An Optimal Decomposition Algorithm for Tree Edit Distance

Abstract

The 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 worstcase O(n 3 )-time algorithm for this problem, improving the previous best O(n 3 log n)time algorithm [6]. 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 decomposition strategy algorithms-which also includes the previous fastest algorithms-by tightening the known lower bound of Ω(n 2 log 2 n) [4] to Ω(n 3 ), matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds of Θ(nm 2 (1 + log n m )) when the two trees have different sizes m and n, where m < n.

Complexity of Computing Distances between Geometric Trees

“…A sequence of edit operations that turn one tree into another is called an edit path between the two trees. When restricted to classes of trees with additional assumptions, such as edge order or bounded size, there exist a number of polynomial time algorithms [3,11,22] for computing the tree edit distance, and with further restrictions on the allowed complexity of the edit paths, there are linear time algorithms available [18]. However, for general, unordered trees, the edit distance computation problem has been shown to be NP complete by Zhang, Statman and Shasha [22].…”

Fast RNA Structure Alignment for Crossing Input Structures

“…For non-crossing input structures, the correspondence of these structures to trees allows for alignment methods that are asymptotically faster than the recurrence used in the first stage [17,2]. In our approach we apply a similar technique, but since our input structures do not correspond to trees, we select a subset P T ⊆ P 1 of the arcs.…”

“…One approach is to predict for every input sequence the minimum free-energy non-crossing structure (in O(n 3 ) time), and then perform pairwise sequence-structure alignments. The problem of aligning two non-crossing structures corresponds to tree editing and can be solved in O(n 3 ) time [2]. However, this approach crucially depends on the quality of the initial structure prediction, which is error-prone.…”

“…In this paper, we introduce a measurement for this deviation (d-crossing), and introduce an efficient alignment algorithm with complexity O(n 3 log n) given that the deviation is small. The fast available structure alignment methods for non-crossing input structures [17,2] rely on a heavy path decomposition which was so far only available for tree-like structures. Our approach generalizes this to d-crossing structures.…”

“…For non-crossing consensus structures, Jiang et al's algorithm [1] computes the alignment in O(n 2 m 2 ) time where n and m denote the lengths of the two input sequences. If the input structures are also non-crossing, the problem corresponds to tree editing which can be solved in O(m 2 n(1+log n m )) time [2]. We present a new algorithm that solves the problem for d-crossing structures in O(dm 2 n log n) time, where d is a parameter that is one for non-crossing structures, bounded by n for crossing structures, and much smaller than n on many practical examples.…”