  Metrics for merge trees that are simultaneously stable, informative, and
efficiently computable have so far eluded researchers. We show in this work
that it is possible to devise such a metric when restricting merge trees to
ordered domains such as the interval and the circle. We present the ``dynamic
ordered persistence editing'' (DOPE) distance, which we prove is stable and
informative 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 accomplish
this by ignoring all of the hierarchical information of the merge tree and
simply focusing on a sequence of ordered critical points, which can be
interpreted as a time series. Thus our algorithm is more similar to string edit
distance and dynamic time warping than it is to more conventional merge tree
comparison algorithms. In the context of time series with the interval as a
domain, we show empirically on the UCR time series classification dataset that
DOPE performs better than bottleneck/Wasserstein distances between persistence
diagrams.
