454-021-00281-9\">10.1007/s00454-021-00281-9</a>.","ama":"Edelsbrunner H, Osang GF. The multi-cover persistence of Euclidean balls. <i>Discrete and Computational Geometry</i>. 2021;65:1296–1313. doi:<a href=\"https://doi.org/10.1007/s00454-021-00281-9\">10.1007/s00454-021-00281-9</a>","ista":"Edelsbrunner H, Osang GF. 2021. The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. 65, 1296–1313.","short":"H. Edelsbrunner, G.F. Osang, Discrete and Computational Geometry 65 (2021) 1296–1313.","chicago":"Edelsbrunner, Herbert, and Georg F Osang. “The Multi-Cover Persistence of Euclidean Balls.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-021-00281-9\">https://doi.org/10.1007/s00454-021-00281-9</a>."},"type":"journal_article","volume":65,"publication":"Discrete and Computational Geometry","publication_status":"published","day":"31","abstract":[{"lang":"eng","text":"Given a locally finite X⊆Rd and a radius r≥0, the k-fold cover of X and r consists of all points in Rd that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in Rd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers."}],"file_date_updated":"2021-12-01T10:56:53Z","corr_author":"1","fulldoi":"https://doi.org/10.1007/s00454-021-00281-9","article_type":"original","author":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"last_name":"Osang","full_name":"Osang, Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","first_name":"Georg F","orcid":"0000-0002-8882-5116"}]},{"_id":"9465","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1420-8997"],"issn":["0047-2468"]},"month":"04","has_accepted_license":"1","year":"2021","quality_controlled":"1","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"Yes (via OA deal)","title":"A step in the Delaunay mosaic of order k","oa_version":"Published Version","department":[{"_id":"HeEd"}],"ddc":["510"],"file":[{"checksum":"e52a832f1def52a2b23d21bcc09e646f","success":1,"date_created":"2021-06-11T13:16:26Z","creator":"kschuh","file_name":"2021_Geometry_Edelsbrunner.pdf","file_size":694706,"relation":"main_file","access_level":"open_access","file_id":"9544","date_updated":"2021-06-11T13:16:26Z","content_type":"application/pdf"}],"date_created":"2021-06-06T22:01:29Z","date_published":"2021-04-01T00:00:00Z","intvolume":"       112","date_updated":"2025-07-10T12:01:46Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"doi":"10.1007/s00022-021-00577-4","scopus_import":"1","publisher":"Springer Nature","oa":1,"type":"journal_article","volume":112,"publication":"Journal of Geometry","issue":"1","citation":{"ieee":"H. Edelsbrunner, A. Nikitenko, and G. F. Osang, “A step in the Delaunay mosaic of order k,” <i>Journal of Geometry</i>, vol. 112, no. 1. Springer Nature, 2021.","apa":"Edelsbrunner, H., Nikitenko, A., &#38; Osang, G. F. (2021). A step in the Delaunay mosaic of order k. <i>Journal of Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00022-021-00577-4\">https://doi.org/10.1007/s00022-021-00577-4</a>","chicago":"Edelsbrunner, Herbert, Anton Nikitenko, and Georg F Osang. “A Step in the Delaunay Mosaic of Order K.” <i>Journal of Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00022-021-00577-4\">https://doi.org/10.1007/s00022-021-00577-4</a>.","mla":"Edelsbrunner, Herbert, et al. “A Step in the Delaunay Mosaic of Order K.” <i>Journal of Geometry</i>, vol. 112, no. 1, 15, Springer Nature, 2021, doi:<a href=\"https://doi.org/10.1007/s00022-021-00577-4\">10.1007/s00022-021-00577-4</a>.","ama":"Edelsbrunner H, Nikitenko A, Osang GF. A step in the Delaunay mosaic of order k. <i>Journal of Geometry</i>. 2021;112(1). doi:<a href=\"https://doi.org/10.1007/s00022-021-00577-4\">10.1007/s00022-021-00577-4</a>","ista":"Edelsbrunner H, Nikitenko A, Osang GF. 2021. A step in the Delaunay mosaic of order k. Journal of Geometry. 112(1), 15.","short":"H. Edelsbrunner, A. Nikitenko, G.F. Osang, Journal of Geometry 112 (2021)."},"article_number":"15","corr_author":"1","file_date_updated":"2021-06-11T13:16:26Z","fulldoi":"https://doi.org/10.1007/s00022-021-00577-4","abstract":[{"text":"Given a locally finite set 𝑋⊆ℝ𝑑 and an integer 𝑘≥0, we consider the function 𝐰𝑘:Del𝑘(𝑋)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.","lang":"eng"}],"publication_status":"published","day":"01","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"orcid":"0000-0002-0659-3201","full_name":"Nikitenko, Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton","last_name":"Nikitenko"},{"last_name":"Osang","full_name":"Osang, Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","first_name":"Georg F","orcid":"0000-0002-8882-5116"}],"article_type":"original"},{"department":[{"_id":"HeEd"}],"ddc":["510"],"date_created":"2021-06-27T22:01:47Z","file":[{"file_id":"9612","content_type":"application/pdf","date_updated":"2021-06-28T13:33:23Z","access_level":"open_access","relation":"main_file","file_size":418168,"file_name":"2021_JournalOfCombinatorialTheory_Pach.pdf","creator":"asandaue","success":1,"date_created":"2021-06-28T13:33:23Z","checksum":"15fbc9064cd9d1c777ac0043b78c8f12"}],"date_published":"2021-06-09T00:00:00Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","oa_version":"Published Version","title":"Erdős-Hajnal-type results for monotone paths","acknowledgement":"We would like to thank the anonymous referees for their useful comments and suggestions. János Pach is partially supported by Austrian Science Fund (FWF) grant Z 342-N31 and by ERC Advanced grant “GeoScape.” István Tomon is partially supported by Swiss National Science Foundation grant no. 200021_196965, and thanks the support of MIPT Moscow. Both authors are partially supported by The Russian Government in the framework of MegaGrant no. 075-15-2019-1926.","isi":1,"external_id":{"isi":["000702280800002"]},"page":"21-37","year":"2021","status":"public","quality_controlled":"1","project":[{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342"}],"_id":"9602","publication_identifier":{"issn":["0095-8956"]},"language":[{"iso":"eng"}],"has_accepted_license":"1","month":"06","author":[{"last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","full_name":"Pach, János","first_name":"János"},{"full_name":"Tomon, István","first_name":"István","last_name":"Tomon"}],"article_type":"original","fulldoi":"https://doi.org/10.1016/j.jctb.2021.05.004","file_date_updated":"2021-06-28T13:33:23Z","corr_author":"1","abstract":[{"text":"An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck > 0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least n^ck . This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs.\r\nA key idea of the above paper was to show that any unordered graph on n vertices that does not contain an induced path of size k, and whose maximum degree is at most c(k)n for some small c(k) > 0, contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for k ≥ 3, by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees.","lang":"eng"}],"publication_status":"published","day":"09","volume":151,"publication":"Journal of Combinatorial Theory. Series B","type":"journal_article","citation":{"mla":"Pach, János, and István Tomon. “Erdős-Hajnal-Type Results for Monotone Paths.” <i>Journal of Combinatorial Theory. Series B</i>, vol. 151, Elsevier, 2021, pp. 21–37, doi:<a href=\"https://doi.org/10.1016/j.jctb.2021.05.004\">10.1016/j.jctb.2021.05.004</a>.","ama":"Pach J, Tomon I. Erdős-Hajnal-type results for monotone paths. <i>Journal of Combinatorial Theory Series B</i>. 2021;151:21-37. doi:<a href=\"https://doi.org/10.1016/j.jctb.2021.05.004\">10.1016/j.jctb.2021.05.004</a>","ista":"Pach J, Tomon I. 2021. Erdős-Hajnal-type results for monotone paths. Journal of Combinatorial Theory. Series B. 151, 21–37.","short":"J. Pach, I. Tomon, Journal of Combinatorial Theory. Series B 151 (2021) 21–37.","chicago":"Pach, János, and István Tomon. “Erdős-Hajnal-Type Results for Monotone Paths.” <i>Journal of Combinatorial Theory. Series B</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.jctb.2021.05.004\">https://doi.org/10.1016/j.jctb.2021.05.004</a>.","apa":"Pach, J., &#38; Tomon, I. (2021). Erdős-Hajnal-type results for monotone paths. <i>Journal of Combinatorial Theory. Series B</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jctb.2021.05.004\">https://doi.org/10.1016/j.jctb.2021.05.004</a>","ieee":"J. Pach and I. Tomon, “Erdős-Hajnal-type results for monotone paths,” <i>Journal of Combinatorial Theory. Series B</i>, vol. 151. Elsevier, pp. 21–37, 2021."