We consider grammar-based text compression with longest first substitution (LFS), where non-overlapping occurrences of a longest repeating factor of the input text are replaced by a new non-terminal symbol. We present the first linear-time algorithm for LFS. Our algorithm employs a new data structure called sparse lazy suffix trees. We also deal with a more sophisticated version of LFS, called LFS2, that allows better compression. The first linear-time algorithm for LFS2 is also presented.

A multi-track string is a tuple of strings of the same length. Given the pattern and text of two multi-track strings, the permuted pattern matching problem is to find the occurrence positions of all permutations of the pattern in the text. In this paper, we propose several algorithms for permuted pattern matching. Our first algorithm, which is based on the Knuth–Morris–Pratt (KMP) algorithm, has a fast theoretical computing time with $O\left(mk\right)$ as the preprocessing time and $O(nklog\sigma )$ as the matching time, where n, m, k, $\sigma$ , and $\mathit{occ}$ denote the length of the text, the length of the pattern, the number of strings in the multi-track, the alphabet size, and the number of occurrences of the pattern, respectively. We then improve the KMP-based algorithm by using an automaton, which has a better experimental running time. The next proposed algorithms are based on the Boyer–Moore algorithm and the Horspool algorithm that try to perform pattern matching. These algorithms are the fastest experimental algorithms. Furthermore, we propose an extension of the AC-automaton algorithm that can solve dictionary matching on multi-tracks, which is a task to find multiple multi-track patterns in a multi-track text. Finally, we propose filtering algorithms that can perform permuted pattern matching quickly in practice.