Annotating simplices with a homology basis and its applications

Let be a simplicial complex and g the rank of its p-th homology group defined with ℤ ₂ coefficients. We show that we can compute a basis H of and annotate each p-simplex of 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 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 , 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 and g the rank of . 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.