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Metrics for merge trees that are simultaneously stable, informative, andefficiently computable have so far eluded researchers. We show in this workthat it is possible to devise such a metric when restricting merge trees toordered domains such as the interval and the circle. We present the ``dynamicordered persistence editing'' (DOPE) distance, which we prove is stable andinformative while satisfying metric properties. We then devise a simple$O(N^2)$ dynamic programming algorithm to compute it on the interval and an$O(N^3)$ algorithm to compute it on the circle. Surprisingly, we accomplishthis by ignoring all of the hierarchical information of the merge tree andsimply focusing on a sequence of ordered critical points, which can beinterpreted as a time series. Thus our algorithm is more similar to string editdistance and dynamic time warping than it is to more conventional merge treecomparison algorithms. In the context of time series with the interval as adomain, we show empirically on the UCR time series classification dataset thatDOPE performs better than bottleneck/Wasserstein distances between persistencediagrams.

The DOPE Distance is SIC: A Stable, Informative, and Computable Metric on Time Series And Ordered Merge Trees