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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Edit distance between trees is a natural generalization of the classical edit distance between strings, in which the allowed elementary operations are contraction, uncontraction and relabeling of an edge. Demaine et al. [ACM Trans. on Algorithms, 6(1), 2009] showed how to compute the edit distance between rooted trees on n nodes in O(n^3) time. However, generalizing their method to unrooted trees seems quite problematic, and the most efficient known solution remains to be the previous O(n^3 log n) time algorithm by Klein [ESA 1998]. Given the lack of progress on improving this complexity, it might appear that unrooted trees are simply more difficult than rooted trees. We show that this is, in fact, not the case, and edit distance between unrooted trees on n nodes can be computed in O(n^3) time. A significantly faster solution is unlikely to exist, as Bringmann et al. [SODA 2018] proved that the complexity of computing the edit distance between rooted trees cannot be decreased to O(n^{3-epsilon}) unless some popular conjecture fails, and the lower bound easily extends to unrooted trees. We also show that for two unrooted trees of size m and n, where m <=n, our algorithm can be modified to run in O(nm^2(1+log(n/m))). This, again, matches the complexity achieved by Demaine et al. for rooted trees, who also showed that this is optimal if we restrict ourselves to the so-called decomposition algorithms.

Bartlomiej Dudek and Pawel Gawrychowski. Edit Distance between Unrooted Trees in Cubic Time. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 45:1-45:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dudek_et_al:LIPIcs.ICALP.2018.45, author = {Dudek, Bartlomiej and Gawrychowski, Pawel}, title = {{Edit Distance between Unrooted Trees in Cubic Time}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.45}, URN = {urn:nbn:de:0030-drops-90492}, doi = {10.4230/LIPIcs.ICALP.2018.45}, annote = {Keywords: tree edit distance, dynamic programming, heavy light decomposition} }

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

We consider the top tree compression scheme introduced by Bille et al. [ICALP 2013] and construct an infinite family of trees on n nodes labeled from an alphabet of size sigma, for which the size of the top DAG is Theta(n/log_sigma n log log_sigma n). Our construction matches a previously known upper bound and exhibits a weakness of this scheme, as the information-theoretic lower bound is Omega(n/log_sigma n}). This settles an open problem stated by Lohrey et al. [arXiv 2017], who designed a more involved version achieving the lower bound. We show that this can be also guaranteed by a very minor modification of the original scheme: informally, one only needs to ensure that different parts of the tree are not compressed too quickly. Arguably, our version is more uniform, and in particular, the compression procedure is oblivious to the value of sigma.

Bartlomiej Dudek and Pawel Gawrychowski. Slowing Down Top Trees for Better Worst-Case Compression. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 16:1-16:8, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dudek_et_al:LIPIcs.CPM.2018.16, author = {Dudek, Bartlomiej and Gawrychowski, Pawel}, title = {{Slowing Down Top Trees for Better Worst-Case Compression}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {16:1--16:8}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2018.16}, URN = {urn:nbn:de:0030-drops-86920}, doi = {10.4230/LIPIcs.CPM.2018.16}, annote = {Keywords: top trees, compression, tree grammars} }

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**Published in:** LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

In the Maximum Duo-Preservation String Mapping problem we are given two strings and wish to map the letters of the former to the letters of the latter as to maximise the number of duos. A duo is a pair of consecutive letters that is mapped to a pair of consecutive letters in the same order. This is complementary to the well-studied Minimum Common String Partition problem, where the goal is to partition the former string into blocks that can be permuted and concatenated to obtain the latter string.
Maximum Duo-Preservation String Mapping is APX-hard. After a series of improvements, Brubach [WABI 2016] showed a polynomial-time 3.25-approximation algorithm. Our main contribution is that, for any eps>0, there exists a polynomial-time (2+eps)-approximation algorithm. Similarly to a previous solution by Boria et al. [CPM 2016], our algorithm uses the local search technique. However, this is used only after a certain preliminary greedy procedure, which gives us more structure and makes a more general local search possible. We complement this with a specialised version of the algorithm that achieves 2.67-approximation in quadratic time.

Bartlomiej Dudek, Pawel Gawrychowski, and Piotr Ostropolski-Nalewaja. A Family of Approximation Algorithms for the Maximum Duo-Preservation String Mapping Problem. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 10:1-10:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dudek_et_al:LIPIcs.CPM.2017.10, author = {Dudek, Bartlomiej and Gawrychowski, Pawel and Ostropolski-Nalewaja, Piotr}, title = {{A Family of Approximation Algorithms for the Maximum Duo-Preservation String Mapping Problem}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {10:1--10:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.10}, URN = {urn:nbn:de:0030-drops-73458}, doi = {10.4230/LIPIcs.CPM.2017.10}, annote = {Keywords: approximation scheme, minimum common string partition, local search} }

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