Annotating Simplices with a Homology Basis and Its Applications

Let K be a simplicial complex andgthe rank of itsp-th homology group Hp(K) defined with ℤ2coefficients. We show that we can compute a basisHof Hp(K) and annotate eachp-simplex of K with a binary vector of lengthgwith the following property: the annotations, summed over allp-simplices in anyp-cyclez, provide the coordinate vector of the homology class [z] in the basisH. The basis and the annotations for all simplices can be computed inO(n omega) time, wherenis the size of K andomega< 2.376 is a quantity so that twon*nmatrices can be multiplied inO(n omega) time. The precomputed annotations permit answering queries about the independence or the triviality ofp-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 K, we improve the previously known time complexity fromO(n4) toO(n omega+n2g omega 1). Herendenotes the size of the 2-skeleton of K andgthe rank of H1(K). Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking 2O(g)nlogntime is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes inO(n omega) + 2O(g)n2logntime using annotations.

让k成为一个简单的复合物，并用ℤ2Ceefficients定义的ITS-Th同源组HP（K）。所有简单都可以计算iNO（n欧米茄（Omega）的时间，k andomega <2.376的大小是一个数量，因此两个*nmatrices可以是多个INO（n omega）的时间，可以回答有关P-Cycles的琐事，以便在2-comple cymity中回答有关独立性的查询。特别是同源物。提高以前已知的时间复杂性（N4）（N Omega+N2G Omega 1），Heredenots k andg skelednots k and g and g and g and g s骨的级别（k）（K） Plexes Ino（N Omega） +使用注释2O（G）N2Logntime。