Interleaving Distance between Merge Trees

Dmitriy Morozov, Kenes Beketayev, Gunther Weber.
Presented at TopoInVis'13.
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Merge trees are topological descriptors of scalar functions. They record how the subsets of the domain where the function value does not exceed a given threshold are connected. We define a distance between merge trees, called an interleaving distance, and prove the stability of these trees, with respect to this distance, to perturbations of the functions that define them. We show that the interleaving distance is never smaller than the bottleneck distance between persistence diagrams.