Lyndon factorization and Lempel-Ziv (LZ) factorization are both important tools for analysing the structure and complexity of strings, but their combinatorial structure is very different. In this paper, we establish the first direct connection between the two by showing that while the Lyndon factorization can be bigger than the non-overlapping LZ factorization (which we demonstrate by describing a new, non-trivial family of strings) it is always less than twice the size.
@InProceedings{karkkainen_et_al:LIPIcs.STACS.2017.45, author = {K\"{a}rkk\"{a}inen, Juha and Kempa, Dominik and Nakashima, Yuto and Puglisi, Simon J. and Shur, Arseny M.}, title = {{On the Size of Lempel-Ziv and Lyndon Factorizations}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {45:1--45:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.45}, URN = {urn:nbn:de:0030-drops-69878}, doi = {10.4230/LIPIcs.STACS.2017.45}, annote = {Keywords: Lempel-Ziv factorization, Lempel-Ziv parsing, LZ, Lyndon word, Lyndon factorization, Standard factorization} }
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