Closed factorization

Abstract

A closed string is a string with a proper substring that occurs in the string as a prefix and a suffix, but not elsewhere. Closed strings were introduced by Fici (Proc. WORDS, 2011) as objects of combinatorial interest in the study of Trapezoidal and Sturmian words. In this paper we present algorithms for computing closed factors (substrings) in strings. First, we consider the problem of greedily factorizing a string into a sequence of longest closed factors. We describe an algorithm for this problem that uses linear time and space. We then consider the related problem of computing, for every position in the string, the longest closed factor starting at that position. We describe a simple algorithm for the problem that runs in O(n log n/ log log n) time, where n is the length of the string. This also leads to an algorithm to compute the maximal closed factor containing (i.e. covering) each position in the string in O(n log n/ log log n) time. We also present linear time algorithms to factorize a string into a sequence of shortest closed factors of length at least two, to compute the shortest closed factor of length at least two starting at each position of the string, and to compute a minimal closed factor of length at least two containing each position of the string.

On the Number of Closed Factors in a Word

“…For more details on closed words and related results see [6,5,9,3,7,1,13]. We end this section by exhibiting some properties of closed words.…”

Off-line and on-line algorithms for closed string factorization

“…Thus the study of closed strings shows potential applications in connection with applications of palindromes [4]. On the algorithmic side Badkobeh et al in [2] presented (among others) an algorithm for the factorisation of a given string of length n into a sequence of longest closed factors (LCFs) in time and space O(n) and another algorithm for computing the longest closed factor starting at every position in the string in O(n log n log log n ) time and O(n) space. Moreover, Iliopoulos et al [5] presented an on-line O(n)-time algorithm to calculate the size of a minimum closed cover for each prefix of a given string X of length n. (A set of closed strings W = {w 1 , · · · , w l } is called a cover of a string X if X can be constructed by concatenations and overlaps of elements of W .…”

The sequence of open and closed prefixes of a Sturmian word

“…Let Σ be a finite nonempty set (the alphabet). A (finite) word w = w [1]w [2] · · · w[n] with w[i] ∈ Σ is closed (also known as periodic-like [6]) if it contains a factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. For example, the words abca, ababa and aabaab are closed -any word of length 1 is closed, the empty word being a factor that occurs both as a prefix and as a suffix but does not have internal occurrences; the words ab, aab and aaba, instead, are open.…”

“…Given a finite or infinite word w = w [1]w [2] · · · , the sequence oc(w) of open/closed prefixes of w, that we refer to as the oc-sequence of w, is the binary sequence c(1)c(2) · · · whose n-th element is 1 if the prefix of w of length n is closed, 0 if it is open. For example, if w = abcab, then oc(w) = 10011.…”

“…Proposition 26. The semicentral prefixes of w are precisely the words of the form u n xyu n , n ≥ 1, where x, y and u n are as in (1).…”