A Deformation-based Edit Distance for Merge Trees

TL;DR

This work introduces a deformation-based edit distance for merge trees that operates directly on the continuous objects and on their abstract representations. By defining continuous deformation retractions and equivalent abstract edits (relabel, contraction, inverse contraction) with a persistence-based cost, it derives path mappings that realize optimal edits and proves that the one-degree distance $\delta_{\text{1}}$ is a metric and branch-decomposition independent. A polynomial-time dynamic programming framework computes the distance, with a practical $\mathcal{O}(n^4)$ algorithm for binary trees and an open-source implementation. The method offers a robust alternative to branch-based distances, enabling metric-based analyses like geodesics or barycenters, while maintaining strong ties to intuitive geometric edits on merge trees.

Abstract

In scientific visualization, scalar fields are often compared through edit distances between their merge trees. Typical tasks include ensemble analysis, feature tracking and symmetry or periodicity detection. Tree edit distances represent how one tree can be transformed into another through a sequence of simple edit operations: relabeling, insertion and deletion of nodes. In this paper, we present a new set of edit operations working directly on the merge tree as an geometrical or topological object: the represented operations are deformation retractions and inverse transformations on merge trees, which stands in contrast to other methods working on branch decomposition trees. We present a quartic time algorithm for the new edit distance, which is branch decomposition-independent and a metric on the set of all merge trees.

Figures (7)

• Figure 1: Illustration of the difference between the classic discrete edit operations and the new continuous edit operations, exemplary for a deletion. If we apply a classic node-deletion or edge-contraction to the middle tree, we get the tree on the left. The result is not a merge tree. The desired result is shown in the right tree. Note that the problem cannot be fixed through deleting the remaining node with another classic edit operation, because a deletion always removes an edge as well.
• Figure 2: Deriving a path mapping from edit operations: If we delete the edge $(F,B)$ in the leftmost tree, this happens in two steps. First, we remove the edge and the leaf node $F$, then we merge the edges $(C,B)$ and $(B,A)$, since $B$ has degree one. The merging step induces a clear correspondence between the path $ABC$ and the edge $(C,A)$. This correspondence is transitive for multiple edit operations, hence, we can map the path $ABCD$ to the edge $(D,A)$ after deleting $(E,C)$. The mappings for insert operations can be derived in the same way.
• Figure 3: Three examples for valid and invalid path mappings. Mapped paths are indicated through the coloring. In (a), an invalid mapping is shown, which contradicts Lemma \ref{['lemma:simplePathProperty']}, since condition 3 is not fulfilled for the path $(B,F)$. A valid mapping can be seen in (b). The mapping in (c) fulfills the property in Lemma \ref{['lemma:simplePathProperty']}, but not Lemma \ref{['lemma:optPathMappingsAreAbstractMergeTrees']}, since the paths $(B,D)$ and $(D,E)$ can be merged to $(B,E)$ without increasing the cost.
• Figure 4: Exemplary illustration of recursive structure of the optimal path mapping of the trees rooted in $(n_1,p_1)$ and $(n_2,p_2)$, denoted by $d_P(n_1,p_1,n_2,p_2)$. In (a) and (b) we stop the currently tracked path (green) and start to track two new paths (blue and red) in both trees. Which of these are mapped is shown by the colors. In (c) we continue the current path in $T_1$ with the left child and delete the right subtree, while staying at the same node in $T_2$. We do the same recursion with left and right child swapped in (d).
• Figure 5: The clustermaps with dendograms for the outlier ensemble using the path mapping distance (left), the branch mapping distance (middle) and the classic constrained edit distance (right).

Theorems & Definitions (16)

• Definition 1
• Theorem 1
• proof
• Theorem 2
• Definition 2
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof