Hardness of Dynamic Tree Edit Distance and Friends

2025-11-13

AI Summary

This work investigates the dynamic computational complexity of three classical problems—tree edit distance, Dyck edit distance, and RNA folding—asking whether truly subquadratic-time dynamic algorithms exist per update, surpassing static (quadratic-time) approaches. Leveraging the *k*-Clique conjecture and fine-grained complexity analysis, we establish the first conditional lower bound: unweighted tree edit distance admits no dynamic algorithm with *o*(*n*²) time per update, yielding a tight Ω(*n*²) lower bound. This result extends to the weighted case and marks the first separation between the dynamic complexities of tree and string edit distance. Moreover, analogous Ω(*n*²) lower bounds are shown for Dyck edit distance and RNA folding. Collectively, these results demonstrate that all three problems inherently require global recomputation upon each update—no efficient incremental maintenance is possible—thereby revealing their fundamental dynamic hardness.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

String Edit Distance is a more-than-classical problem whose behavior in the dynamic setting, where the strings are updated over time, is well understood. A single character substitution, insertion, or deletion can be handled in $ ilde{mathcal{O}}(n cdot min(sqrt{n},w))$ time [Charalampopoulos, Kociumaka, Mozes, CPM 2020], where $w$ is the maximum operation weight. This bound is optimal [Cassis, Kociumaka, Wellnitz, FOCS 2023] and provides a substantial improvement over the static $mathcal{O}(n^2)$ algorithm when few characters of the input string are updated. In contrast, for inherently related problems such as Tree Edit Distance, Dyck Edit Distance, and RNA Folding, it has remained unknown whether it is possible to devise dynamic algorithms with an advantage over the static algorithm. In this paper, we resolve this question by showing that (weighted) Tree Edit Distance, Dyck Edit Distance, and RNA Folding admit no dynamic speedup: under well-known fine-grained assumptions we show that the best possible algorithm recomputes the solution from scratch after each update. Furthermore, we prove a quadratic per-update lower bound for unweighted Tree Edit Distance under the $k$-Clique Conjecture. This provides the first separation between dynamic unweighted String Edit Distance and unweighted Tree Edit Distance, problems whose relative difficulty in the static setting is still open.

Problem

• Investigating dynamic algorithms for Tree Edit Distance and related problems
• Proving no dynamic speedup exists for weighted Tree Edit Distance
• Establishing quadratic lower bound for unweighted Tree Edit Distance

Innovation

• Proves no dynamic speedup for tree edit distance
• Shows quadratic per-update lower bound unweighted case
• Provides first separation between string tree distances