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The Runs Theorem and Beyond

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Developments in Language Theory (DLT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11088))

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Abstract

Repetitions in a string are fundamental properties the string possesses, which have been extensively studied in word combinatorics and utilized in many efficient string processing algorithms. Particularly maximal repetitions, also known as runs, are useful for representing all the repetitions in the string. Since it was shown that the number of runs in a string of length n is upper bounded by O(n) [Kolpakov and Kucherov, FOCS, pp. 596–604, 1999], the following conjecture (known as the “runs” conjecture) have been attracting the attention of many researchers: The number of runs in a string of length n is upper bounded by n. This conjecture was recently solved affirmatively using a characterization based on Lyndon words [Bannai et al., SIAM J Comput, pp. 1501–1514, 2017]. The characterization not only gives a surprisingly simple proof to the 15-years open problem but also provides completely new insights on how repetitions are packed into a string. In this article, we will briefly review the runs theorem and some related topics.

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Authors and Affiliations

  1. Kyushu Institute of Technology, Iizuka, Fukuoka, Japan

    Tomohiro I

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  1. Tomohiro I
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Correspondence to Tomohiro I .

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  1. National Archives of Japan, Tokyo, Japan

    Mizuho Hoshi

  2. The University of Electro-Communications, Tokyo, Japan

    Shinnosuke Seki

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I, T. (2018). The Runs Theorem and Beyond. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_2

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  • DOI: https://doi.org/10.1007/978-3-319-98654-8_2

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