# Intrinsic Interleaving Distance for Merge Trees

arXiv: https://arxiv.org/abs/1908.00063 · HTML: https://mlsanity.com/paper/1908.00063/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jul 2019.

## Built on by (most cited first, 5 of 5)

- [Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2207.10960/) (2022)
- [Sketching Merge Trees for Scientific Data Visualization](https://mlsanity.com/paper/2101.03196/) (2021)
- [Rapid and Precise Topological Comparison with Merge Tree Neural Networks](https://mlsanity.com/paper/2404.05879/) (2024)
- [Algebraic and Geometric Models for Space Networking](https://mlsanity.com/paper/2304.01150/) (2023)
- [MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport](https://mlsanity.com/paper/2606.08258/) (2026)

## Abstract (arXiv)

Merge trees are a type of graph-based topological summary that tracks the evolution of connected components in the sublevel sets of scalar functions. They enjoy widespread applications in data analysis and scientific visualization. In this paper, we consider the problem of comparing two merge trees via the notion of interleaving distance in the metric space setting. We investigate various theoretical properties of such a metric. In particular, we show that the interleaving distance is intrinsic on the space of labeled merge trees and provide an algorithm to construct metric 1-centers for collections of labeled merge trees. We further prove that the intrinsic property of the interleaving distance also holds for the space of unlabeled merge trees. Our results are a first step toward performing statistics on graph-based topological summaries.