In this paper, we have extended the concept of unordered tree inclusion to take the costs of insertions and substitutions into account. The resulting algorithm, MinCostIncl, has the same time complexity as the original algorithm of [4] for unordered tree inclusion (O(22Dmn)). Computational experiments on a large synthetic dataset as well as real datasets showed that our proposed algorithm is fast and scalable. Source codes of the implemented algorithms are available upon request.

BACKGROUND: Many tree structures are found in nature and organisms. Such trees are believed to be constructed on the basis of certain rules. We have previously developed grammar-based compression methods for ordered and unordered single trees, based on bisection-type tree grammars. Here, these methods find construction rules for one single tree. On the other hand, specified construction rules can be utilized to generate multiple similar trees. RESULTS: Therefore, in this paper, we develop novel methods to discover common rules for the construction of multiple distinct trees, by improving and extending the previous methods using integer programming. We apply our proposed methods to several sets of glycans and RNA secondary structures, which play important roles in cellular systems, and can be regarded as tree structures. The results suggest that our method can be successfully applied to determining the minimum grammar and several common rules among glycans and RNAs. CONCLUSIONS: We propose integer programming-based methods MinSEOTGMul and MinSEUTGMul for the determination of the minimum grammars constructing multiple ordered and unordered trees, respectively. The proposed methods can provide clues for the determination of hierarchical structures contained in tree-structured biological data, beyond the extraction of frequent patterns.

This paper considers the problem of identifying all locations of subtrees in a large tree or in a large collection of trees that are similar to a specified pattern tree, where all trees are assumed to be rooted and node-labeled. The tree edit distance is a widely-used measure of tree (dis-)similarity, but is NP-hard to compute for unordered trees. To cope with this issue, we propose a new similarity measure which extends the concept of unordered tree inclusion by taking the costs of insertion and substitution operations on the pattern tree into account, and present an algorithm for computing it. Our algorithm has the same time complexity as the original one for unordered tree inclusion, i.e., it runs in O(|T1∥T2|) time, where T1and T2denote the pattern tree and the text tree, respectively, when the maximum outdegree of T1is bounded by a constant. Our experimental evaluation using synthetic and real datasets confirms that the proposed algorithm is fast and scalable and very useful for bibliographic matching, which is a typical entity resolution problem for tree-structured data. Furthermore, we extend our algorithm to also allow a constant number of deletion operations on T1while still running in O(|T1∥T2|) time.