Order-Preserving Squares in Strings

TL;DR

This paper establishes tight bounds on the number of order-preserving squares in strings and provides an optimal algorithm to find all such squares efficiently.

Contribution

It improves the upper bound on the number of order-preserving squares and presents an optimal algorithm for their enumeration.

Findings

Upper bound of O(σn) on distinct order-preserving squares

Existence of strings with Ω(σn) such squares, matching the upper bound

An O(σn) time algorithm to list all order-preserving squares

Abstract

An order-preserving square in a string is a fragment of the form $uv$ where $u\neq v$ and $u$ is order-isomorphic to $v$. We show that a string $w$ of length $n$ over an alphabet of size $\sigma$ contains $\mathcal{O}(\sigma n)$ order-preserving squares that are distinct as words. This improves the upper bound of $\mathcal{O}(\sigma^{2}n)$ by Kociumaka, Radoszewski, Rytter, and Wale\'n [TCS 2016]. Further, for every $\sigma$ and $n$ we exhibit a string with $\Omega(\sigma n)$ order-preserving squares that are distinct as words, thus establishing that our upper bound is asymptotically tight. Finally, we design an $\mathcal{O}(\sigma n)$ time algorithm that outputs all order-preserving squares that occur in a given string and are distinct as words. By our lower bound, this is optimal in the worst case.