},"date_updated":"2025-04-15T07:16:52Z","intvolume":"       151","doi":"10.1016/j.jctb.2021.05.004","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"scopus_import":"1","publisher":"Elsevier","oa":1},{"project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"name":"Persistent Homology, Algorithms and Stochastic Geometry","grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"}],"year":"2021","status":"public","quality_controlled":"1","publication_identifier":{"isbn":["9783959771849"],"issn":["1868-8969"]},"language":[{"iso":"eng"}],"month":"06","has_accepted_license":"1","ec_funded":1,"_id":"9604","date_created":"2021-06-27T22:01:48Z","file":[{"access_level":"open_access","date_updated":"2021-06-28T13:11:39Z","content_type":"application/pdf","file_id":"9611","relation":"main_file","file_size":727817,"creator":"asandaue","file_name":"2021_LIPIcs_Biswas.pdf","checksum":"22b11a719018b22ecba2471b51f2eb40","date_created":"2021-06-28T13:11:39Z","success":1}],"date_published":"2021-06-02T00:00:00Z","department":[{"_id":"HeEd"}],"ddc":["516"],"title":"Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory","oa_version":"Published Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","citation":{"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.","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>","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>.","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.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, in:, Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.","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>","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>."},"alternative_title":["LIPIcs"],"article_number":"16","type":"conference","publication":"Leibniz International Proceedings in Informatics","volume":189,"scopus_import":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","oa":1,"date_updated":"2025-07-10T12:01:56Z","intvolume":"       189","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"doi":"10.4230/LIPIcs.SoCG.2021.16","author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita","first_name":"Ranita","orcid":"0000-0002-5372-7890","last_name":"Biswas"},{"full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","orcid":"0000-0001-6249-0832","last_name":"Cultrera di Montesano"},{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"last_name":"Saghafian","first_name":"Morteza","full_name":"Saghafian, Morteza"}],"conference":{"name":"SoCG: International Symposium on Computational Geometry","location":"Online","start_date":"2021-06-07","end_date":"2021-06-11"},"publication_status":"published","day":"02","file_date_updated":"2021-06-28T13:11:39Z","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2021.16","abstract":[{"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.","lang":"eng"}]},{"_id":"9605","has_accepted_license":"1","month":"06","language":[{"iso":"eng"}],"publication_identifier":{"issn":["1868-8969"],"isbn":["9783959771849"]},"status":"public","quality_controlled":"1","year":"2021","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["2103.07823"]},"related_material":{"link":[{"relation":"extended_version","url":"https://arxiv.org/abs/2103.07823"}],"record":[{"status":"public","relation":"later_version","id":"12709"}]},"oa_version":"Published Version","acknowledgement":"The authors want to thank the reviewers for many helpful comments and suggestions.","title":"Computing the multicover bifiltration","ddc":["516"],"department":[{"_id":"HeEd"}],"date_published":"2021-06-02T00:00:00Z","date_created":"2021-06-27T22:01:49Z","file":[{"file_id":"9610","content_type":"application/pdf","date_updated":"2021-06-28T12:40:47Z","access_level":"open_access","relation":"main_file","file_size":"1367983","file_name":"2021_LIPIcs_Corbet.pdf","creator":"cziletti","success":1,"date_created":"2021-06-28T12:40:47Z","checksum":"0de217501e7ba8b267d58deed0d51761"}],"doi":"10.4230/LIPIcs.SoCG.2021.27","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"intvolume":"       189","date_updated":"2025-07-10T12:01:57Z","oa":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","scopus_import":"1","publication":"Leibniz International Proceedings in Informatics","volume":189,"type":"conference","article_number":"27","alternative_title":["LIPIcs"],"citation":{"apa":"Corbet, R., Kerber, M., Lesnick, M., &#38; Osang, G. F. (2021). Computing the multicover bifiltration. 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.2021.27\">https://doi.org/10.4230/LIPIcs.SoCG.2021.27</a>","ieee":"R. Corbet, M. Kerber, M. Lesnick, and G. F. Osang, “Computing the multicover bifiltration,” in <i>Leibniz International Proceedings in Informatics</i>, Online, 2021, vol. 189.","mla":"Corbet, René, et al. “Computing the Multicover Bifiltration.” <i>Leibniz International Proceedings in Informatics</i>, vol. 189, 27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.27\">10.4230/LIPIcs.SoCG.2021.27</a>.","ama":"Corbet R, Kerber M, Lesnick M, Osang GF. Computing the multicover bifiltration. 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.2021.27\">10.4230/LIPIcs.SoCG.2021.27</a>","short":"R. Corbet, M. Kerber, M. Lesnick, G.F. Osang, in:, Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.","ista":"Corbet R, Kerber M, Lesnick M, Osang GF. 2021. Computing the multicover bifiltration. Leibniz International Proceedings in Informatics. SoCG: International Symposium on Computational Geometry, LIPIcs, vol. 189, 27.","chicago":"Corbet, René, Michael Kerber, Michael Lesnick, and Georg F Osang. “Computing the Multicover Bifiltration.” 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.2021.27\">https://doi.org/10.4230/LIPIcs.SoCG.2021.27</a>."},"abstract":[{"lang":"eng","text":"Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness. "}],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2021.27","file_date_updated":"2021-06-28T12:40:47Z","conference":{"start_date":"2021-06-07","end_date":"2021-06-11","name":"SoCG: International Symposium on Computational Geometry","location":"Online"},"publication_status":"published","arxiv":1,"day":"02","author":[{"full_name":"Corbet, René","first_name":"René","last_name":"Corbet"},{"first_name":"Michael","full_name":"Kerber, Michael","last_name":"Kerber"},{"full_name":"Lesnick, Michael","first_name":"Michael","last_name":"Lesnick"},{"last_name":"Osang","orcid":"0000-0002-8882-5116","first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","full_name":"Osang, Georg F"}]},{"has_accepted_license":"1","month":"07","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1932-6203"]},"_id":"9821","status":"public","quality_controlled":"1","year":"2021","isi":1,"external_id":{"isi":["000678124900050"],"pmid":["34292957"]},"oa_version":"Published Version","title":"Persistent homology as a new method of the assessment of heart rate variability","acknowledgement":"We express our gratitude to the anonymous referees who provided constructive comments that helped us improve the quality of the paper.","article_processing_charge":"Yes","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","date_published":"2021-07-01T00:00:00Z","date_created":"2021-08-08T22:01:28Z","file":[{"creator":"asandaue","file_name":"2021_PLoSONE_Graff.pdf","checksum":"0277aa155d5db1febd2cb384768bba5f","success":1,"date_created":"2021-08-09T09:25:41Z","access_level":"open_access","file_id":"9832","date_updated":"2021-08-09T09:25:41Z","content_type":"application/pdf","file_size":2706919,"relation":"main_file"}],"ddc":["006"],"department":[{"_id":"HeEd"}],"oa":1,"pmid":1,"publisher":"Public Library of Science","scopus_import":"1","doi":"10.1371/journal.pone.0253851","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"date_updated":"2026-04-02T13:56:42Z","intvolume":"        16","article_number":"e0253851","citation":{"mla":"Graff, Grzegorz, et al. “Persistent Homology as a New Method of the Assessment of Heart Rate Variability.” <i>PLoS ONE</i>, vol. 16, no. 7, e0253851, Public Library of Science, 2021, doi:<a href=\"https://doi.org/10.1371/journal.pone.0253851\">10.1371/journal.pone.0253851</a>.","ama":"Graff G, Graff B, Pilarczyk P, Jablonski G, Gąsecki D, Narkiewicz K. Persistent homology as a new method of the assessment of heart rate variability. <i>PLoS ONE</i>. 