Triplet Merge Trees

Merge trees are fundamental data structures in computational topology. They track connected components in sublevel sets of scalar functions and can be used to compute 0-dimensional persistence diagrams, to construct contour trees on simply connected domains, and to quickly query the relationship between connected components in different sublevel sets. We introduce a representation of merge trees that tracks the nesting of their branches. We present algorithms to construct and manipulate the trees in this representation directly. We show that our algorithms are not only fast, outperforming Kruskal’s algorithm, but they are easy to parallelize in shared memory using double-word compare-and-swap operations. We present experiments that illustrate the scaling of our algorithms as functions of the data size and of the number of threads.