Detecting regularities on grammar-compressed strings

Abstract

We address the problems of detecting and counting various forms of regularities in a string represented as a straight-line program (SLP) which is essentially a context free grammar in the Chomsky normal form. Given an SLP of size n that represents a string s of length N, our algorithm computes all runs and squares in s inh is the height of the derivation tree of the SLP. We also show an algorithm to compute all gapped-palindromes in O (n 3 h + gnh log N) time and O (n 2 ) space, where g is the length of the gap. As one of the main components of the above solution, we propose a new technique called approximate doubling which seems to be a useful tool for a wide range of algorithms on SLPs. Indeed, we show that the technique can be used to compute the periods and covers of the string in O (n 2 h) time and O (nh(n + log 2 N)) time, respectively.

Dynamic index and LZ factorization in compressed space

“…Using Theorem 6, we obtain a faster algorithm, as follows: A non-empty string s is called a Lyndon word if s is the lexicographically smallest suffix of s. The Lyndon factorization of a non-empty string w is a sequence of pairs (|f i |, p i ) where each f i is a Lyndon word and p i is a positive integer such that w = f p1 1 • • • f pm m and f i−1 is lexicographically smaller than f i for all 1 ≤ i < m. I et al [14] proposed a Lyndon factorization algorithm running in O(nh(n + log n log N )) time and O(n 2 ) space. Their algorithm use the LCE data structure on SLPs [16] which requires O(n 2 h) preprocessing time, O(n 2 ) working space, and O(h log N ) time for LCE queries. We can obtain a faster algorithm using Theorem 6.…”