Computing minimal and maximal suffixes of a substring

Abstract

We consider the problems of computing the maximal and the minimal non-empty suffixes of substrings of a longer text of length n. For the minimal suffix problem we show that for every τ , 1 ≤ τ ≤ log n, there exists a linear-space data structure with O(τ) query time and O(n log n/τ) preprocessing time. As a sample application, we show that this data structure can be used to compute the Lyndon decomposition of any substring of the text in O(kτ) time, where k is the number of distinct factors in the decomposition. For the maximal suffix problem, we give a linear-space structure with O(1) query time and O(n) preprocessing time. In other words, we simultaneously achieve both the optimal query time and the optimal construction time. $ This article is based on a study first reported at 24th and 25th Symposiums on Combinatorial Pattern Matching.

Minimal Suffix and Rotation of a Substring in Optimal Time

“…For both problems we obtain optimal solutions with linear construction time and constant query time. For Minimal Suffix Queries this improves upon the results of Babenko et al [5], who developed a trade-off solution, which for a text of length n has Θ(n log n) product of preprocessing and query time. We are not aware of any results for Minimal Rotation Queries except for a data structure only testing cyclic equivalence of two subwords [26].…”

“…An optimal solution for the Maximal Suffix Queries was already obtained in [5], while the Maximal Rotation Queries are equivalent to Minimal Rotation Queries subject to alphabet reversal. Hence, we do not focus on the maximization variants of our problems.…”

“…The last factor of the Lyndon factorization of a string is its minimal suffix. As noted in [5], this can be used to reduce computing the factorization…”

An Almost Optimal Edit Distance Oracle

“…A natural direction after having determined the complexity of a particular problem on strings is to consider the more general version in which we need to answer queries on substrings of the input string. This has been done for alignment [34,35], pattern matching [24,28], approximate pattern matching [17], dictionary matching [14,15], compression [24], periodicity [27,28], counting palindromes [33], longest common substring [3], computing minimal and maximal suffixes [5,26], and computing the lexicographically k-th suffix [4].…”

Counting Lyndon Subsequences

“…Lyndon factors enjoy a rich class of algorithmic and stringology applications including: counting and finding the maximal repetitions (a.k.a. runs) in a string [2] and in a trie [8], constant-space pattern matching [3], comparison of the sizes of run-length Burrows-Wheeler Transform of a sting and its reverse [4], substring minimal suffix queries [1], the shortest common superstring problem [7], and grammar-compressed self-index (Lyndon-SLP) [9].…”