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Twins in words and long common subsequences in permutations

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Abstract

A large family of words must contain two words that are similar. We investigate several problems where the measure of similarity is the length of a common subsequence.

We construct a family of n 1/3 permutations on n letters, such that LCS of any two of them is only cn 1/3, improving a construction of Beame, Blais, and Huynh-Ngoc. We relate the problem of constructing many permutations with small LCS to the twin word problem of Axenovich, Person and Puzynina. In particular, we show that every word of length n over a k-letter alphabet contains two disjoint equal subsequences of length cnk -2/3. Connections to other extremal questions on common subsequences are exhibited.

Many problems are left open.

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Authors and Affiliations

  1. Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA, 15213, USA

    Boris Bukh

  2. 901 Cherry Ave, San Bruno, CA, 94066, USA

    Lidong Zhou

Authors
  1. Boris Bukh
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  2. Lidong Zhou
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Correspondence to Boris Bukh.

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Bukh, B., Zhou, L. Twins in words and long common subsequences in permutations. Isr. J. Math. 213, 183–209 (2016). https://doi.org/10.1007/s11856-016-1323-8

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  • DOI: https://doi.org/10.1007/s11856-016-1323-8

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