Annotating Simplices with a Homology Basis and Its Applications

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 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.

Research was partially supported by the Slovenian Research Agency, program P1-0297 and NSF grant CCF 1064416.