Homology algorithm based on acyclic subspace

Abstract

We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.

Computation of cubical homology, cohomology, and (co)homological operations via chain contraction

“…Although this is too expensive for processing large datasets that appear in practical applications, various reduction techniques help decrease the size of the data considerably without loss of homological information (see e.g. [43]), and thus achieve much better performance in practice. These reduction techniques may come from geometric representation of the data, or be derived from other heuristics.…”

Homological Computation Using Spanning Trees

“…To compute homology for a nD digital object (with n ≥ 3) is cubic in time with regards to the number n of cells [9,2,8]. Classical homology algorithms reduce the problem to Smith diagonalization, where the best available algorithms have supercubical complexity [12].…”

“…An alternative to these solutions are the reduction methods. They iteratively reduce the input data by a smaller one with the same homology, and compute the homolgy when no more reductions are possible [8,10].…”

“…To have at hand an algorithm computing homology in O(n) is one of the main challenge in this area and has been recently conjectured in [8].…”

Getting Topological Information for a 80-Adjacency Doxel-Based 4D Volume through a Polytopal Cell Complex

“…A boundary isosurface extraction algorithm can be derived from this framework. Different homology computation techniques [2,3,6,7,15,22] can be applied to K(V ). Starting from K(V ) and using vector fields [20] or spanning-like trees [24,21], an algorithm (based on configuration look-up table) for constructing a global homology operator and, hence, a 4D AT-model of V appears as a feasible task and will be our objective in a near future.…”