Longest Property-Preserved Common Factor

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 ≤ 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.

On Longest Common Property Preserved Substring Queries

“…substring, suffix) if it is shorter than the string. Let x R denote the reverse of x, i.e., x R = x[|x|] · · · x [1]. A string x is said to be a palindrome if x = x R .…”

“…A node in ST (x) is called an explicit node, while a position on the edges corresponding to proper prefixes of the edge label are called implicit nodes. For a (possibly implicit) node v in ST (x), let str (v) denote the string obtained by concatenating the edge labels on the path from the root to v. The locus of a string p in ST (x) is a (possibly implicit) node v in ST (x) such that str (v) = p. Each explicit node v of the suffix tree can be augmented with a suffix link, that points to the node u, such that str (v) = str (v) [1]str (u). It is easy to see that because v is an explicit node, u is also always an explicit node.…”

“….|y|] as long as possible to compute the locus corresponding to MS y,x [1]. Let this prefix be y[1..e 1 ].…”

“…Ayad et al [1,2] gave a solution to the on-line longest common square-free substring query problem for a single string (Problem 3), in O(|x|) time and space preprocessing and O(|y| log σ) time and O(1) space query. We note that their algorithm easily extends to the generalized version (Problem 4).…”

“…Ayad et al [1,2] proposed a class of problems called longest common property preserved substring, where the aim is to find the longest substring that has some property and is common to a subset of the input strings. They considered several problems in two different settings.…”