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

Lightweight Lempel-Ziv Parsing

We introduce a new approach to LZ77 factorization that uses O(n/d) words of working space and O(dn) time for any d >= 1 (for polylogarithmic alphabet sizes). We also describe carefully engineered implementations of alternative approaches to lightweight LZ77 factorization. Extensive experiments show that the new algorithm is superior in most cases, particularly at the lowest memory levels and for highly repetitive data. As a part of the algorithm, we describe new methods for computing matching statistics which may be of independent interest.Comment: 12 page

A really simple approximation of smallest grammar

In this paper we present a really simple linear-time algorithm constructing a context-free grammar of size O(g log (N/g)) for the input string, where N is the size of the input string and g the size of the optimal grammar generating this string. The algorithm works for arbitrary size alphabets, but the running time is linear assuming that the alphabet Sigma of the input string can be identified with numbers from 1,ldots, N^c for some constant c. Algorithms with such an approximation guarantee and running time are known, however all of them were non-trivial and their analyses were involved. The here presented algorithm computes the LZ77 factorisation and transforms it in phases to a grammar. In each phase it maintains an LZ77-like factorisation of the word with at most l factors as well as additional O(l) letters, where l was the size of the original LZ77 factorisation. In one phase in a greedy way (by a left-to-right sweep and a help of the factorisation) we choose a set of pairs of consecutive letters to be replaced with new symbols, i.e. nonterminals of the constructed grammar. We choose at least 2/3 of the letters in the word and there are O(l) many different pairs among them. Hence there are O(log N) phases, each of them introduces O(l) nonterminals to a grammar. A more precise analysis yields a bound O(l log(N/l)). As l \leq g, this yields the desired bound O(g log(N/g)).Comment: Accepted for CPM 201

OPTIMIZING LEMPEL-ZIV FACTORIZATION FOR THE GPU ARCHITECTURE

Lossless data compression is used to reduce storage requirements, allowing for the relief of I/O channels and better utilization of bandwidth. The Lempel-Ziv lossless compression algorithms form the basis for many of the most commonly used compression schemes. General purpose computing on graphic processing units (GPGPUs) allows us to take advantage of the massively parallel nature of GPUs for computations other that their original purpose of rendering graphics. Our work targets the use of GPUs for general lossless data compression. Specifically, we developed and ported an algorithm that constructs the Lempel-Ziv factorization directly on the GPU. Our implementation bypasses the sequential nature of the LZ factorization and attempts to compute the factorization in parallel. By breaking down the LZ factorization into what we call the PLZ, we are able to outperform the fastest serial CPU implementations by up to 24x and perform comparatively to a parallel multicore CPU implementation. To achieve these speeds, our implementation outputted LZ factorizations that were on average only 0.01 percent greater than the optimal solution that what could be computed sequentially. We are also able to reevaluate the fastest GPU suffix array construction algorithm, which is needed to compute the LZ factorization. We are able to find speedups of up to 5x over the fastest CPU implementations