[{"acknowledgement":"This research is partially supported by ESF under the ACAT Research Network Programme, and by the Russian Government under mega project 11.G34.31.0053","oa":1,"publisher":"ACM","quality_controlled":"1","publication":"Proceedings of the Annual Symposium on Computational Geometry","day":"01","year":"2014","date_created":"2018-12-11T11:56:01Z","date_published":"2014-06-01T00:00:00Z","doi":"10.1145/2582112.2582167","page":"484 - 490","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"318493","name":"Topological Complex Systems"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","citation":{"apa":"Bauer, U., & Edelsbrunner, H. (2014). The morse theory of Čech and Delaunay filtrations. In Proceedings of the Annual Symposium on Computational Geometry (pp. 484–490). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582167","ama":"Bauer U, Edelsbrunner H. The morse theory of Čech and Delaunay filtrations. In: Proceedings of the Annual Symposium on Computational Geometry. ACM; 2014:484-490. doi:10.1145/2582112.2582167","short":"U. Bauer, H. Edelsbrunner, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 484–490.","ieee":"U. Bauer and H. Edelsbrunner, “The morse theory of Čech and Delaunay filtrations,” in Proceedings of the Annual Symposium on Computational Geometry, Kyoto, Japan, 2014, pp. 484–490.","mla":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Filtrations.” Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 484–90, doi:10.1145/2582112.2582167.","ista":"Bauer U, Edelsbrunner H. 2014. The morse theory of Čech and Delaunay filtrations. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 484–490.","chicago":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Filtrations.” In Proceedings of the Annual Symposium on Computational Geometry, 484–90. ACM, 2014. https://doi.org/10.1145/2582112.2582167."},"title":"The morse theory of Čech and Delaunay filtrations","author":[{"last_name":"Bauer","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"publist_id":"4851","oa_version":"Submitted Version","abstract":[{"lang":"eng","text":"Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Copyright is held by the owner/author(s)."}],"month":"06","main_file_link":[{"url":"http://arxiv.org/abs/1312.1231","open_access":"1"}],"scopus_import":1,"language":[{"iso":"eng"}],"publication_status":"published","ec_funded":1,"_id":"2155","status":"public","conference":{"name":"SoCG: Symposium on Computational Geometry","end_date":"2014-06-11","location":"Kyoto, Japan","start_date":"2014-06-08"},"type":"conference","date_updated":"2021-01-12T06:55:38Z","department":[{"_id":"HeEd"}]},{"status":"public","type":"conference","conference":{"start_date":"2014-01-05","end_date":"2014-01-07","location":"Portland, USA","name":"SODA: Symposium on Discrete Algorithms"},"_id":"2177","department":[{"_id":"HeEd"}],"title":"On the computational complexity of betti numbers reductions from matrix rank","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"first_name":"Salman","id":"4BDBD4F2-F248-11E8-B48F-1D18A9856A87","last_name":"Parsa","full_name":"Parsa, Salman"}],"publist_id":"4805","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","date_updated":"2021-01-12T06:55:48Z","citation":{"apa":"Edelsbrunner, H., & Parsa, S. (2014). On the computational complexity of betti numbers reductions from matrix rank. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 152–160). Portland, USA: SIAM. https://doi.org/10.1137/1.9781611973402.11","ama":"Edelsbrunner H, Parsa S. On the computational complexity of betti numbers reductions from matrix rank. In: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM; 2014:152-160. doi:10.1137/1.9781611973402.11","ieee":"H. Edelsbrunner and S. Parsa, “On the computational complexity of betti numbers reductions from matrix rank,” in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, Portland, USA, 2014, pp. 152–160.","short":"H. Edelsbrunner, S. Parsa, in:, Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2014, pp. 152–160.","mla":"Edelsbrunner, Herbert, and Salman Parsa. “On the Computational Complexity of Betti Numbers Reductions from Matrix Rank.” Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2014, pp. 152–60, doi:10.1137/1.9781611973402.11.","ista":"Edelsbrunner H, Parsa S. 2014. On the computational complexity of betti numbers reductions from matrix rank. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms, 152–160.","chicago":"Edelsbrunner, Herbert, and Salman Parsa. “On the Computational Complexity of Betti Numbers Reductions from Matrix Rank.” In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, 152–60. SIAM, 2014. https://doi.org/10.1137/1.9781611973402.11."