Longest Property-Preserved Common Factor

TL;DR

This paper introduces efficient algorithms for finding the longest common factors with specific properties—square-free, periodic, or palindromic—across multiple strings, with potential for broader applications.

Contribution

The paper presents the first linear-time algorithms for computing longest property-preserving common factors in various string settings.

Findings

Linear-time algorithms for square-free common factors.

Efficient solutions for periodic common factors among multiple strings.

Fast computation of palindromic common factors between two strings.

Abstract

In this paper we introduce a new family of string processing problems. We are given two or more strings and we are asked to compute a factor common to all strings that preserves a specific property and has maximal length. Here we consider three fundamental string properties: square-free factors, periodic factors, and palindromic factors under three different settings, one per property. In the first setting, we are given a string $x$ and we are asked to construct a data structure over $x$ answering the following type of on-line queries: given string $y$, find a longest square-free factor common to $x$ and $y$. In the second setting, we are given $k$ strings and an integer $1 < k'\leq k$ and we are asked to find a longest periodic factor common to at least $k'$ strings. In the third setting, we are given two strings and we are asked to find a longest palindromic factor common to the two strings. We present linear-time solutions for all settings. We anticipate that our paradigm can be extended to other string properties or settings.