{"scopus_import":1,"title":"Annotating simplices with a homology basis and its applications","alternative_title":["LNCS"],"oa_version":"Preprint","main_file_link":[{"url":"http://arxiv.org/abs/1107.3793","open_access":"1"}],"_id":"3129","status":"public","publist_id":"3569","oa":1,"quality_controlled":"1","day":"19","date_updated":"2021-01-12T07:41:15Z","language":[{"iso":"eng"}],"month":"06","author":[{"full_name":"Busaryev, Oleksiy","first_name":"Oleksiy","last_name":"Busaryev"},{"first_name":"Sergio","full_name":"Cabello, Sergio","last_name":"Cabello"},{"id":"3E92416E-F248-11E8-B48F-1D18A9856A87","last_name":"Chen","full_name":"Chen, Chao","first_name":"Chao"},{"last_name":"Dey","first_name":"Tamal","full_name":"Dey, Tamal"},{"last_name":"Wang","first_name":"Yusu","full_name":"Wang, Yusu"}],"date_created":"2018-12-11T12:01:33Z","publisher":"Springer","publication_status":"published","abstract":[{"text":"Let K be a simplicial complex and g the rank of its p-th homology group Hp(K) defined with ℤ2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each p-simplex of K with a binary vector of length g with the following property: the annotations, summed over all p-simplices in any p-cycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω ) time, where n is the size of K and ω &lt; 2.376 is a quantity so that two n×n matrices can be multiplied in O(n ω ) time. The precomputed annotations permit answering queries about the independence or the triviality of p-cycles efficiently.\r\n\r\nUsing annotations of edges in 2-complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1 - dimensional homology. Specifically, for computing an optimal basis of H1(K) , we improve the previously known time complexity from O(n 4) to O(n ω  + n 2 g ω − 1). Here n denotes the size of the 2-skeleton of K and g the rank of H1(K) . Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking 2 O(g) nlogn time is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes in O(n ω ) + 2 O(g) n 2logn time using annotations.\r\n","lang":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"date_published":"2012-06-19T00:00:00Z","external_id":{"arxiv":["1107.3793"]},"volume":7357,"type":"conference","conference":{"start_date":"2012-07-04","name":"SWAT: Symposium and Workshops on Algorithm Theory","location":"Helsinki, Finland","end_date":"2012-07-06"},"page":"189 - 200","doi":"10.1007/978-3-642-31155-0_17","intvolume":"      7357","year":"2012","citation":{"ista":"Busaryev O, Cabello S, Chen C, Dey T, Wang Y. 2012. Annotating simplices with a homology basis and its applications. SWAT: Symposium and Workshops on Algorithm Theory, LNCS, vol. 7357, 189–200.","short":"O. Busaryev, S. Cabello, C. Chen, T. Dey, Y. Wang, in:, Springer, 2012, pp. 189–200.","chicago":"Busaryev, Oleksiy, Sergio Cabello, Chao Chen, Tamal Dey, and Yusu Wang. “Annotating Simplices with a Homology Basis and Its Applications,” 7357:189–200. Springer, 2012. <a href=\"https://doi.org/10.1007/978-3-642-31155-0_17\">https://doi.org/10.1007/978-3-642-31155-0_17</a>.","ieee":"O. Busaryev, S. Cabello, C. Chen, T. Dey, and Y. Wang, “Annotating simplices with a homology basis and its applications,” presented at the SWAT: Symposium and Workshops on Algorithm Theory, Helsinki, Finland, 2012, vol. 7357, pp. 189–200.","apa":"Busaryev, O., Cabello, S., Chen, C., Dey, T., &#38; Wang, Y. (2012). Annotating simplices with a homology basis and its applications (Vol. 7357, pp. 189–200). Presented at the SWAT: Symposium and Workshops on Algorithm Theory, Helsinki, Finland: Springer. <a href=\"https://doi.org/10.1007/978-3-642-31155-0_17\">https://doi.org/10.1007/978-3-642-31155-0_17</a>","mla":"Busaryev, Oleksiy, et al. <i>Annotating Simplices with a Homology Basis and Its Applications</i>. Vol. 7357, Springer, 2012, pp. 189–200, doi:<a href=\"https://doi.org/10.1007/978-3-642-31155-0_17\">10.1007/978-3-642-31155-0_17</a>.","ama":"Busaryev O, Cabello S, Chen C, Dey T, Wang Y. Annotating simplices with a homology basis and its applications. In: Vol 7357. Springer; 2012:189-200. doi:<a href=\"https://doi.org/10.1007/978-3-642-31155-0_17\">10.1007/978-3-642-31155-0_17</a>"}}