An O(n^3)-Time Algorithm for Tree Edit Distance

TL;DR

This paper introduces an improved $O(n^3)$-time algorithm for computing tree edit distance, utilizing a novel adaptive dynamic programming strategy and establishing its optimality among decomposition strategies.

Contribution

The paper presents the first $O(n^3)$-time algorithm for tree edit distance and proves its optimality within a class of algorithms, improving upon previous bounds.

Findings

Achieved a worst-case $O(n^3)$ algorithm for tree edit distance.

Proved the algorithm's optimality among decomposition strategy algorithms.

Established tight bounds for trees of different sizes.

Abstract

The {\em edit distance} between two ordered trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. In this paper, we present a worst-case $O(n^3)$-time algorithm for this problem, improving the previous best $O(n^3\log n)$-time algorithm~\cite{Klein}. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems (which is interesting in its own right), together with a deeper understanding of the previous algorithms for the problem. We also prove the optimality of our algorithm among the family of \emph{decomposition strategy} algorithms--which also includes the previous fastest algorithms--by tightening the known lower bound of $\Omega(n^2\log^2 n)$~\cite{Touzet} to $\Omega(n^3)$, matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds of $\Theta(n m^2 (1 + \log \frac{n}{m}))$ when the two trees have different sizes $m$ and~$n$, where $m < n$.