Linear-Size CDAWG: New Repetition-Aware Indexing and Grammar Compression

Abstract

In this paper, we propose a novel approach to combine compact directed acyclic word graphs (CDAWGs) and grammar-based compression. This leads us to an efficient self-index, called Linear-size CDAWGs (L-CDAWGs), which can be represented with $O(\tilde{e}_T \log n)$ bits of space allowing for $O(\log n)$-time random and O(1)-time sequential accesses to edge labels, and $O(m \log \sigma + occ)$-time pattern matching. Here, $\tilde{e}_T$ is the number of all extensions of maximal repeats in T, n and m are respectively the lengths of the text T and a given pattern, $\sigma$ is the alphabet size, and $occ$ is the number of occurrences of the pattern in T. The repetitiveness measure $\tilde{e}_T$ is known to be much smaller than the text length n for highly repetitive text. For constant alphabets, our L-CDAWGs achieve $O(m + occ )$ pattern matching time with $O(e_T^r \log n)$ bits of space, which improves the pattern matching time of Belazzougui et al.’s run-length BWT-CDAWGs by a factor of $\log \log n$, with the same space complexity. Here, $e_T^r$ is the number of right extensions of maximal repeats in T. As a byproduct, our result gives a way of constructing a straight-line program (SLP) of size $O(\tilde{e}_T)$ for a given text T in $O(n + \tilde{e}_T \log \sigma )$ time.