Lempel-Ziv (LZ) factorization and Lyndon factorization are well-known factorizations of strings. Recently, Kärkkäinen et al. studied the relation between the sizes of the two factorizations, and showed that the size of the Lyndon factorization is always smaller than twice the size of the non-overlapping LZ factorization [STACS 2017]. In this paper, we consider a similar problem for the overlapping version of the LZ factorization. Since the size of the overlapping LZ factorization is always smaller than the size of the non-overlapping LZ factorization and, in fact, can even be an O(log n) factor smaller, it is not immediately clear whether a similar bound as in previous work would hold. Nevertheless, in this paper, we prove that the size of the Lyndon factorization is always smaller than four times the size of the overlapping LZ factorization.
@InProceedings{urabe_et_al:LIPIcs.CPM.2019.29, author = {Urabe, Yuki and Nakashima, Yuto and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{On the Size of Overlapping Lempel-Ziv and Lyndon Factorizations}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {29:1--29:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-103-0}, ISSN = {1868-8969}, year = {2019}, volume = {128}, editor = {Pisanti, Nadia and P. Pissis, Solon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.29}, URN = {urn:nbn:de:0030-drops-105008}, doi = {10.4230/LIPIcs.CPM.2019.29}, annote = {Keywords: Lyndon factorization, Lyndon words, Lempel-Ziv factorization} }
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