conference paper
Annotating simplices with a homology basis and its applications
LNCS
published
yes
Oleksiy
Busaryev
author
Sergio
Cabello
author
Chao
Chen
author 3E92416E-F248-11E8-B48F-1D18A9856A87
Tamal
Dey
author
Yusu
Wang
author
HeEd
department
SWAT: Symposium and Workshops on Algorithm Theory
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.
Springer2012Helsinki, Finland
eng
1107.379310.1007/978-3-642-31155-0_17
7357189 - 200
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>
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>.
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.
Busaryev, O., Cabello, S., Chen, C., Dey, T., & 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>
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>.
O. Busaryev, S. Cabello, C. Chen, T. Dey, Y. Wang, in:, Springer, 2012, pp. 189–200.
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.
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