Tree Compression with Top Trees Revisited

We revisit tree compression with top trees (Bille et al, ICALP'13) and present several improvements to the compressor and its analysis. By significantly reducing the amount of information stored and guiding the compression step using a RePair-inspired heuristic, we obtain a fast compressor achieving good compression ratios, addressing an open problem posed by Bille et al. We show how, with relatively small overhead, the compressed file can be converted into an in-memory representation that supports basic navigation operations in worst-case logarithmic time without decompression. We also show a much improved worst-case bound on the size of the output of top-tree compression (answering an open question posed in a talk on this algorithm by Weimann in 2012).Comment: SEA 201

Searching for Smallest Grammars on Large Sequences and Application to DNA

International audienceMotivated by the inference of the structure of genomic sequences, we address here the smallest grammar problem. In previous work, we introduced a new perspective on this problem, splitting the task into two different optimization problems: choosing which words will be considered constituents of the final grammar and finding a minimal parsing with these constituents. Here we focus on making these ideas applicable on large sequences. First, we improve the complexity of existing algorithms by using the concept of maximal repeats when choosing which substrings will be the constituents of the grammar. Then, we improve the size of the grammars by cautiously adding a minimal parsing optimization step. Together, these approaches enable us to propose new practical algorithms that return smaller grammars (up to 10\%) in approximately the same amount of time than their competitors on a classical set of genomic sequences and on whole genomes of model organisms

Fast Label Extraction in the CDAWG

The compact directed acyclic word graph (CDAWG) of a string $T$ of length $n$ takes space proportional just to the number $e$ of right extensions of the maximal repeats of $T$, and it is thus an appealing index for highly repetitive datasets, like collections of genomes from similar species, in which $e$ grows significantly more slowly than $n$. We reduce from $O(m\log{\log{n}})$ to $O(m)$ the time needed to count the number of occurrences of a pattern of length $m$, using an existing data structure that takes an amount of space proportional to the size of the CDAWG. This implies a reduction from $O(m\log{\log{n}}+\mathtt{occ})$ to $O(m+\mathtt{occ})$ in the time needed to locate all the $\mathtt{occ}$ occurrences of the pattern. We also reduce from $O(k\log{\log{n}})$ to $O(k)$ the time needed to read the $k$ characters of the label of an edge of the suffix tree of $T$, and we reduce from $O(m\log{\log{n}})$ to $O(m)$ the time needed to compute the matching statistics between a query of length $m$ and $T$, using an existing representation of the suffix tree based on the CDAWG. All such improvements derive from extracting the label of a vertex or of an arc of the CDAWG using a straight-line program induced by the reversed CDAWG.Comment: 16 pages, 1 figure. In proceedings of the 24th International Symposium on String Processing and Information Retrieval (SPIRE 2017). arXiv admin note: text overlap with arXiv:1705.0864

Lempel-Ziv Parsing in External Memory

For decades, computing the LZ factorization (or LZ77 parsing) of a string has been a requisite and computationally intensive step in many diverse applications, including text indexing and data compression. Many algorithms for LZ77 parsing have been discovered over the years; however, despite the increasing need to apply LZ77 to massive data sets, no algorithm to date scales to inputs that exceed the size of internal memory. In this paper we describe the first algorithm for computing the LZ77 parsing in external memory. Our algorithm is fast in practice and will allow the next generation of text indexes to be realised for massive strings and string collections.Comment: 10 page

Fast and Space-Efficient Construction of AVL Grammars from the LZ77 Parsing

Grammar compression is, next to Lempel-Ziv (LZ77) and run-length Burrows-Wheeler transform (RLBWT), one of the most flexible approaches to representing and processing highly compressible strings. The main idea is to represent a text as a context-free grammar whose language is precisely the input string. This is called a straight-line grammar (SLG). An AVL grammar, proposed by Rytter [Theor. Comput. Sci., 2003] is a type of SLG that additionally satisfies the AVL property: the heights of parse trees for children of every nonterminal differ by at most one. In contrast to other SLG constructions, AVL grammars can be constructed from the LZ77 parsing in compressed time: ?(z log n) where z is the size of the LZ77 parsing and n is the length of the input text. Despite these advantages, AVL grammars are thought to be too large to be practical. We present a new technique for rapidly constructing a small AVL grammar from an LZ77 or LZ77-like parse. Our algorithm produces grammars that are always at least five times smaller than those produced by the original algorithm, and usually not more than double the size of grammars produced by the practical Re-Pair compressor [Larsson and Moffat, Proc. IEEE, 2000]. Our algorithm also achieves low peak RAM usage. By combining this algorithm with recent advances in approximating the LZ77 parsing, we show that our method has the potential to construct a run-length BWT in about one third of the time and peak RAM required by other approaches. Overall, we show that AVL grammars are surprisingly practical, opening the door to much faster construction of key compressed data structures