An algorithm for finding the largest approximately common substructures of two trees

Abstract

Ordered, labeled trees are trees in which each node has a label and the left-to-right order of its children (if it has any) is xed. Such trees have many applications in vision, pattern recognition, molecular biology and natural language processing. We consider a substructure of an ordered labeled tree T to be a connected subgraph of T. Given two ordered labeled trees T1 and T2 and an integer d, the largest approximately common substructure problem is to nd a substructure U1 of T1 and a substructure U2 of T2 such that U1 is within edit distance d of U2 and where there does not exist any other substructure V1 of T1 and V2 of T2 such that V1 and V2 satisfy the distance constraint and the sum of the sizes of V1 and V2 is greater than the sum of the sizes of U1 and U2. We present a dynamic programming algorithm to solve this problem, which runs as fast as the fastest known algorithm for computing the edit distance of two trees when the distance allowed in the common substructures is a constant independent of the input trees. To demonstrate the utility of our algorithm, we discuss its application to discovering motifs in multiple RNA secondary structures (which are ordered labeled trees).

Design of an RNA structural motif database

“…For example, in Chang et al (1998), Wang et al (1996Wang et al ( , 1998 we represented an RNA secondary structure using an ordered labelled tree and designed a tree matching algorithm to find motifs in multiple RNA secondary structures.…”

Local similarity in RNA secondary structures

“…Edit distance models on unordered trees are considered in [32,29]. Problem variations on rooted and/or unrooted trees are considered in [15,31,26]. Algorithms that calculate local similarity of trees in the tree editing model are presented in [26,28].…”

Validating web content with senser

“…The pairwise tree alignment problem can be solved in O(|T1| × |T2| × h1 × h2) time, where |Ti| is the size of tree i and hi is the height of tree i [25]. Wang et al [28] improve upon this algorithm to solve the problem in O(|T1| × |T2| × min(h1, l1) × min(h2, l2)) time, where li is the number of leaves in tree i. Followup work [4] applies the center star approximation algorithm [11] for multiple string alignment in order to approximately align multiple HTML trees.…”

“…The algorithm starts from the roots of the MHTs, and traverses recursively through the MHTs in a top-down fashion. For each tree node, we compute the majority consensus for the full-hashes and tag-hashes (lines 10-13): if a majority of the proxies agree on the same full-hash, which indicates that a majority consensus has been reached for the complete subtree rooted by that tree node, then the whole subtree is copied into the final consensus tree (lines 14-16); otherwise, if the corresponding tree nodes in a majority of the summaries have the same tag-hash, we heuristically assume that these tree nodes correspond to the same fragment in the HTML but disagree on the contents, in which case, that tree node is copied into the final consensus tree (lines [18][19][20], and the BFS algorithm will construct the consensus version of the corresponding subtree when the children nodes are visited (lines [21][22][23][24][25][26][27][28]. If neither a tree node's full-hash nor its tag-hash are present in a majority of the MHTs, no consensus can be drawn, and the node is marked as NON-CONSENSUS (lines 29-31).…”