A Fast Algorithm for Computing Reeb Graph of 2-Manifold

The notion of Reeb graph is part of the Morse theory which provides an analysis of the relationship between the topology and the geometrical information of a space given by a suitable function, a Morse function. Until recently, the main drawback for using Reeb graph analysis was the incapacity of constructing adequate Morse functions. Before the work of Ni et al. [2004], the used Morse functions gave rise to many unnecessary critical points, making the construction and the analysis of the Reeb graph expensive. The work presented in [X. Ni and Hart 2004] now allows us to precompute Morse functions with just the required number of critical points in time linear relative to the size of the model. An algorithm which computes a Reeb graph in a time that depends on the number of critical points would thus be appreciated. This paper presents such an algorithm, which in most cases runs in O(n + m. log(m)) time, where n is the number of vertices and m the number of critical points.