Applied Topology Seminar

Merge trees are a topological descriptor of a filtered space that enriches the degree zero barcode with its merge structure. The space of merge trees comes equipped with an interleaving distance d_I , which prompts a naive question: is the interleaving distance between two merge trees equal to the bottleneck distance between their corresponding barcodes? As the map from merge trees to barcodes is not injective, the answer as posed is no, but as proposed by Gasparovic et al., we explore intrinsic metrics d_I and d_B realized by infinitesimal path length in merge tree space, which do indeed coincide. This result suggests that in some special cases the bottleneck distance (which can be computed quickly) can be substituted for the interleaving distance (in general, NP-hard).