On Longest Common Property Preserved Substring Queries

Abstract

We revisit the problem of longest common property preserving substring queries introduced by Ayad et al. (SPIRE 2018, arXiv 2018). We consider a generalized and unified on-line setting, where we are given a set X of k strings of total length n that can be pre-processed so that, given a query string y and a positive integer $k'\le k$, we can determine the longest substring of y that satisfies some specific property and is common to at least $k'$ strings in X. Ayad et al. considered the longest square-free substring in an on-line setting and the longest periodic and palindromic substring in an off-line setting. In this paper, we give efficient solutions in the on-line setting for finding the longest common square, periodic, palindromic, and Lyndon substrings. More precisely, we show that X can be pre-processed in O(n) time resulting in a data structure of O(n) size that answers queries in $O(|y|\log \sigma )$ time and O(1) working space, where $\sigma$ is the size of the alphabet, and the common substring must be a square, a periodic substring, a palindrome, or a Lyndon word.

This work was supported by JSPS KAKENHI Grant Numbers JP18K18002 (YN), JP17H01697 (SI), JP16H02783 (HB), and JP18H04098 (MT). Tomasz Kociumaka was supported by ISF grants no. 824/17 and 1278/16 and by an ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064).