2021;16(7). doi:<a href=\"https://doi.org/10.1371/journal.pone.0253851\">10.1371/journal.pone.0253851</a>","ista":"Graff G, Graff B, Pilarczyk P, Jablonski G, Gąsecki D, Narkiewicz K. 2021. Persistent homology as a new method of the assessment of heart rate variability. PLoS ONE. 16(7), e0253851.","short":"G. Graff, B. Graff, P. Pilarczyk, G. Jablonski, D. Gąsecki, K. Narkiewicz, PLoS ONE 16 (2021).","chicago":"Graff, Grzegorz, Beata Graff, Pawel Pilarczyk, Grzegorz Jablonski, Dariusz Gąsecki, and Krzysztof Narkiewicz. “Persistent Homology as a New Method of the Assessment of Heart Rate Variability.” <i>PLoS ONE</i>. Public Library of Science, 2021. <a href=\"https://doi.org/10.1371/journal.pone.0253851\">https://doi.org/10.1371/journal.pone.0253851</a>.","apa":"Graff, G., Graff, B., Pilarczyk, P., Jablonski, G., Gąsecki, D., &#38; Narkiewicz, K. (2021). Persistent homology as a new method of the assessment of heart rate variability. <i>PLoS ONE</i>. Public Library of Science. <a href=\"https://doi.org/10.1371/journal.pone.0253851\">https://doi.org/10.1371/journal.pone.0253851</a>","ieee":"G. Graff, B. Graff, P. Pilarczyk, G. Jablonski, D. Gąsecki, and K. Narkiewicz, “Persistent homology as a new method of the assessment of heart rate variability,” <i>PLoS ONE</i>, vol. 16, no. 7. Public Library of Science, 2021."},"issue":"7","volume":16,"publication":"PLoS ONE","type":"journal_article","day":"01","publication_status":"published","abstract":[{"text":"Heart rate variability (hrv) is a physiological phenomenon of the variation in the length of the time interval between consecutive heartbeats. In many cases it could be an indicator of the development of pathological states. The classical approach to the analysis of hrv includes time domain methods and frequency domain methods. However, attempts are still being made to define new and more effective hrv assessment tools. Persistent homology is a novel data analysis tool developed in the recent decades that is rooted at algebraic topology. The Topological Data Analysis (TDA) approach focuses on examining the shape of the data in terms of connectedness and holes, and has recently proved to be very effective in various fields of research. In this paper we propose the use of persistent homology to the hrv analysis. We recall selected topological descriptors used in the literature and we introduce some new topological descriptors that reflect the specificity of hrv, and we discuss their relation to the standard hrv measures. In particular, we show that this novel approach provides a collection of indices that might be at least as useful as the classical parameters in differentiating between series of beat-to-beat intervals (RR-intervals) in healthy subjects and patients suffering from a stroke episode.","lang":"eng"}],"fulldoi":"https://doi.org/10.1371/journal.pone.0253851","file_date_updated":"2021-08-09T09:25:41Z","article_type":"original","author":[{"last_name":"Graff","first_name":"Grzegorz","full_name":"Graff, Grzegorz"},{"first_name":"Beata","full_name":"Graff, Beata","last_name":"Graff"},{"last_name":"Pilarczyk","id":"3768D56A-F248-11E8-B48F-1D18A9856A87","full_name":"Pilarczyk, Pawel","first_name":"Pawel"},{"first_name":"Grzegorz","full_name":"Jablonski, Grzegorz","id":"4483EF78-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-3536-9866","last_name":"Jablonski"},{"full_name":"Gąsecki, Dariusz","first_name":"Dariusz","last_name":"Gąsecki"},{"full_name":"Narkiewicz, Krzysztof","first_name":"Krzysztof","last_name":"Narkiewicz"}]},{"author":[{"last_name":"Čomić","full_name":"Čomić, Lidija","first_name":"Lidija"},{"full_name":"Zrour, Rita","first_name":"Rita","last_name":"Zrour"},{"last_name":"Largeteau-Skapin","first_name":"Gaëlle","full_name":"Largeteau-Skapin, Gaëlle"},{"orcid":"0000-0002-5372-7890","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita","last_name":"Biswas"},{"first_name":"Eric","full_name":"Andres, Eric","last_name":"Andres"}],"publication_status":"published","day":"16","conference":{"name":"DGMM: International Conference on Discrete Geometry and Mathematical Morphology","location":"Uppsala, Sweden","start_date":"2021-05-24","end_date":"2021-05-27"},"fulldoi":"https://doi.org/10.1007/978-3-030-76657-3_10","abstract":[{"text":"We define a new compact coordinate system in which each integer triplet addresses a voxel in the BCC grid, and we investigate some of its properties. We propose a characterization of 3D discrete analytical planes with their topological features (in the Cartesian and in the new coordinate system) such as the interrelation between the thickness of the plane and the separability constraint we aim to obtain.","lang":"eng"}],"alternative_title":["LNCS"],"citation":{"apa":"Čomić, L., Zrour, R., Largeteau-Skapin, G., Biswas, R., &#38; Andres, E. (2021). Body centered cubic grid - coordinate system and discrete analytical plane definition. In <i>Discrete Geometry and Mathematical Morphology</i> (Vol. 12708, pp. 152–163). Uppsala, Sweden: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-030-76657-3_10\">https://doi.org/10.1007/978-3-030-76657-3_10</a>","ieee":"L. Čomić, R. Zrour, G. Largeteau-Skapin, R. Biswas, and E. Andres, “Body centered cubic grid - coordinate system and discrete analytical plane definition,” in <i>Discrete Geometry and Mathematical Morphology</i>, Uppsala, Sweden, 2021, vol. 12708, pp. 152–163.","short":"L. Čomić, R. Zrour, G. Largeteau-Skapin, R. Biswas, E. Andres, in:, Discrete Geometry and Mathematical Morphology, Springer Nature, 2021, pp. 152–163.","ista":"Čomić L, Zrour R, Largeteau-Skapin G, Biswas R, Andres E. 2021. Body centered cubic grid - coordinate system and discrete analytical plane definition. Discrete Geometry and Mathematical Morphology. DGMM: International Conference on Discrete Geometry and Mathematical Morphology, LNCS, vol. 12708, 152–163.","ama":"Čomić L, Zrour R, Largeteau-Skapin G, Biswas R, Andres E. Body centered cubic grid - coordinate system and discrete analytical plane definition. In: <i>Discrete Geometry and Mathematical Morphology</i>. Vol 12708. Springer Nature; 2021:152-163. doi:<a href=\"https://doi.org/10.1007/978-3-030-76657-3_10\">10.1007/978-3-030-76657-3_10</a>","mla":"Čomić, Lidija, et al. “Body Centered Cubic Grid - Coordinate System and Discrete Analytical Plane Definition.” <i>Discrete Geometry and Mathematical Morphology</i>, vol. 12708, Springer Nature, 2021, pp. 152–63, doi:<a href=\"https://doi.org/10.1007/978-3-030-76657-3_10\">10.1007/978-3-030-76657-3_10</a>.","chicago":"Čomić, Lidija, Rita Zrour, Gaëlle Largeteau-Skapin, Ranita Biswas, and Eric Andres. “Body Centered Cubic Grid - Coordinate System and Discrete Analytical Plane Definition.” In <i>Discrete Geometry and Mathematical Morphology</i>, 12708:152–63. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/978-3-030-76657-3_10\">https://doi.org/10.1007/978-3-030-76657-3_10</a>."