arXiv subjects

Michał Gańczorz

Publications and source records attributed to Michał Gańczorz.

3 recordsLinked to original sources

arXiv2020

Entropy bounds for grammar compression

Grammar compression represents a string as a context free grammar. Achieving compression requires encoding such grammar as a binary string; there are a few commonly used encodings. We bound the size of practically used encodings for several heuristical compression methods, including \RePair and \Greedy algorithms: the standard encoding of \RePair, which combines entropy coding and special encoding of a grammar, achieves $1.5|S|H_k(S)$, where $H_k(S)$ is $k$-th order entropy of $S$. We also show that by stopping after some iteration we can achieve $|S|H_k(S)$. This is particularly interesting, as it explains a phenomenon observed in practice: introducing too many nonterminals causes the bit-size to grow. We generalize our approach to other compression methods like \Greedy and a wide class of irreducible grammars as well as to other practically used bit encodings (including naive, which uses fixed-length codes). Our approach not only proves the bounds but also partially explains why \Greedy and \RePair are much better in practice than other grammar based methods. In some cases we argue that our estimations are optimal. The tools used in our analysis are of independent interest: we prove the new, optimal, bounds on the zeroth order entropy of parsing of a string.

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arXiv2019

Towards Better Compressed Representations

We introduce the problem of computing a parsing where each phrase is of length at most $m$ and which minimizes the zeroth order entropy of parsing. Based on the recent theoretical results we devise a heuristic for this problem. The solution has straightforward application in succinct text representations and gives practical improvements. Moreover the proposed heuristic yields structure whose size can be bounded both by $|S|H_{m-1}(S)$ and by $|S|/m(H_0(S) + \cdots + H_{m-1})$, where $H_{k}(S)$ is the $k$-th order empirical entropy of $S$. We also consider a similar problem in which the first-order entropy is minimized.

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arXiv2018

Using statistical encoding to achieve tree succinctness never seen before

We propose a new succinct representation of labeled trees which represents a tree T using |T|H_k(T) number of bits (plus some smaller order terms), where |T|H_k(T) denotes the k-th order (tree label) entropy, as defined by Ferragina at al. 2005. Our representation employs a new, simple method of partitioning the tree, which preserves both tree shape and node degrees. Previously, the only representation that used |T|H_k(T) bits was based on XBWT, a transformation that linearizes tree labels into a single string, combined with compression boosting. The proposed representation is much simpler than the one based on XBWT, which used additional linear space (bounded by 0.01n) hidden in the "smaller order terms" notion, as an artifact of using zeroth order entropy coder; our representation uses sublinear additional space (for reasonable values of k and size of the label alphabet σ). The proposed representation can be naturally extended to a succinct data structure for trees, which uses |T|H_k(T) plus additional O(|T|k log_σ/ log_σ |T| + |T| log log_σ |T|/ log_σ |T|) bits and supports all the usual navigational queries in constant time. At the cost of increasing the query time to O(log log |T|/ log |T|) we can further reduce the space redundancy to O(|T| log log |T|/ log_σ |T|) bits, assuming k <= log_σ |T|. This is a major improvement over representation based on XBWT: even though XBWT-based representation uses |T|H_k(T) bits, the space needed for structure supporting navigational queries is much larger: (...)

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