Online Grammar Compression for Frequent Pattern Discovery

Various grammar compression algorithms have been proposed in the last decade. A grammar compression is a restricted CFG deriving the string deterministically. An efficient grammar compression develops a smaller CFG by finding duplicated patterns and removing them. This process is just a frequent pattern discovery by grammatical inference. While we can get any frequent pattern in linear time using a preprocessed string, a huge working space is required for longer patterns, and the whole string must be loaded into the memory preliminarily. We propose an online algorithm approximating this problem within a compressed space. The main contribution is an improvement of the previously best known approximation ratio $\Omega(\frac{1}{\lg^2m})$ to $\Omega(\frac{1}{\lg^*N\lg m})$ where $m$ is the length of an optimal pattern in a string of length $N$ and $\lg^*$ is the iteration of the logarithm base $2$. For a sufficiently large $N$, $\lg^*N$ is practically constant. The experimental results show that our algorithm extracts nearly optimal patterns and achieves a significant improvement in memory consumption compared to the offline algorithm.Comment: 14 page

Deterministic Sparse Suffix Sorting in the Restore Model

Given a text $T$ of length $n$, we propose a deterministic online algorithm computing the sparse suffix array and the sparse longest common prefix array of $T$ in $O(c \sqrt{\lg n} + m \lg m \lg n \lg^* n)$ time with $O(m)$ words of space under the premise that the space of $T$ is rewritable, where $m \le n$ is the number of suffixes to be sorted (provided online and arbitrarily), and $c$ is the number of characters with $m \le c \le n$ that must be compared for distinguishing the designated suffixes