},"month":"01","publisher":"SIAM","scopus_import":1,"quality_controlled":"1","oa_version":"None","abstract":[{"text":"We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with to non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times to simplices. Together with the known reduction in the other direction, this implies that the two problems have the same computational complexity.","lang":"eng"}],"date_published":"2014-01-01T00:00:00Z","doi":"10.1137/1.9781611973402.11","date_created":"2018-12-11T11:56:09Z","page":"152 - 160","day":"01","language":[{"iso":"eng"}],"publication":"Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms","year":"2014","publication_status":"published"},{"department":[{"_id":"UlWa"},{"_id":"HeEd"}],"date_updated":"2021-01-12T06:55:50Z","type":"journal_article","status":"public","_id":"2184","issue":"3","volume":61,"publication_status":"published","language":[{"iso":"eng"}],"scopus_import":1,"main_file_link":[{"url":"http://arxiv.org/abs/1105.6257","open_access":"1"}],"month":"05","intvolume":" 61","abstract":[{"text":"Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X→ Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of suchmaps.We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d-2 and Y is (d-1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A→ Y and ask whether it extends to a map X → Y, or computing the Z2-index-everything in the stable range. Outside the stable range, the extension problem is undecidable.","lang":"eng"}],"oa_version":"Preprint","publist_id":"4797","author":[{"full_name":"Čadek, Martin","last_name":"Čadek","first_name":"Martin"},{"id":"33E21118-F248-11E8-B48F-1D18A9856A87","first_name":"Marek","last_name":"Krcál","full_name":"Krcál, Marek"},{"full_name":"Matoušek, Jiří","last_name":"Matoušek","first_name":"Jiří"},{"full_name":"Sergeraert, Francis","last_name":"Sergeraert","first_name":"Francis"},{"first_name":"Lukáš","full_name":"Vokřínek, Lukáš","last_name":"Vokřínek"},{"full_name":"Wagner, Uli","orcid":"0000-0002-1494-0568","last_name":"Wagner","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87"}],"title":"Computing all maps into a sphere","citation":{"mla":"Čadek, Martin, et al. “Computing All Maps into a Sphere.” Journal of the ACM, vol. 61, no. 3, 17, ACM, 2014, doi:10.1145/2597629.","apa":"Čadek, M., Krcál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., & Wagner, U. (2014). Computing all maps into a sphere. Journal of the ACM. ACM. https://doi.org/10.1145/2597629","ama":"Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. Computing all maps into a sphere. Journal of the ACM. 2014;61(3). doi:10.1145/2597629","ieee":"M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, and U. Wagner, “Computing all maps into a sphere,” Journal of the ACM, vol. 61, no. 3. ACM, 2014.","short":"M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, U. Wagner, Journal of the ACM 61 (2014).","chicago":"Čadek, Martin, Marek Krcál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, and Uli Wagner. “Computing All Maps into a Sphere.” Journal of the ACM. ACM, 2014. https://doi.org/10.1145/2597629.","ista":"Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. 2014. Computing all maps into a sphere. Journal of the ACM. 61(3), 17."},"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","article_number":"17 ","date_published":"2014-05-01T00:00:00Z","doi":"10.1145/2597629","date_created":"2018-12-11T11:56:12Z","year":"2014","day":"01","publication":"Journal of the ACM","publisher":"ACM","quality_controlled":"1","oa":1,"acknowledgement":"The research by M. K. was supported by project GAUK 49209. The research by M. K. was also supported by project 1M0545 by the Ministry of Education of the Czech Republic and by Center of Excellence { Inst. for Theor. Comput. Sci., Prague (project P202/12/G061 of GACR). The research by U. W. was supported by the Swiss National Science Foundation (SNF Projects 200021-125309, 200020-138230, and PP00P2-138948)."},{"title":"Persistent homology: Theory and practice","publist_id":"3842","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"full_name":"Morozovy, Dmitriy","last_name":"Morozovy","first_name":"Dmitriy"}],"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Edelsbrunner, Herbert, and Dmitriy Morozovy. Persistent Homology: Theory and Practice. European Mathematical Society Publishing House, 2014, pp. 31–50, doi:10.4171/120-1/3.","ieee":"H. Edelsbrunner and D. Morozovy, “Persistent homology: Theory and practice,” presented at the ECM: European Congress of Mathematics, Kraków, Poland, 2014, pp. 31–50.","short":"H. Edelsbrunner, D. Morozovy, in:, European Mathematical Society Publishing House, 2014, pp. 31–50.","apa":"Edelsbrunner, H., & Morozovy, D. (2014). Persistent homology: Theory and practice (pp. 31–50). Presented at the ECM: European Congress of Mathematics, Kraków, Poland: European Mathematical Society Publishing House. https://doi.org/10.4171/120-1/3","ama":"Edelsbrunner H, Morozovy D. Persistent homology: Theory and practice. In: European Mathematical Society Publishing House; 2014:31-50. doi:10.4171/120-1/3","chicago":"Edelsbrunner, Herbert, and Dmitriy Morozovy. “Persistent Homology: Theory and Practice,” 31–50. European Mathematical Society Publishing House, 2014. https://doi.org/10.4171/120-1/3.","ista":"Edelsbrunner H, Morozovy D. 2014. Persistent homology: Theory and practice. ECM: European Congress of Mathematics, 31–50."