This paper considers the problem of inferring a graph from the numbers of occurrences of vertex-labeled paths, which is closely related to the pre-image problem for graphs: to reconstruct a graph from its feature space representation. It is shown that both exact and approximate versions of the problem can be solved in polynomial time in the size of an output graph by using dynamic programming algorithms if the graphs are trees whose maximum degree is bounded by a constant and the lengths of given paths and alphabet size are bounded by constants. On the other hand, it is shown that this problem is strongly NP-hard even for trees of bounded degree if the maximum length of paths is not bounded. The problem of inferring a string from the number of occurrences of fixed size substrings is also studied.

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Many kinds of tree-structured data, such as RNA secondary structures, have become available due to the progress of techniques in the field of molecular biology. To analyze the treestructured data, various measures for computing the similarity between them have been developed and applied. Among them, tree edit distance is one of the most widely used measures. However, the tree edit distance problem for unordered trees is NP-hard. Therefore, it is required to develop efficient algorithms for the problem. Recently, a practical method called clique-based algorithm has been proposed, but it is not fast for large trees.This article presents an improved clique-based method for the tree edit distance problem for unordered trees. The improved method is obtained by introducing a dynamic programming scheme and heuristic techniques to the previous clique-based method. To evaluate the efficiency of the improved method, we applied the method to comparison of real tree structured data such as glycan structures. For large tree-structures, the improved method is much faster than the previous method. In particular, for hard instances, the improved method achieved more than 100 times speed-up.

BackgroundMeasuring similarities between tree structured data is important for analysis of RNA secondary structures, phylogenetic trees, glycan structures, and vascular trees. The edit distance is one of the most widely used measures for comparison of tree structured data. However, it is known that computation of the edit distance for rooted unordered trees is NP-hard. Furthermore, there is almost no available software tool that can compute the exact edit distance for unordered trees.ResultsIn this paper, we present a practical method for computing the edit distance between rooted unordered trees. In this method, the edit distance problem for unordered trees is transformed into the maximum clique problem and then efficient solvers for the maximum clique problem are applied. We applied the proposed method to similar structure search for glycan structures. The result suggests that our proposed method can efficiently compute the edit distance for moderate size unordered trees. It also suggests that the proposed method has the accuracy comparative to those by the edit distance for ordered trees and by an existing method for glycan search.ConclusionsThe proposed method is simple but useful for computation of the edit distance between unordered trees. The object code is available upon request.