Finger Search in Grammar-Compressed Strings

Grammar-based compression, where one replaces a long string by a small context-free grammar that generates the string, is a simple and powerful paradigm that captures many popular compression schemes. Given a grammar, the random access problem is to compactly represent the grammar while supporting random access, that is, given a position in the original uncompressed string report the character at that position. In this paper we study the random access problem with the finger search property, that is, the time for a random access query should depend on the distance between a specified index $f$, called the \emph{finger}, and the query index $i$. We consider both a static variant, where we first place a finger and subsequently access indices near the finger efficiently, and a dynamic variant where also moving the finger such that the time depends on the distance moved is supported. Let $n$ be the size the grammar, and let $N$ be the size of the string. For the static variant we give a linear space representation that supports placing the finger in $O(\log N)$ time and subsequently accessing in $O(\log D)$ time, where $D$ is the distance between the finger and the accessed index. For the dynamic variant we give a linear space representation that supports placing the finger in $O(\log N)$ time and accessing and moving the finger in $O(\log D + \log \log N)$ time. Compared to the best linear space solution to random access, we improve a $O(\log N)$ query bound to $O(\log D)$ for the static variant and to $O(\log D + \log \log N)$ for the dynamic variant, while maintaining linear space. As an application of our results we obtain an improved solution to the longest common extension problem in grammar compressed strings. To obtain our results, we introduce several new techniques of independent interest, including a novel van Emde Boas style decomposition of grammars

Bicriteria data compression

The advent of massive datasets (and the consequent design of high-performing distributed storage systems) have reignited the interest of the scientific and engineering community towards the design of lossless data compressors which achieve effective compression ratio and very efficient decompression speed. Lempel-Ziv's LZ77 algorithm is the de facto choice in this scenario because of its decompression speed and its flexibility in trading decompression speed versus compressed-space efficiency. Each of the existing implementations offers a trade-off between space occupancy and decompression speed, so software engineers have to content themselves by picking the one which comes closer to the requirements of the application in their hands. Starting from these premises, and for the first time in the literature, we address in this paper the problem of trading optimally, and in a principled way, the consumption of these two resources by introducing the Bicriteria LZ77-Parsing problem, which formalizes in a principled way what data-compressors have traditionally approached by means of heuristics. The goal is to determine an LZ77 parsing which minimizes the space occupancy in bits of the compressed file, provided that the decompression time is bounded by a fixed amount (or vice-versa). This way, the software engineer can set its space (or time) requirements and then derive the LZ77 parsing which optimizes the decompression speed (or the space occupancy, respectively). We solve this problem efficiently in O(n log^2 n) time and optimal linear space within a small, additive approximation, by proving and deploying some specific structural properties of the weighted graph derived from the possible LZ77-parsings of the input file. The preliminary set of experiments shows that our novel proposal dominates all the highly engineered competitors, hence offering a win-win situation in theory&practice