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Tai Mapping Hierarchy for Rooted Labeled Trees Through Common Subforest

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Abstract

A Tai mapping between two rooted labeled trees (trees, for short) is a one-to-one node correspondence preserving ancestors and siblings (if trees are ordered). The variations of the Tai mapping are known to provide a hierarchy, called a Tai mapping hierarchy. In this paper, we characterize the Tai mapping hierarchy as a common subforest by focusing on the connections of nodes and the arrangements of subtrees in a common subforest. Then, we fill a gap in the Tai mapping hierarchy by introducing several new variations. Furthermore, we summarize and investigate the time complexity of computing the variations of the edit distance as the minimum cost of the variations of the Tai mapping.

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Notes

  1. The largest common subtree in the standard definition [1, 5, 9, 16, 21] is corresponding to the minimum cost of the LCA-preserving segmental mapping (LcaSg, defined below) which consists of pairs of nodes with the same label.

  2. In contrast to the standard definition of deletion and insertion [9, 14], this paper allows the deletion of a root, in order to characterize the bottom-up distance explicitly. Then, this paper also allows that the resulting tree applying to the deletion becomes a forest.

  3. Whereas Valiente [15] has introduced a bottom-up mapping that requires an isolated-subtree mapping, his algorithm computes a bottom-up distance that is not an isolated-subtree distance. Hence, we adopt the revised definition here not to be an isolated-subtree mapping. See [9, 18].

  4. In the definition of an alignable mapping [9], F is not a forest but a tree, because we can assume that the alignable mapping always contains the pair of the roots of two trees, corresponding to the alignment tree [6]. On the other hand, since the intersection of the alignable mapping to other mappings does not always contain the pair of the roots of two trees, we use a forest F in the definition of an alignable mapping.

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Acknowledgments

The authors would like to thank Prof. Tetsuji Kuboyama at Gakushuin University, Prof. Kilho Shin at University of Hyogo and Prof. Tetsuhiro Miyahara at Hiroshima City University for fruitful discussion about tree edit distance and its variations. Also they would like to thank anonymous referees of Theory of Computing Systems for valuable comments to revise the submitted version of this paper.

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  1. Kyushu Institute of Technology, Kawazu 680-4, Iizuka, 820-8502, Japan

    Takuya Yoshino & Kouichi Hirata

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  1. Takuya Yoshino
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  2. Kouichi Hirata
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Correspondence to Kouichi Hirata.

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This work is partially supported by Grant-in-Aid for Scientific Research 16H02870, 16H01743, 15K12102, 26280085 and 24300060 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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Yoshino, T., Hirata, K. Tai Mapping Hierarchy for Rooted Labeled Trees Through Common Subforest. Theory Comput Syst 60, 759–783 (2017). https://doi.org/10.1007/s00224-016-9705-1

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