Twins in words and long common subsequences in permutations

TL;DR

This paper explores the similarity of words through longest common subsequences, constructing permutations with small LCS and relating the problem to twin words, revealing new bounds and open questions.

Contribution

It introduces a new permutation construction with small LCS and connects the problem to twin words, advancing understanding of word similarity measures.

Findings

Constructed permutations with LCS of size cn^{1/3}

Every word over a k-letter alphabet contains twin subsequences of length cnk^{-2/3}

Improved previous bounds on permutation similarity measures

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}. Many problems are left open.