Computing homology and persistent homology using iterated Morse   decomposition
  In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph theoretical operations. We use iterated Morse decomposition, which allows us to sidetrack many problems related to the standard discrete Morse theory. In particular, this approach is provably correct in any dimension. 
Pawe{\l} D{\l}otko, Hubert Wagner
math.AT cs.DS