},"volume":12708,"publication":"Discrete Geometry and Mathematical Morphology","type":"conference","publisher":"Springer Nature","scopus_import":"1","intvolume":"     12708","date_updated":"2026-04-16T09:26:30Z","doi":"10.1007/978-3-030-76657-3_10","date_created":"2021-08-08T22:01:29Z","date_published":"2021-05-16T00:00:00Z","department":[{"_id":"HeEd"}],"oa_version":"None","acknowledgement":"This work has been partially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project no. 451-03-68/2020-14/200156: “Innovative scientific and artistic research from the FTS (activity) domain” (LČ), the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183 (RB), and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35 (RB).","title":"Body centered cubic grid - coordinate system and discrete analytical plane definition","isi":1,"external_id":{"isi":["001286400400010"]},"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","article_processing_charge":"No","year":"2021","status":"public","quality_controlled":"1","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"page":"152-163","language":[{"iso":"eng"}],"publication_identifier":{"issn":["0302-9743"],"eissn":["1611-3349"],"isbn":["9783030766566"]},"month":"05","ec_funded":1,"_id":"9824"},{"publication_identifier":{"issn":["1744-683X"],"eissn":["1744-6848"]},"language":[{"iso":"eng"}],"month":"10","has_accepted_license":"1","ec_funded":1,"_id":"10204","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"}],"quality_controlled":"1","year":"2021","status":"public","page":"9107-9115","acknowledgement":"MS acknowledges the support by Australian Research Council funding through the ARC Training Centre for M3D Innovation (IC180100008). MS thanks M. Hanifpour and N. Francois for their input and valuable discussions. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, grant no. 788183 and from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","title":"Topological signatures and stability of hexagonal close packing and Barlow stackings","oa_version":"Submitted Version","external_id":{"isi":["000700090000001"],"pmid":["34569592"]},"isi":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","date_created":"2021-10-31T23:01:30Z","file":[{"access_level":"open_access","file_id":"14385","date_updated":"2023-10-03T09:21:42Z","content_type":"application/pdf","file_size":4678788,"relation":"main_file","creator":"dernst","file_name":"2021_SoftMatter_acceptedversion_Osang.pdf","checksum":"b4da0c420530295e61b153960f6cb350","success":1,"date_created":"2023-10-03T09:21:42Z"}],"date_published":"2021-10-20T00:00:00Z","department":[{"_id":"HeEd"}],"ddc":["540"],"publisher":"Royal Society of Chemistry","scopus_import":"1","pmid":1,"oa":1,"intvolume":"        17","date_updated":"2026-07-06T13:55:05Z","doi":"10.1039/d1sm00774b","citation":{"chicago":"Osang, Georg F, Herbert Edelsbrunner, and Mohammad Saadatfar. “Topological Signatures and Stability of Hexagonal Close Packing and Barlow Stackings.” <i>Soft Matter</i>. Royal Society of Chemistry, 2021. <a href=\"https://doi.org/10.1039/d1sm00774b\">https://doi.org/10.1039/d1sm00774b</a>.","ista":"Osang GF, Edelsbrunner H, Saadatfar M. 2021. Topological signatures and stability of hexagonal close packing and Barlow stackings. Soft Matter. 17(40), 9107–9115.","short":"G.F. Osang, H. Edelsbrunner, M. Saadatfar, Soft Matter 17 (2021) 9107–9115.","mla":"Osang, Georg F., et al. “Topological Signatures and Stability of Hexagonal Close Packing and Barlow Stackings.” <i>Soft Matter</i>, vol. 17, no. 40, Royal Society of Chemistry, 2021, pp. 9107–15, doi:<a href=\"https://doi.org/10.1039/d1sm00774b\">10.1039/d1sm00774b</a>.","ama":"Osang GF, Edelsbrunner H, Saadatfar M. Topological signatures and stability of hexagonal close packing and Barlow stackings. <i>Soft Matter</i>. 2021;17(40):9107-9115. doi:<a href=\"https://doi.org/10.1039/d1sm00774b\">10.1039/d1sm00774b</a>","ieee":"G. F. Osang, H. Edelsbrunner, and M. Saadatfar, “Topological signatures and stability of hexagonal close packing and Barlow stackings,” <i>Soft Matter</i>, vol. 17, no. 40. Royal Society of Chemistry, pp. 9107–9115, 2021.","apa":"Osang, G. F., Edelsbrunner, H., &#38; Saadatfar, M. (2021). Topological signatures and stability of hexagonal close packing and Barlow stackings. <i>Soft Matter</i>. Royal Society of Chemistry. <a href=\"https://doi.org/10.1039/d1sm00774b\">https://doi.org/10.1039/d1sm00774b</a>"},"type":"journal_article","publication":"Soft Matter","volume":17,"das_tickbox":"1","issue":"40","day":"20","publication_status":"published","file_date_updated":"2023-10-03T09:21:42Z","fulldoi":"https://doi.org/10.1039/d1sm00774b","abstract":[{"lang":"eng","text":"Two common representations of close packings of identical spheres consisting of hexagonal layers, called Barlow stackings, appear abundantly in minerals and metals. These motifs, however, occupy an identical portion of space and bear identical first-order topological signatures as measured by persistent homology. Here we present a novel method based on k-fold covers that unambiguously distinguishes between these patterns. Moreover, our approach provides topological evidence that the FCC motif is the more stable of the two in the context of evolving experimental sphere packings during the transition from disordered to an ordered state. We conclude that our approach can be generalised to distinguish between various Barlow stackings manifested in minerals and metals."}],"article_type":"original","author":[{"last_name":"Osang","full_name":"Osang, Georg F","first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8882-5116"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"last_name":"Saadatfar","first_name":"Mohammad","full_name":"Saadatfar, Mohammad"}]},{"das_tickbox":"1","type":"journal_article","publication":"Experimental Mathematics","citation":{"chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, and Anton Nikitenko. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” <i>Experimental Mathematics</i>. Taylor &#38; Francis, 2021. <a href=\"https://doi.org/10.1080/10586458.2021.1980459\">https://doi.org/10.1080/10586458.2021.1980459</a>.","ama":"Akopyan A, Edelsbrunner H, Nikitenko A. The beauty of random polytopes inscribed in the 2-sphere. <i>Experimental Mathematics</i>. 2021:1-15. doi:<a href=\"https://doi.org/10.1080/10586458.2021.1980459\">10.1080/10586458.2021.1980459</a>","mla":"Akopyan, Arseniy, et al. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” <i>Experimental Mathematics</i>, Taylor &#38; Francis, 2021, pp. 1–15, doi:<a href=\"https://doi.org/10.1080/10586458.2021.1980459\">10.1080/10586458.2021.1980459</a>.","short":"A. Akopyan, H. Edelsbrunner, A. Nikitenko, Experimental Mathematics (2021) 1–15.","ista":"Akopyan A, Edelsbrunner H, Nikitenko A. 2021. The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics., 1–15.","ieee":"A. Akopyan, H. Edelsbrunner, and A. Nikitenko, “The beauty of random polytopes inscribed in the 2-sphere,” <i>Experimental Mathematics</i>. Taylor &#38; Francis, pp. 1–15, 2021.","apa":"Akopyan, A., Edelsbrunner, H., &#38; Nikitenko, A. (2021). The beauty of random polytopes inscribed in the 2-sphere. <i>Experimental Mathematics</i>. Taylor &#38; Francis. <a href=\"https://doi.org/10.1080/10586458.2021.1980459\">https://doi.org/10.1080/10586458.2021.1980459</a>"},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"doi":"10.1080/10586458.2021.1980459","date_updated":"2026-07-07T05:33:35Z","oa":1,"publisher":"Taylor & Francis","scopus_import":"1","author":[{"orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","full_name":"Akopyan, Arseniy","last_name":"Akopyan"},{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"orcid":"0000-0002-0659-3201","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","last_name":"Nikitenko"}],"article_type":"original","abstract":[{"lang":"eng","text":"Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density."