$\tilde{O}(n+\mathrm{poly}(k))$-time Algorithm for Bounded Tree Edit Distance

TL;DR

This paper presents an efficient algorithm with near-linear time complexity in the size of the input for computing the tree edit distance when the distance is bounded by a parameter k, improving over previous methods.

Contribution

The paper introduces the first $ ilde{O}(n+ ext{poly}(k))$-time algorithm for bounded tree edit distance, resolving an open problem in the field.

Findings

Achieves near-linear time complexity for bounded tree edit distance

Improves upon previous $ ilde{O}(nk^2)$-time algorithms

Provides theoretical foundation for faster tree similarity computations

Abstract

Computing the edit distance of two strings is one of the most basic problems in computer science and combinatorial optimization. Tree edit distance is a natural generalization of edit distance in which the task is to compute a measure of dissimilarity between two (unweighted) rooted trees with node labels. Perhaps the most notable recent application of tree edit distance is in NoSQL big databases, such as MongoDB, where each row of the database is a JSON document represented as a labeled rooted tree, and finding dissimilarity between two rows is a basic operation. Until recently, the fastest algorithm for tree edit distance ran in cubic time (Demaine, Mozes, Rossman, Weimann; TALG'10); however, Mao (FOCS'21) broke the cubic barrier for the tree edit distance problem using fast matrix multiplication. Given a parameter $k$ as an upper bound on the distance, an $O(n+k^2)$-time algorithm for edit distance has been known since the 1980s due to the works of Myers (Algorithmica'86) and Landau and Vishkin (JCSS'88). The existence of an $\tilde{O}(n+\mathrm{poly}(k))$-time algorithm for tree edit distance has been posed as an open question, e.g., by Akmal and Jin (ICALP'21), who gave a state-of-the-art $\tilde{O}(nk^2)$-time algorithm. In this paper, we answer this question positively.