Towards Computing Average Merge Tree Based on the Interleaving Distance

TL;DR

This paper introduces a method to compute an average merge tree based on the interleaving distance, providing a way to summarize and compare topological data structures of scalar functions.

Contribution

It defines and constructs a representative average merge tree using the interleaving distance, addressing non-uniqueness and proving its natural averaging properties.

Findings

Proposed a method to construct an average merge tree.

Proved the average merge tree satisfies natural averaging properties.

Included illustrative examples demonstrating the structure of the average merge tree.

Abstract

The interleaving distance is a key tool for comparing merge trees, which provide topological summaries of scalar functions. In this work, we define an average merge tree for a pair of merge trees using the interleaving distance. Since such an average is not unique, we propose a method to construct a representative average merge tree. We further prove that the resulting merge tree indeed satisfies a natural notion of averaging for the two given merge trees. To demonstrate the structure of the average merge tree, we include illustrative examples.