Longest property-preserved common factor: A new string-processing framework

Lorraine A. K. Ayad, Giulia Bernardini, Roberto Grossi, Costas S. Iliopoulos, Nadia Pisanti, Solon P. Pissis, Giovanna Rosone

Abstract: We introduce a new family of string processing problems. Given two or more strings, we are asked to compute a factor common to all strings that preserves a specific property and has maximal length. 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 online queries: given a 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′≤k<math><mn is="true">1</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mo is="true">≤</mo><mi is="true">k</mi></math> and we are asked to find a longest periodic factor common to at least k′<math><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup></math> strings. In the third one, 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.This is a full and extended version of a paper from SPIRE 2018.