Merging discrete Morse vector fields: a case of stubborn geometric parallelization

Abstract: We address the basic question in discrete Morse theory of combining discrete gradient fields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field $V$ is generated using a fixed algorithm which has a local nature. One example is ProcessLowerStars, a widely used algorithm for computing persistent homology associated to a grey-scale image in 2D or 3D. While the algorithm for $V$ may be inherently local, being computed within stars of vertices and so embarrassingly parallelizable, in practical use it is natural to want to distribute the computation over patches $P_{i}$, apply the chosen algorithm to compute the fields $V_{i}$ associated to each patch, and then assemble the ambient field $V$ from these. Simply merging the fields from the patches, even when that makes sense, gives a wrong answer. We develop both very general merging procedures and leaner versions designed for specific, easy to arrange covering patterns.