}],"file_date_updated":"2023-08-14T11:55:10Z","corr_author":"1","fulldoi":"https://doi.org/10.1080/10586458.2021.1980459","publication_status":"published","day":"25","arxiv":1,"page":"1-15","project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887","name":"Persistent Homology, Algorithms and Stochastic Geometry"},{"grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes"}],"status":"public","quality_controlled":"1","year":"2021","_id":"10222","ec_funded":1,"month":"10","has_accepted_license":"1","publication_identifier":{"eissn":["1944-950X"],"issn":["1058-6458"]},"language":[{"iso":"eng"}],"ddc":["510"],"department":[{"_id":"HeEd"}],"date_published":"2021-10-25T00:00:00Z","file":[{"checksum":"3514382e3a1eb87fa6c61ad622874415","date_created":"2023-08-14T11:55:10Z","success":1,"creator":"dernst","file_name":"2023_ExperimentalMath_Akopyan.pdf","relation":"main_file","file_size":1966019,"access_level":"open_access","date_updated":"2023-08-14T11:55:10Z","content_type":"application/pdf","file_id":"14053"}],"date_created":"2021-11-07T23:01:25Z","article_processing_charge":"Yes (via OA deal)","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["2007.07783"],"isi":["000710893500001"]},"isi":1,"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.\r\nWe are grateful to Dmitry Zaporozhets and Christoph Thäle for valuable comments and for directing us to relevant references. We also thank to Anton Mellit for a useful discussion on Bessel functions.","title":"The beauty of random polytopes inscribed in the 2-sphere","oa_version":"Published Version"},{"oa_version":"Published Version","related_material":{"record":[{"id":"12960","relation":"later_version","status":"public"}]},"acknowledgement":"We thank Dominique Attali, Guilherme de Fonseca, Arijit Ghosh, Vincent Pilaud and Aurélien Alvarez for their comments and suggestions. We also acknowledge the reviewers.","title":"Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","place":"Dagstuhl, Germany","date_created":"2021-06-02T10:10:55Z","file":[{"date_created":"2021-06-02T10:22:33Z","success":1,"checksum":"c322aa48d5d35a35877896cc565705b6","file_name":"LIPIcs-SoCG-2021-17.pdf","creator":"mwintrae","relation":"main_file","file_size":1972902,"date_updated":"2021-06-02T10:22:33Z","content_type":"application/pdf","file_id":"9442","access_level":"open_access"}],"date_published":"2021-06-02T00:00:00Z","series_title":"Leibniz International Proceedings in Informatics (LIPIcs)","department":[{"_id":"HeEd"}],"ddc":["005","516","514"],"language":[{"iso":"eng"}],"publication_identifier":{"isbn":["978-3-95977-184-9"],"issn":["1868-8969"]},"has_accepted_license":"1","month":"06","ec_funded":1,"_id":"9441","year":"2021","status":"public","quality_controlled":"1","project":[{"name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411"}],"page":"17:1-17:16","publication_status":"published","conference":{"location":"Virtual","name":"SoCG: Symposium on Computational Geometry","end_date":"2021-06-11","start_date":"2021-06-07"},"day":"02","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2021.17","file_date_updated":"2021-06-02T10:22:33Z","abstract":[{"text":"Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. ","lang":"eng"}],"author":[{"last_name":"Boissonnat","first_name":"Jean-Daniel","full_name":"Boissonnat, Jean-Daniel"},{"full_name":"Kachanovich, Siargey","first_name":"Siargey","last_name":"Kachanovich"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220","last_name":"Wintraecken"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","scopus_import":"1","oa":1,"intvolume":"       189","date_updated":"2026-07-07T13:43:40Z","doi":"10.4230/LIPIcs.SoCG.2021.17","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"alternative_title":["LIPIcs"],"citation":{"ieee":"J.-D. Boissonnat, S. Kachanovich, and M. Wintraecken, “Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations,” in <i>37th International Symposium on Computational Geometry</i>, Virtual, 2021, vol. 189, p. 17:1-17:16.","apa":"Boissonnat, J.-D., Kachanovich, S., &#38; Wintraecken, M. (2021). Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations. In <i>37th International Symposium on Computational Geometry</i> (Vol. 189, p. 17:1-17:16). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.17\">https://doi.org/10.4230/LIPIcs.SoCG.2021.17</a>","chicago":"Boissonnat, Jean-Daniel, Siargey Kachanovich, and Mathijs Wintraecken. “Tracing Isomanifolds in Rd in Time Polynomial in d Using Coxeter-Freudenthal-Kuhn Triangulations.” In <i>37th International Symposium on Computational Geometry</i>, 189:17:1-17:16. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.17\">https://doi.org/10.4230/LIPIcs.SoCG.2021.17</a>.","ama":"Boissonnat J-D, Kachanovich S, Wintraecken M. Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations. In: <i>37th International Symposium on Computational Geometry</i>. Vol 189. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021:17:1-17:16. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.17\">10.4230/LIPIcs.SoCG.2021.17</a>","mla":"Boissonnat, Jean-Daniel, et al. “Tracing Isomanifolds in Rd in Time Polynomial in d Using Coxeter-Freudenthal-Kuhn Triangulations.” <i>37th International Symposium on Computational Geometry</i>, vol. 189, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, p. 17:1-17:16, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.17\">10.4230/LIPIcs.SoCG.2021.17</a>.","ista":"Boissonnat J-D, Kachanovich S, Wintraecken M. 2021. Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations. 37th International Symposium on Computational Geometry. SoCG: Symposium on Computational GeometryLeibniz International Proceedings in Informatics (LIPIcs), LIPIcs, vol. 189, 17:1-17:16.","short":"J.-D. Boissonnat, S. Kachanovich, M. Wintraecken, in:, 37th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2021, p. 17:1-17:16."},"volume":189,"publication":"37th International Symposium on Computational Geometry","type":"conference","das_tickbox":"1"},{"main_file_link":[{"url":"https://arxiv.org/abs/1910.09917v3","open_access":"1"}],"year":"2021","quality_controlled":"1","status":"public","project":[{"name":"Mathematics, Computer Science","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"}],"_id":"8317","month":"02","publication_identifier":{"eissn":["1879-081X"],"issn":["0925-7721"]},"language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"date_published":"2021-02-01T00:00:00Z","date_created":"2020-08-30T22:01:09Z","article_processing_charge":"No","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","isi":1,"external_id":{"isi":["000579185100004"],"arxiv":["1910.09917"]},"related_material":{"record":[{"id":"6989","status":"public","relation":"shorter_version"}]},"oa_version":"Preprint","acknowledgement":"This research was performed in part at the 33rd Bellairs Winter Workshop on Computational Geometry. We thank all other participants for a fruitful atmosphere. H. Akitaya was supported by NSF CCF-1422311 & 1423615. Z. Masárová was partially funded by Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","title":"Folding polyominoes with holes into a cube","publication":"Computational Geometry: Theory and Applications","volume":93,"type":"journal_article","article_number":"101700","citation":{"apa":"Aichholzer, O., Akitaya, H. A., Cheung, K. C., Demaine, E. D., Demaine, M. L., Fekete, S. P., … Schmidt, C. (2021). Folding polyominoes with holes into a cube. <i>Computational Geometry: Theory and Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2020.101700\">https://doi.org/10.1016/j.comgeo.2020.101700</a>","ieee":"O. Aichholzer <i>et al.</i>, “Folding polyominoes with holes into a cube,” <i>Computational Geometry: Theory and Applications</i>, vol. 93. Elsevier, 2021.","ama":"Aichholzer O, Akitaya HA, Cheung KC, et al. Folding polyominoes with holes into a cube. <i>Computational Geometry: Theory and Applications</i>. 2021;93. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2020.101700\">10.1016/j.comgeo.2020.101700</a>","mla":"Aichholzer, Oswin, et al. “Folding Polyominoes with Holes into a Cube.” <i>Computational Geometry: Theory and Applications</i>, vol. 93, 101700, Elsevier, 2021, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2020.101700\">10.1016/j.comgeo.2020.101700</a>.","ista":"Aichholzer O, Akitaya HA, Cheung KC, Demaine ED, Demaine ML, Fekete SP, Kleist L, Kostitsyna I, Löffler M, Masárová Z, Mundilova K, Schmidt C. 2021. Folding polyominoes with holes into a cube. Computational Geometry: Theory and Applications. 