The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(n log n + n log g(log log g) 3 ) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O(n log n(log log n) 3 ) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O(n 2 ) algorithm. Finally, we present experimental results of our implementation and demonstrate that, in practice, our algorithm for 3-manifolds performs better than what the theoretical bound suggests.
Doraiswamy and Natarajan REF further extended the sweeping idea to handle 3-manifolds, and presented an O(m log m+m log g(log log g)
Efficient Algorithms for Computing Reeb Graphs
Comput. Geom.
Comput. Geom.
Mathematics Computer Science