Height-bounded Lempel-Ziv encodings

We introduce height-bounded LZ encodings (LZHB), a new family of compressed representations that are variants of Lempel-Ziv parsings with a focus on bounding the worst-case access time to arbitrary positions in the text directly via the compressed representation. An LZ-like encoding is a partitioning of the string into phrases of length $1$ which can be encoded literally, or phrases of length at least $2$ which have a previous occurrence in the string and can be encoded by its position and length. An LZ-like encoding induces an implicit referencing forest on the set of positions of the string. An LZHB encoding is an LZ-like encoding where the height of the implicit referencing forest is bounded. An LZHB encoding with height constraint $h$ allows access to an arbitrary position of the underlying text using $O(h)$ predecessor queries. While computing the smallest LZHB encoding efficiently seems to be difficult [Cicalese \& Ugazio 2024, arxiv], we give the first linear time algorithm for strings over a constant size alphabet that computes the greedy LZHB encoding, i.e., the string is processed from beginning to end, and the longest prefix of the remaining string that can satisfy the height constraint is taken as the next phrase. Our algorithms significantly improve both theoretically and practically, the very recently and independently proposed algorithms by Lipt\'ak et al. (arxiv, to appear at CPM 2024). We also analyze the size of height bounded LZ encodings in the context of repetitiveness measures, and show for some constant $c$, the size $z_{HB}$ of the optimal LZHB encoding with height bound $c\log n$ is $O(g_{rl})$, where $g_{rl}$ is the size of the smallest run-length grammar. We also show $z_{HB} = o(g_{rl})$ for some family of strings, making $z_{HB}$ one of the smallest known repetitiveness measures for which $O({\sf polylog} n)$ time access is possible using linear space.