93, 101700.","short":"O. Aichholzer, H.A. Akitaya, K.C. Cheung, E.D. Demaine, M.L. Demaine, S.P. Fekete, L. Kleist, I. Kostitsyna, M. Löffler, Z. Masárová, K. Mundilova, C. Schmidt, Computational Geometry: Theory and Applications 93 (2021).","chicago":"Aichholzer, Oswin, Hugo A. Akitaya, Kenneth C. Cheung, Erik D. Demaine, Martin L. Demaine, Sándor P. Fekete, Linda Kleist, et al. “Folding Polyominoes with Holes into a Cube.” <i>Computational Geometry: Theory and Applications</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.comgeo.2020.101700\">https://doi.org/10.1016/j.comgeo.2020.101700</a>."},"doi":"10.1016/j.comgeo.2020.101700","date_updated":"2026-07-28T13:08:48Z","intvolume":"        93","oa":1,"publisher":"Elsevier","scopus_import":"1","author":[{"full_name":"Aichholzer, Oswin","first_name":"Oswin","last_name":"Aichholzer"},{"last_name":"Akitaya","full_name":"Akitaya, Hugo A.","first_name":"Hugo A."},{"last_name":"Cheung","first_name":"Kenneth C.","full_name":"Cheung, Kenneth C."},{"first_name":"Erik D.","full_name":"Demaine, Erik D.","last_name":"Demaine"},{"last_name":"Demaine","first_name":"Martin L.","full_name":"Demaine, Martin L."},{"last_name":"Fekete","full_name":"Fekete, Sándor P.","first_name":"Sándor P."},{"first_name":"Linda","full_name":"Kleist, Linda","last_name":"Kleist"},{"full_name":"Kostitsyna, Irina","first_name":"Irina","last_name":"Kostitsyna"},{"first_name":"Maarten","full_name":"Löffler, Maarten","last_name":"Löffler"},{"first_name":"Zuzana","id":"45CFE238-F248-11E8-B48F-1D18A9856A87","full_name":"Masárová, Zuzana","orcid":"0000-0002-6660-1322","last_name":"Masárová"},{"last_name":"Mundilova","first_name":"Klara","full_name":"Mundilova, Klara"},{"last_name":"Schmidt","full_name":"Schmidt, Christiane","first_name":"Christiane"}],"article_type":"original","abstract":[{"text":"When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.","lang":"eng"}],"fulldoi":"https://doi.org/10.1016/j.comgeo.2020.101700","corr_author":"1","arxiv":1,"day":"01","publication_status":"published"},{"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"doi":"10.4230/LIPIcs.SoCG.2021.32","intvolume":"       189","date_updated":"2026-10-02T11:24:33Z","oa":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","scopus_import":"1","das_tickbox":"1","type":"conference","publication":"37th International Symposium on Computational Geometry","volume":189,"citation":{"short":"H. Edelsbrunner, T. Heiss, V.  Kurlin , P. Smith, M. Wintraecken, in:, 37th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, p. 32:1-32:16.","ista":"Edelsbrunner H, Heiss T,  Kurlin  V, Smith P, Wintraecken M. 2021. The density fingerprint of a periodic point set. 37th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 189, 32:1-32:16.","mla":"Edelsbrunner, Herbert, et al. “The Density Fingerprint of a Periodic Point Set.” <i>37th International Symposium on Computational Geometry</i>, vol. 189, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, p. 32:1-32:16, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">10.4230/LIPIcs.SoCG.2021.32</a>.","ama":"Edelsbrunner H, Heiss T,  Kurlin  V, Smith P, Wintraecken M. The density fingerprint of a periodic point set. In: <i>37th International Symposium on Computational Geometry</i>. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021:32:1-32:16. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">10.4230/LIPIcs.SoCG.2021.32</a>","chicago":"Edelsbrunner, Herbert, Teresa Heiss, Vitaliy  Kurlin , Philip Smith, and Mathijs Wintraecken. “The Density Fingerprint of a Periodic Point Set.” In <i>37th International Symposium on Computational Geometry</i>, 189:32:1-32:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">https://doi.org/10.4230/LIPIcs.SoCG.2021.32</a>.","apa":"Edelsbrunner, H., Heiss, T.,  Kurlin , V., Smith, P., &#38; Wintraecken, M. (2021). The density fingerprint of a periodic point set. In <i>37th International Symposium on Computational Geometry</i> (Vol. 189, p. 32:1-32:16). Virtual: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2021.32\">https://doi.org/10.4230/LIPIcs.SoCG.2021.32</a>","ieee":"H. Edelsbrunner, T. Heiss, V.  Kurlin , P. Smith, and M. Wintraecken, “The density fingerprint of a periodic point set,” in <i>37th International Symposium on Computational Geometry</i>, Virtual, 2021, vol. 189, p. 32:1-32:16."},"alternative_title":["LIPIcs"],"abstract":[{"text":"Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction.","lang":"eng"}],"file_date_updated":"2021-04-22T08:08:14Z","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2021.32","day":"02","conference":{"start_date":"2021-06-07","end_date":"2021-06-11","name":"SoCG: Symposium on Computational Geometry","location":"Virtual"},"publication_status":"published","author":[{"last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"orcid":"0000-0002-1780-2689","full_name":"Heiss, Teresa","first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","last_name":"Heiss"},{"last_name":" Kurlin ","full_name":" Kurlin , Vitaliy","first_name":"Vitaliy"},{"last_name":"Smith","full_name":"Smith, Philip","first_name":"Philip"},{"last_name":"Wintraecken","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","full_name":"Wintraecken, Mathijs","orcid":"0000-0002-7472-2220"}],"_id":"9345","ec_funded":1,"month":"06","has_accepted_license":"1","publication_identifier":{"issn":["1868-8969"]},"language":[{"iso":"eng"}],"page":"32:1-32:16","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887","name":"Persistent Homology, Algorithms and Stochastic Geometry"},{"call_identifier":"FWF","_id":"25C5A090-B435-11E9-9278-68D0E5697425","grant_number":"Z00312","name":"Synaptic communication in neuronal microcircuits"},{"_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"status":"public","year":"2021","quality_controlled":"1","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"The authors thank Janos Pach for insightful discussions on the topic of thispaper, Morteza Saghafian for finding the one-dimensional counterexample mentioned in Section 5,and Larry Andrews for generously sharing his crystallographic perspective.","title":"The density fingerprint of a periodic point set","related_material":{"record":[{"id":"18667","relation":"dissertation_contains","status":"public"}]},"oa_version":"Published Version","ddc":["004","516"],"department":[{"_id":"HeEd"}],"date_published":"2021-06-02T00:00:00Z","file":[{"creator":"mwintrae","file_name":"df_socg_final_version.pdf","checksum":"1787baef1523d6d93753b90d0c109a6d","date_created":"2021-04-22T08:08:14Z","success":1,"access_level":"open_access","date_updated":"2021-04-22T08:08:14Z","content_type":"application/pdf","file_id":"9346","file_size":3117435,"relation":"main_file"}],"date_created":"2021-04-22T08:09:58Z"},{"_id":"10867","month":"02","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1687-0247"],"issn":["1073-7928"]},"main_file_link":[{"url":"https://arxiv.org/abs/1702.07513","open_access":"1"}],"page":"669-697","quality_controlled":"1","status":"public","year":"2020","article_processing_charge":"No","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","isi":1,"external_id":{"isi":["000522852700002"],"arxiv":["1702.07513"]},"oa_version":"Preprint","title":"Waist of balls in hyperbolic and spherical spaces","acknowledgement":" Supported by the Russian Foundation for Basic Research grant 18-01-00036.","department":[{"_id":"HeEd"}],"date_published":"2020-02-01T00:00:00Z","date_created":"2022-03-18T11:39:30Z","doi":"10.1093/imrn/rny037","date_updated":"2023-08-24T14:19:55Z","intvolume":"      2020","oa":1,"scopus_import":"1","publisher":"Oxford University Press","issue":"3","publication":"International Mathematics Research Notices","volume":2020,"type":"journal_article","citation":{"short":"A. Akopyan, R. Karasev, International Mathematics Research Notices 2020 (2020) 669–697.","ista":"Akopyan A, Karasev R. 2020. Waist of balls in hyperbolic and spherical spaces. International Mathematics Research Notices. 