Breaking the Cubic Barrier for (Unweighted) Tree Edit Distance

TL;DR

This paper presents a novel algorithm that computes the unweighted tree edit distance faster than the previous cubic time, breaking the long-standing barrier and achieving near-quadratic time complexity.

Contribution

The authors develop the first sub-cubic algorithm for unweighted tree edit distance, surpassing the previous $O(n^3)$ time barrier using advanced matrix techniques.

Findings

Achieved $O(n^{2.9546})$ time complexity for unweighted tree edit distance.

Reduced the problem to max-plus product of bounded-difference matrices.

Demonstrated the applicability of combinatorial techniques to improve classical algorithms.

Abstract

The (unweighted) tree edit distance problem for $n$ node trees asks to compute a measure of dissimilarity between two rooted trees with node labels. The current best algorithm from more than a decade ago runs in $O(n ^ 3)$ time [Demaine, Mozes, Rossman, and Weimann, ICALP 2007]. The same paper also showed that $O(n ^ 3)$ is the best possible running time for any algorithm using the so-called decomposition strategy, which underlies almost all the known algorithms for this problem. These algorithms would also work for the weighted tree edit distance problem, which cannot be solved in truly sub-cubic time under the APSP conjecture [Bringmann, Gawrychowski, Mozes, and Weimann, SODA 2018]. In this paper, we break the cubic barrier by showing an $O(n ^ {2.9546})$ time algorithm for the unweighted tree edit distance problem. We consider an equivalent maximization problem and use a dynamic programming scheme involving matrices with many special properties. By using a decomposition scheme as well as several combinatorial techniques, we reduce tree edit distance to the max-plus product of bounded-difference matrices, which can be solved in truly sub-cubic time [Bringmann, Grandoni, Saha, and Vassilevska Williams, FOCS 2016].