Annotating simplices with a homology basis and its applications
LNCS
Busaryev, Oleksiy
Cabello, Sergio
Chen, Chao
Dey, Tamal
Wang, Yusu
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 < 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.
Using 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.
Springer
2012
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doc-type:conferenceObject
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http://purl.org/coar/resource_type/c_5794
https://research-explorer.app.ist.ac.at/record/3129
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>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/978-3-642-31155-0_17
info:eu-repo/semantics/altIdentifier/arxiv/1107.3793
info:eu-repo/semantics/openAccess