Abstract
Let \({\cal K}\) be a simplicial complex and g the rank of its p-th homology group \({\sf H}_{p}({\cal K})\) defined with ℤ2 coefficients. We show that we can compute a basis H of \({\sf H}_{p}({\cal K})\) and annotate each p-simplex of \({\cal K}\) 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 \({\cal K}\) 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 \({\sf H}_{1}({\cal K})\), 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 \({\cal K}\) and g the rank of \({\sf H}_{1}({\cal K})\). Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking 2O(g) nlogn time is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes in O(n ω) + 2O(g) n 2logn time using annotations.
Research was partially supported by the Slovenian Research Agency, program P1-0297 and NSF grant CCF 1064416.
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Busaryev, O., Cabello, S., Chen, C., Dey, T.K., Wang, Y. (2012). Annotating Simplices with a Homology Basis and Its Applications. In: Fomin, F.V., Kaski, P. (eds) Algorithm Theory – SWAT 2012. SWAT 2012. Lecture Notes in Computer Science, vol 7357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31155-0_17
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DOI: https://doi.org/10.1007/978-3-642-31155-0_17
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