An Optimal Decomposition Algorithm for Tree Edit Distance

Abstract

The edit distance between two ordered rooted 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 ³logn)-time algorithm [7]. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems, together with a deeper understanding of the previous algorithms for the problem. We prove the optimality of our algorithm among the family of decomposition strategy algorithms—which also includes the previous fastest algorithms—by tightening the known lower bound of Ω(n ²log² n) [4] to Ω(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 sizes m and n where m < n.