2020(3), 669–697.","mla":"Akopyan, Arseniy, and Roman Karasev. “Waist of Balls in Hyperbolic and Spherical Spaces.” <i>International Mathematics Research Notices</i>, vol. 2020, no. 3, Oxford University Press, 2020, pp. 669–97, doi:<a href=\"https://doi.org/10.1093/imrn/rny037\">10.1093/imrn/rny037</a>.","ama":"Akopyan A, Karasev R. Waist of balls in hyperbolic and spherical spaces. <i>International Mathematics Research Notices</i>. 2020;2020(3):669-697. doi:<a href=\"https://doi.org/10.1093/imrn/rny037\">10.1093/imrn/rny037</a>","chicago":"Akopyan, Arseniy, and Roman Karasev. “Waist of Balls in Hyperbolic and Spherical Spaces.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2020. <a href=\"https://doi.org/10.1093/imrn/rny037\">https://doi.org/10.1093/imrn/rny037</a>.","apa":"Akopyan, A., &#38; Karasev, R. (2020). Waist of balls in hyperbolic and spherical spaces. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rny037\">https://doi.org/10.1093/imrn/rny037</a>","ieee":"A. Akopyan and R. Karasev, “Waist of balls in hyperbolic and spherical spaces,” <i>International Mathematics Research Notices</i>, vol. 2020, no. 3. Oxford University Press, pp. 669–697, 2020."},"abstract":[{"text":"In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces.","lang":"eng"}],"fulldoi":"https://doi.org/10.1093/imrn/rny037","arxiv":1,"publication_status":"published","keyword":["General Mathematics"],"day":"01","author":[{"last_name":"Akopyan","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy"},{"last_name":"Karasev","full_name":"Karasev, Roman","first_name":"Roman"}],"article_type":"original"},{"page":"455-480","year":"2020","status":"public","quality_controlled":"1","_id":"15064","language":[{"iso":"eng"}],"publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"month":"12","has_accepted_license":"1","department":[{"_id":"HeEd"}],"ddc":["500"],"file":[{"checksum":"eed1168b6e66cd55272c19bb7fca8a1c","date_created":"2024-03-04T10:52:42Z","success":1,"creator":"dernst","file_name":"2020_JourApplCompTopology_Bauer.pdf","file_size":851190,"relation":"main_file","access_level":"open_access","date_updated":"2024-03-04T10:52:42Z","content_type":"application/pdf","file_id":"15065"}],"date_created":"2024-03-04T10:47:49Z","date_published":"2020-12-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"Yes (via OA deal)","acknowledgement":"This research has been supported by the DFG Collaborative Research Center SFB/TRR 109 “Discretization in Geometry and Dynamics”, by Polish MNiSzW Grant No. 2621/7.PR/12/2013/2, by the Polish National Science Center under Maestro Grant No. 2014/14/A/ST1/00453 and Grant No. DEC-2013/09/N/ST6/02995. Open Access funding provided by Projekt DEAL.","title":"Čech-Delaunay gradient flow and homology inference for self-maps","oa_version":"Published Version","type":"journal_article","volume":4,"publication":"Journal of Applied and Computational Topology","issue":"4","citation":{"ieee":"U. Bauer, H. Edelsbrunner, G. Jablonski, and M. Mrozek, “Čech-Delaunay gradient flow and homology inference for self-maps,” <i>Journal of Applied and Computational Topology</i>, vol. 4, no. 4. Springer Nature, pp. 455–480, 2020.","apa":"Bauer, U., Edelsbrunner, H., Jablonski, G., &#38; Mrozek, M. (2020). Čech-Delaunay gradient flow and homology inference for self-maps. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-020-00058-8\">https://doi.org/10.1007/s41468-020-00058-8</a>","chicago":"Bauer, U., Herbert Edelsbrunner, Grzegorz Jablonski, and M. Mrozek. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s41468-020-00058-8\">https://doi.org/10.1007/s41468-020-00058-8</a>.","short":"U. Bauer, H. Edelsbrunner, G. Jablonski, M. Mrozek, Journal of Applied and Computational Topology 4 (2020) 455–480.","ista":"Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. 2020. Čech-Delaunay gradient flow and homology inference for self-maps. Journal of Applied and Computational Topology. 4(4), 455–480.","mla":"Bauer, U., et al. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.” <i>Journal of Applied and Computational Topology</i>, vol. 4, no. 4, Springer Nature, 2020, pp. 455–80, doi:<a href=\"https://doi.org/10.1007/s41468-020-00058-8\">10.1007/s41468-020-00058-8</a>.","ama":"Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. Čech-Delaunay gradient flow and homology inference for self-maps. <i>Journal of Applied and Computational Topology</i>. 2020;4(4):455-480. doi:<a href=\"https://doi.org/10.1007/s41468-020-00058-8\">10.1007/s41468-020-00058-8</a>"},"date_updated":"2024-03-04T10:54:04Z","intvolume":"         4","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"doi":"10.1007/s41468-020-00058-8","scopus_import":"1","publisher":"Springer Nature","oa":1,"author":[{"last_name":"Bauer","full_name":"Bauer, U.","first_name":"U."},{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert"},{"last_name":"Jablonski","orcid":"0000-0002-3536-9866","id":"4483EF78-F248-11E8-B48F-1D18A9856A87","full_name":"Jablonski, Grzegorz","first_name":"Grzegorz"},{"first_name":"M.","full_name":"Mrozek, M.","last_name":"Mrozek"}],"article_type":"original","file_date_updated":"2024-03-04T10:52:42Z","fulldoi":"https://doi.org/10.1007/s41468-020-00058-8","abstract":[{"text":"We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspaces of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.","lang":"eng"}],"publication_status":"published","day":"01"},{"citation":{"apa":"Choudhary, A., Kachanovich, S., &#38; Wintraecken, M. (2020). Coxeter triangulations have good quality. <i>Mathematics in Computer Science</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11786-020-00461-5\">https://doi.org/10.1007/s11786-020-00461-5</a>","ieee":"A. Choudhary, S. Kachanovich, and M. Wintraecken, “Coxeter triangulations have good quality,” <i>Mathematics in Computer Science</i>, vol. 14. Springer Nature, pp. 141–176, 2020.","ista":"Choudhary A, Kachanovich S, Wintraecken M. 2020. Coxeter triangulations have good quality. Mathematics in Computer Science. 14, 141–176.","short":"A. Choudhary, S. Kachanovich, M. Wintraecken, Mathematics in Computer Science 14 (2020) 141–176.","mla":"Choudhary, Aruni, et al. “Coxeter Triangulations Have Good Quality.” <i>Mathematics in Computer Science</i>, vol. 14, Springer Nature, 2020, pp. 141–76, doi:<a href=\"https://doi.org/10.1007/s11786-020-00461-5\">10.1007/s11786-020-00461-5</a>.","ama":"Choudhary A, Kachanovich S, Wintraecken M. Coxeter triangulations have good quality. <i>Mathematics in Computer Science</i>. 2020;14:141-176. doi:<a href=\"https://doi.org/10.1007/s11786-020-00461-5\">10.1007/s11786-020-00461-5</a>","chicago":"Choudhary, Aruni, Siargey Kachanovich, and Mathijs Wintraecken. “Coxeter Triangulations Have Good Quality.” <i>Mathematics in Computer Science</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s11786-020-00461-5\">https://doi.org/10.1007/s11786-020-00461-5</a>."},"type":"journal_article","volume":14,"publication":"Mathematics in Computer Science","oa":1,"publisher":"Springer Nature","scopus_import":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"doi":"10.1007/s11786-020-00461-5","date_updated":"2025-04-14T07:44:03Z","intvolume":"        14","article_type":"original","author":[{"last_name":"Choudhary","full_name":"Choudhary, Aruni","first_name":"Aruni"},{"last_name":"Kachanovich","full_name":"Kachanovich, Siargey","first_name":"Siargey"},{"last_name":"Wintraecken","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87"}],"day":"01","publication_status":"published","abstract":[{"text":"Coxeter triangulations are triangulations of Euclidean space based on a single simplex. By this we mean that given an individual simplex we can recover the entire triangulation of Euclidean space by inductively reflecting in the faces of the simplex. In this paper we establish that the quality of the simplices in all Coxeter triangulations is O(1/d−−√) of the quality of regular simplex. We further investigate the Delaunay property for these triangulations. Moreover, we consider an extension of the Delaunay property, namely protection, which is a measure of non-degeneracy of a Delaunay triangulation. In particular, one family of Coxeter triangulations achieves the protection O(1/d2). We conjecture that both bounds are optimal for triangulations in Euclidean space.","lang":"eng"}],"corr_author":"1","file_date_updated":"2020-11-20T10:18:02Z","fulldoi":"https://doi.org/10.1007/s11786-020-00461-5","project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411"}],"status":"public","year":"2020","quality_controlled":"1","page":"141-176","month":"03","has_accepted_license":"1","publication_identifier":{"eissn":["1661-8289"],"issn":["1661-8270"]},"language":[{"iso":"eng"}],"_id":"7567","ec_funded":1,"date_published":"2020-03-01T00:00:00Z","date_created":"2020-03-05T13:30:18Z","file":[{"file_name":"2020_MathCompScie_Choudhary.pdf","creator":"dernst","date_created":"2020-11-20T10:18:02Z","success":1,"checksum":"1d145f3ab50ccee735983cb89236e609","content_type":"application/pdf","date_updated":"2020-11-20T10:18:02Z","file_id":"8783","access_level":"open_access","relation":"main_file","file_size":872275}],"ddc":["510"],"department":[{"_id":"HeEd"}],"title":"Coxeter triangulations have good quality","oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"external_id":{"isi":["000520918800001"]},"isi":1,"acknowledgement":"This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF).","title":"Tri-partitions and bases of an ordered complex","oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_published":"2020-03-20T00:00:00Z","date_created":"2020-04-19T22:00:56Z","file":[{"date_created":"2020-11-20T13:22:21Z","success":1,"checksum":"f8cc96e497f00c38340b5dafe0cb91d7","file_name":"2020_DiscreteCompGeo_Edelsbrunner.pdf","creator":"dernst","relation":"main_file","file_size":701673,"content_type":"application/pdf","date_updated":"2020-11-20T13:22:21Z","file_id":"8786","access_level":"open_access"}],"ddc":["510"],"department":[{"_id":"HeEd"}],"month":"03","has_accepted_license":"1","language":[{"iso":"eng"}],"publication_identifier":{"issn":["01795376"],"eissn":["14320444"]},"_id":"7666","ec_funded":1,"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"quality_controlled":"1","year":"2020","status":"public","page":"759-775","day":"20","publication_status":"published","abstract":[{"lang":"eng","text":"Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups."}],"file_date_updated":"2020-11-20T13:22:21Z","corr_author":"1","fulldoi":"https://doi.org/10.1007/s00454-020-00188-x","article_type":"original","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"orcid":"0000-0002-4672-8297","full_name":"Ölsböck, Katharina","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","first_name":"Katharina","last_name":"Ölsböck"}],"oa":1,"scopus_import":"1","publisher":"Springer Nature","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"doi":"10.1007/s00454-020-00188-x","intvolume":"        64","date_updated":"2025-04-14T07:48:36Z","citation":{"ieee":"H. Edelsbrunner and K. Ölsböck, “Tri-partitions and bases of an ordered complex,” <i>Discrete and Computational Geometry</i>, vol. 64. Springer Nature, pp. 759–775, 2020.","apa":"Edelsbrunner, H., &#38; Ölsböck, K. (2020). Tri-partitions and bases of an ordered complex. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00188-x\">https://doi.org/10.1007/s00454-020-00188-x</a>","chicago":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s00454-020-00188-x\">https://doi.org/10.1007/s00454-020-00188-x</a>.","ista":"Edelsbrunner H, Ölsböck K. 2020. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 64, 759–775.","short":"H. Edelsbrunner, K. Ölsböck, Discrete and Computational Geometry 64 (2020) 759–775.","ama":"Edelsbrunner H, Ölsböck K. Tri-partitions and bases of an ordered complex. <i>Discrete and Computational Geometry</i>. 2020;64:759-775. doi:<a href=\"https://doi.org/10.1007/s00454-020-00188-x\">10.1007/s00454-020-00188-x</a>","mla":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” <i>Discrete and Computational Geometry</i>, vol. 64, Springer Nature, 2020, pp. 759–75, doi:<a href=\"https://doi.org/10.1007/s00454-020-00188-x\">10.1007/s00454-020-00188-x</a>."},"type":"journal_article","publication":"Discrete and Computational Geometry","volume":64},{"doi":"10.15479/AT:ISTA:7944","tmp":{"name":"Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0)","image":"/images/cc_by_sa.png","legal_code_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","short":"CC BY-SA (4.0)"},"date_updated":"2026-04-08T07:23:01Z","degree_awarded":"PhD","oa":1,"publisher":"Institute of Science and Technology Austria","type":"dissertation","alternative_title":["ISTA Thesis"],"citation":{"ista":"Masárová Z. 2020. Reconfiguration problems. Institute of Science and Technology Austria.","short":"Z. Masárová, Reconfiguration Problems, Institute of Science and Technology Austria, 2020.","ama":"Masárová Z. Reconfiguration problems. 2020. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:7944\">10.15479/AT:ISTA:7944</a>","mla":"Masárová, Zuzana. <i>Reconfiguration Problems</i>. Institute of Science and Technology Austria, 2020, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:7944\">10.15479/AT:ISTA:7944</a>.","chicago":"Masárová, Zuzana. “Reconfiguration Problems.” Institute of Science and Technology Austria, 2020. <a href=\"https://doi.org/10.15479/AT:ISTA:7944\">https://doi.org/10.15479/AT:ISTA:7944</a>.","apa":"Masárová, Z. (2020). <i>Reconfiguration problems</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:7944\">https://doi.org/10.15479/AT:ISTA:7944</a>","ieee":"Z. Masárová, “Reconfiguration problems,” Institute of Science and Technology Austria, 2020."},"abstract":[{"text":"This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph.\r\n\r\nFor triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton.\r\n\r\nIn the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars.","lang":"eng"}],"fulldoi":"https://doi.org/10.15479/AT:ISTA:7944","corr_author":"1","supervisor":[{"last_name":"Wagner","orcid":"0000-0002-1494-0568","full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli"},{"last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"}],"OA_place":"publisher","file_date_updated":"2020-07-14T12:48:05Z","day":"09","publication_status":"published","keyword":["reconfiguration","reconfiguration graph","triangulations","flip","constrained triangulations","shellability","piecewise-linear balls","token swapping","trees","coloured weighted token swapping"],"author":[{"last_name":"Masárová","id":"45CFE238-F248-11E8-B48F-1D18A9856A87","first_name":"Zuzana","full_name":"Masárová, Zuzana","orcid":"0000-0002-6660-1322"}],"_id":"7944","has_accepted_license":"1","month":"06","language":[{"iso":"eng"}],"publication_identifier":{"isbn":["978-3-99078-005-3"],"issn":["2663-337X"]},"page":"160","license":"https://creativecommons.org/licenses/by-sa/4.0/","status":"public","year":"2020","article_processing_charge":"No","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","oa_version":"Published Version","related_material":{"record":[{"id":"7950","status":"public","relation":"part_of_dissertation"},{"id":"5986","relation":"part_of_dissertation","status":"public"}]},"title":"Reconfiguration problems","ddc":["516","514"],"department":[{"_id":"HeEd"},{"_id":"UlWa"}],"date_published":"2020-06-09T00:00:00Z","file":[{"creator":"zmasarov","file_name":"THESIS_Zuzka_Masarova.pdf","checksum":"df688bc5a82b50baee0b99d25fc7b7f0","date_created":"2020-06-08T00:34:00Z","access_level":"open_access","file_id":"7945","content_type":"application/pdf","date_updated":"2020-07-14T12:48:05Z","file_size":13661779,"relation":"main_file"},{"creator":"zmasarov","file_name":"THESIS_Zuzka_Masarova_SOURCE_FILES.zip","checksum":"45341a35b8f5529c74010b7af43ac188","date_created":"2020-06-08T00:35:30Z","access_level":"closed","content_type":"application/zip","date_updated":"2020-07-14T12:48:05Z","file_id":"7946","file_size":32184006,"relation":"source_file"}],"date_created":"2020-06-08T00:49:46Z"}]