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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

In this paper, we propose tree edit distance with variables, which is an extension of the tree edit distance to handle trees with variables and has a potential application to measuring the similarity between mathematical formulas. We analyze the computational complexity of several variants of this model. In particular, we show that the problem is NP-complete for ordered trees. We also show for unordered trees that the problem of deciding whether or not the distance is 0 is graph isomorphism complete but can be solved in polynomial time if the maximum outdegree of input trees is bounded by a constant. We also present parameterized and exponential-time algorithms for ordered and unordered cases, respectively.

Tatsuya Akutsu, Tomoya Mori, Naotoshi Nakamura, Satoshi Kozawa, Yuhei Ueno, and Thomas N. Sato. On the Complexity of Tree Edit Distance with Variables. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{akutsu_et_al:LIPIcs.ISAAC.2022.44, author = {Akutsu, Tatsuya and Mori, Tomoya and Nakamura, Naotoshi and Kozawa, Satoshi and Ueno, Yuhei and Sato, Thomas N.}, title = {{On the Complexity of Tree Edit Distance with Variables}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {44:1--44:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.44}, URN = {urn:nbn:de:0030-drops-173295}, doi = {10.4230/LIPIcs.ISAAC.2022.44}, annote = {Keywords: Tree edit distance, unification, parameterized algorithms} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

The tree inclusion problem is, given two node-labeled trees P and T (the "pattern tree" and the "text tree"), to locate every minimal subtree in T (if any) that can be obtained by applying a sequence of node insertion operations to P. Although the ordered tree inclusion problem is solvable in polynomial time, the unordered tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is from 1995 and runs in O(poly(m,n) * 2^{2d}) = O^*(2^{2d}) time, where m and n are the sizes of the pattern and text trees, respectively, and d is the maximum outdegree of the pattern tree. Here, we develop a new algorithm that improves the exponent 2d to d by considering a particular type of ancestor-descendant relationships and applying dynamic programming, thus reducing the time complexity to O^*(2^d). We then study restricted variants of the unordered tree inclusion problem where the number of occurrences of different node labels and/or the input trees' heights are bounded. We show that although the problem remains NP-hard in many such cases, it can be solved in polynomial time for c = 2 and in O^*(1.8^d) time for c = 3 if the leaves of P are distinctly labeled and each label occurs at most c times in T. We also present a randomized O^*(1.883^d)-time algorithm for the case that the heights of P and T are one and two, respectively.

Tatsuya Akutsu, Jesper Jansson, Ruiming Li, Atsuhiro Takasu, and Takeyuki Tamura. New and Improved Algorithms for Unordered Tree Inclusion. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 27:1-27:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{akutsu_et_al:LIPIcs.ISAAC.2018.27, author = {Akutsu, Tatsuya and Jansson, Jesper and Li, Ruiming and Takasu, Atsuhiro and Tamura, Takeyuki}, title = {{New and Improved Algorithms for Unordered Tree Inclusion}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {27:1--27:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.27}, URN = {urn:nbn:de:0030-drops-99752}, doi = {10.4230/LIPIcs.ISAAC.2018.27}, annote = {Keywords: parameterized algorithms, tree inclusion, unordered trees, dynamic programming} }

Document

**Published in:** LIPIcs, Volume 105, 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)

We present a simple linear-time algorithm for computing the topological centroid and the canonical form of a plane graph. Although the targets are restricted to plane graphs, it is much simpler than the linear-time algorithm by Hopcroft and Wong for determination of the canonical form and isomorphism of planar graphs. By utilizing a modified centroid for outerplanar graphs, we present a linear-time algorithm for a geometric version of the maximum common connected edge subgraph (MCCES) problem for the special case in which input geometric graphs have outerplanar structures, MCCES can be obtained by deleting at most a constant number of edges from each input graph, and both the maximum degree and the maximum face degree are bounded by constants.

Tatsuya Akutsu, Colin de la Higuera, and Takeyuki Tamura. A Simple Linear-Time Algorithm for Computing the Centroid and Canonical Form of a Plane Graph and Its Applications. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 10:1-10:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{akutsu_et_al:LIPIcs.CPM.2018.10, author = {Akutsu, Tatsuya and de la Higuera, Colin and Tamura, Takeyuki}, title = {{A Simple Linear-Time Algorithm for Computing the Centroid and Canonical Form of a Plane Graph and Its Applications}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {10:1--10:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-074-3}, ISSN = {1868-8969}, year = {2018}, volume = {105}, editor = {Navarro, Gonzalo and Sankoff, David and Zhu, Binhai}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2018.10}, URN = {urn:nbn:de:0030-drops-86992}, doi = {10.4230/LIPIcs.CPM.2018.10}, annote = {Keywords: Plane graph, Graph isomorphism, Maximum common subgraph} }

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