Optimal dynamic update time for tree edit distance approximation

Determine the minimum update time required to dynamically maintain exact tree edit distance or a (1+ε)-approximation under node insertions, deletions, and label substitutions, and establish whether dynamic maintenance can substantially outperform recomputing tree edit distance from scratch after every update.

Background

Dynamic tree edit distance asks for efficient maintenance of the distance between two forests undergoing node edits. Prior to this work, no nontrivial dynamic exact TED algorithm was known, and no dynamic algorithm maintained a (1+ε)-approximation.

The paper gives dynamic exact and approximate algorithms with update times depending polynomially or nearly linearly on a distance threshold k. These results leave open the optimal dependence on k and whether substantially faster dynamic algorithms than repeated static recomputation are possible.

References

To date, no known dynamic algorithm maintains a $(1+\varepsilon)$-approximation of TED. Question 3: What is the best update time needed to dynamically maintain TED or its $(1+\varepsilon)$ approximation? Is there any approach significantly better than recomputing TED from scratch after each update?

— Tree Edit Distance: Near-Linear-Time Approximate, Sublinear, and Dynamic  (2610.22250 - Hu et al., 15 Oct 2026) in Section 1, Question 3 (Introduction)