# The Universal ℓp-Metric on Merge Trees

arXiv: https://arxiv.org/abs/2112.12165 · HTML: https://mlsanity.com/paper/2112.12165/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 53 references that are in the map.
- First posted on arXiv 22 Dec 2021.

## Builds on (most cited first, 20 of 20)

- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Hypothesis Testing for Topological Data Analysis](https://mlsanity.com/paper/1310.7467/) (2013)
- [Measuring Distance between Reeb Graphs](https://mlsanity.com/paper/1307.2839/) (2013)
- [Computing the Gromov-Hausdorff Distance for Metric Trees](https://mlsanity.com/paper/1509.05751/) (2015)
- [Metrics for generalized persistence modules](https://mlsanity.com/paper/1312.3829/) (2013)
- [Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem](https://mlsanity.com/paper/1610.10085/) (2016)
- [Reeb Graph Metrics from the Ground Up](https://mlsanity.com/paper/2110.05631/) (2021)
- [Comparative Analysis of Merge Trees using Local Tree Edit Distance](https://mlsanity.com/paper/2111.04382/) (2021)
- [The edit distance for Reeb graphs of surfaces](https://mlsanity.com/paper/1411.1544/) (2014)
- [Categorified Reeb Graphs](https://mlsanity.com/paper/1501.04147/) (2015)
- [The Reeb Graph Edit Distance is Universal](https://mlsanity.com/paper/1801.01866/) (2018)
- [FPT-algorithms for computing Gromov-Hausdorff and interleaving distances between trees](https://mlsanity.com/paper/1811.02425/) (2018)
- [On the persistent homology of almost surely C0 stochastic processes](https://mlsanity.com/paper/2012.09459/) (2020)
- [Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs](https://mlsanity.com/paper/1412.6646/) (2014)
- [Generalized Cluster Trees and Singular Measures](https://mlsanity.com/paper/1611.02762/) (2016)
- [Universality of the Homotopy Interleaving Distance](https://mlsanity.com/paper/1705.01690/) (2017)
- [The ℓ^∞-Cophenetic Metric for Phylogenetic Trees as an Interleaving Distance](https://mlsanity.com/paper/1803.07609/) (2018)
- [A family of metrics from the truncated smoothing of Reeb graphs](https://mlsanity.com/paper/2007.07795/) (2020)
- [On C0-persistent homology and trees](https://mlsanity.com/paper/2012.02634/) (2020)
- [Tight quasi-universality of Reeb graph distances](https://mlsanity.com/paper/2112.00720/) (2021)

## Abstract (arXiv)

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an ℓp-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the p-Wasserstein distance between the associated barcodes. For each p∈[1,∞], we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the p=∞ case, this gives a novel proof of universality for the interleaving distance on merge trees.