Residual Interleaving Distance

• Residual interleaving distance is a metric for merge trees that integrates global alignment with mandatory local constraints to identify fine-grained structural similarities.
• It interpolates between unconstrained bottleneck matching and fully-specified correspondences, offering a rigorous framework for stratified, locally optimal matchings.
• Dynamic programming techniques, enhanced from standard methods, efficiently compute this metric, making it valuable for applications like terrain evolution and hierarchical data analysis.

The residual interleaving distance is a generalization of the classical interleaving distance for merge trees, designed to capture not only global similarity but also the local fit between substructures. This metric quantifies the “slack” in an isomorphism between merge trees that respects both global alignment and imposed local constraints. By interpolating between unconstrained bottleneck matching and fully specified correspondences, the residual interleaving distance enables the detection of local similarities that the standard interleaving distance may obscure. Its formulation provides a rigorous foundation for stratified, locally optimal matchings critical for the comparative analysis of hierarchical structures in topological data analysis (Beurskens et al., 18 Dec 2025).

1. Merge Trees and Classical Interleaving Distance

A merge tree is a pair $(T, f)$, where $T$ is a finite rooted tree with unit-length edges, and $f: T \to \mathbb{R} \cup \{\infty\}$ is a continuous height function, strictly increasing from leaves toward the root, which is mapped to infinity. The shortest-path metric induces a natural notion of distance on $T$, and the ancestor relation $x \preceq y$ designates that $x$ is a descendant of $y$ along an $f$-monotone path.

The interleaving distance $d_I(T_1, T_2)$ measures similarity between two merge trees by the minimal vertical shift $\delta$ for which there exist two continuous, ancestry-preserving maps $\alpha: T_1 \to T_2$, $\beta: T_2 \to T_1$ such that:

• $f_2(\alpha(x)) = f_1(x) + \delta$ for all $x \in T_1$,
• $f_1(\beta(y)) = f_2(y) + \delta$ for $y \in T_2$,
• $$\beta \circ \alpha(x) = \#_1\{x, f_1(x) + 2\delta\}$$,
• $$\alpha \circ \beta(y) = \#_1\{y, f_2(y) + 2\delta\}$$,

where $\#_1\{x, h\}$ denotes the ancestor of $x$ at height $h \ge f(x)$.

$d_I(T_1, T_2)$ is realized as the smallest possible $\delta$ for which such maps exist, and always lies in a finite set of critical values derived from the node heights of the input trees (Beurskens et al., 18 Dec 2025).

2. Motivation and Definition of Residual Interleaving Distance

While the classical interleaving distance minimizes the largest required shift between matched points—yielding a global bottleneck measure—it can exaggerate the required deformation away from locally similar substructures. To enable finer-grained analysis, additional constraints specifying mandatory matches between certain nodes are imposed.

Given such a set of constraints, formalized as a finite partial interleaving $P = (\rho, \psi)$ (where $\rho: S_1 \to T_2$, $\psi: S_2 \to T_1$ for $S_1 \subseteq T_1$, $S_2 \subseteq T_2$), the residual interleaving distance $d_P(T_1, T_2)$ is the infimum of the maximal shift needed by any complete interleaving that extends $P$. This construction interpolates between the unconstrained case ($d_P = d_I$) and fully specified matchings (Beurskens et al., 18 Dec 2025).

For a complete interleaving $I = (\alpha, \beta)$ that extends $P$, the $P$-residual shift for an arrow $a = (x, y)$ is:

• $\xi_P(a) = 0$ if $a$ is within the fan $F[P]$ of $P$ (i.e., matches forced by ancestry),
• $\xi_P(a) = f_2(y) - f_1(x)$ otherwise.

The residual shift of $I$ is then $$\xi_P(I) = \max\{ \xi_P(a): a \in \text{arrows}(I) \}$$.

The residual interleaving distance is formally defined as:

$$d_P(T_1, T_2) = \inf \{ \xi_P(I) : I \text{ is a complete interleaving extending } P \}.$$

3. Critical Values and Existence of Optimal Extensions

Just as in the classical case, $d_P(T_1, T_2)$ always attains its value at a critical point derived from the node-set and constraints of the partial interleaving. The set of $P$-critical points includes all vertices of $T_1$ and $T_2$ involved in $P$, together with marked heights derived from candidate matches.

A finite set $$\Delta_1[P] = \{ |f_1(v) - f_2(w)| : v \in C_1[P], w \in C_2[P] \}$$, with $C_1[P]$ and $C_2[P]$ the $P$-critical node sets, together with the original $\Delta_2$ and $\Delta_3$ from the unconstrained measure, ensures $$d_P(T_1, T_2) \in \Delta_1[P] \cup \Delta_2 \cup \Delta_3$$. An optimal complete interleaving that realizes this infimum always exists (Beurskens et al., 18 Dec 2025).

4. Locally Correct Interleavings

A complete interleaving $I = (\alpha, \beta)$ is locally correct if for every restriction $R = (\alpha|_{S_1}, \beta|_{S_2})$, the residual shift $\xi_R(I)$ coincides with the residual interleaving distance $d_R(T_1, T_2)$. This condition guarantees that any partial matching imposed by $I$ cannot be improved without altering the rest of the correspondence, thus yielding a stratified, locally optimal match at every substructure.

A central existence theorem establishes that every pair of finite merge trees admits at least one locally correct interleaving. The constructive proof iteratively augments the set of forced matchings, always realizing residual distances at critical shifts, terminating with a complete interleaving that is locally optimal everywhere (Beurskens et al., 18 Dec 2025).

5. Properties and Theoretical Implications

Key properties of the residual interleaving distance include:

• If $P$ is the empty set, $d_P$ recovers the classical interleaving distance.
• For $P \subseteq Q$, monotonicity $d_P \le d_Q$ holds.
• The residual interleaving distance is always realized at a finite critical value and by some extension achieving that value.
• Local optimality is precisely captured by the requirement that any restriction of a locally correct interleaving is itself optimally residual given its forced matching.

A plausible implication is that the residual interleaving distance enables not only summary metrics but also a stratified “shift profile” of matchings across the tree, reflecting intrinsic local correspondence patterns.

6. Algorithmic Computation of the Residual Interleaving Distance

To compute $d_P(T_1, T_2)$, standard dynamic programming techniques, such as those of Touli–Wang, can be extended. The state space is augmented to track matched points under $P$, transitions are constrained to be consistent with the fans $F[P]$ and a target threshold $\delta$, and binary search is performed over the discrete set of critical values computed above. Once an optimal extension is found, one can iterate the process—extracting bottleneck arrows, augmenting $P$, and repeating—to construct a locally correct interleaving (Beurskens et al., 18 Dec 2025).

7. Illustrative Example and Applications

An illustrative example involves two merge trees, each with three leaves at heights $0, 1, 2$ merging to a common root at $\infty$. The classical interleaving distance might align disparate leaves globally with a shift of $2$, neglecting tighter local correspondences. By incrementally fixing correspondences as constraints in $P$ and recomputing $d_P$, the shifts required for subtrees can be locally minimized, obtaining a matching where shifts $\{1, 1, 2\}$ are used as needed, rather than a uniform bottleneck value. Such a matching accurately captures local similarity structure and constitutes a locally correct interleaving.

Residual interleaving distance thus enables deeper comparative analysis of hierarchical and topological structures, with particular relevance for studying temporal sequences of terrains and other evolving topological data (Beurskens et al., 18 Dec 2025).