},"date_published":"2014-01-01T00:00:00Z","doi":"10.4171/120-1/3","date_created":"2018-12-11T12:00:16Z","page":"31 - 50","day":"01","has_accepted_license":"1","year":"2014","quality_controlled":"1","publisher":"European Mathematical Society Publishing House","oa":1,"acknowledgement":"This research is partially supported by NSF under grant DBI-0820624, by ESF under the Research Networking Programme, and by the Russian Government Project 11.G34.31.0053.","file_date_updated":"2020-07-14T12:45:52Z","department":[{"_id":"HeEd"}],"ddc":["000"],"date_updated":"2021-01-12T07:00:36Z","status":"public","pubrep_id":"544","type":"conference","conference":{"start_date":"2012-07-02","end_date":"2012-07-07","location":"Kraków, Poland","name":"ECM: European Congress of Mathematics"},"_id":"2905","file":[{"relation":"main_file","access_level":"open_access","content_type":"application/pdf","checksum":"1d4a046f1af945c407c5c4d411d4c5e4","file_id":"5232","creator":"system","file_size":435320,"date_updated":"2020-07-14T12:45:52Z","file_name":"IST-2016-544-v1+1_2012-P-11-PHTheoryPractice.pdf","date_created":"2018-12-12T10:16:43Z"}],"language":[{"iso":"eng"}],"publication_status":"published","month":"01","oa_version":"Submitted Version","abstract":[{"text":"Persistent homology is a recent grandchild of homology that has found use in\r\nscience and engineering as well as in mathematics. This paper surveys the method as well\r\nas the applications, neglecting completeness in favor of highlighting ideas and directions.","lang":"eng"}]},{"citation":{"mla":"Biedl, Therese, et al. “Planar Matchings for Weighted Straight Skeletons.” 25th International Symposium, ISAAC 2014, vol. 8889, Springer Nature, 2014, pp. 117–27, doi:10.1007/978-3-319-13075-0_10.","ieee":"T. Biedl, S. Huber, and P. Palfrader, “Planar matchings for weighted straight skeletons,” in 25th International Symposium, ISAAC 2014, Jeonju, Korea, 2014, vol. 8889, pp. 117–127.","short":"T. Biedl, S. Huber, P. Palfrader, in:, 25th International Symposium, ISAAC 2014, Springer Nature, 2014, pp. 117–127.","ama":"Biedl T, Huber S, Palfrader P. Planar matchings for weighted straight skeletons. In: 25th International Symposium, ISAAC 2014. Vol 8889. Springer Nature; 2014:117-127. doi:10.1007/978-3-319-13075-0_10","apa":"Biedl, T., Huber, S., & Palfrader, P. (2014). Planar matchings for weighted straight skeletons. In 25th International Symposium, ISAAC 2014 (Vol. 8889, pp. 117–127). Jeonju, Korea: Springer Nature. https://doi.org/10.1007/978-3-319-13075-0_10","chicago":"Biedl, Therese, Stefan Huber, and Peter Palfrader. “Planar Matchings for Weighted Straight Skeletons.” In 25th International Symposium, ISAAC 2014, 8889:117–27. Springer Nature, 2014. https://doi.org/10.1007/978-3-319-13075-0_10.","ista":"Biedl T, Huber S, Palfrader P. 2014. Planar matchings for weighted straight skeletons. 25th International Symposium, ISAAC 2014. ISAAC: International Symposium on Algorithms and Computation, LNCS, vol. 8889, 117–127."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"full_name":"Biedl, Therese","last_name":"Biedl","first_name":"Therese"},{"last_name":"Huber","orcid":"0000-0002-8871-5814","full_name":"Huber, Stefan","id":"4700A070-F248-11E8-B48F-1D18A9856A87","first_name":"Stefan"},{"last_name":"Palfrader","full_name":"Palfrader, Peter","first_name":"Peter"}],"article_processing_charge":"No","title":"Planar matchings for weighted straight skeletons","acknowledgement":"T. Biedl was supported by NSERC and the Ross and Muriel Cheriton Fellowship. P. Palfrader was supported by Austrian Science Fund (FWF): P25816-N15.","quality_controlled":"1","publisher":"Springer Nature","year":"2014","day":"08","publication":"25th International Symposium, ISAAC 2014","page":"117-127","doi":"10.1007/978-3-319-13075-0_10","date_published":"2014-11-08T00:00:00Z","date_created":"2022-03-21T07:09:03Z","_id":"10892","type":"conference","conference":{"name":"ISAAC: International Symposium on Algorithms and Computation","location":"Jeonju, Korea","end_date":"2014-12-17","start_date":"2014-12-15"},"status":"public","date_updated":"2023-02-23T12:20:55Z","department":[{"_id":"HeEd"}],"abstract":[{"text":"In this paper, we introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist.\r\nUsing our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.","lang":"eng"}],"oa_version":"None","scopus_import":"1","alternative_title":["LNCS"],"month":"11","intvolume":" 8889","publication_identifier":{"issn":["0302-9743"],"eissn":["1611-3349"],"isbn":["9783319130743"],"eisbn":["9783319130750"]},"publication_status":"published","language":[{"iso":"eng"}],"volume":8889,"related_material":{"record":[{"relation":"later_version","id":"481","status":"public"}]}}]