Constructing small tree grammars and small circuits for formulas

Moses Ganardi; Danny Hucke; Artur Jez; Markus Lohrey; Eric Noeth

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arXiv · 1407.4286

Constructing small tree grammars and small circuits for formulas

Moses Ganardi·Danny Hucke·Artur Jez·Markus Lohrey·Eric Noeth

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Abstract

It is shown that every tree of size $n$ over a fixed set of $\sigma$ different ranked symbols can be decomposed (in linear time as well as in logspace) into $O\big(\frac{n}{\log_\sigma n}\big) = O\big(\frac{n \log \sigma}{\log n}\big)$ many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size $O\big(\frac{n}{\log_\sigma n}\big)$, which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case size of the computed grammar, except for the top dag of Bille et al., for which only the weaker upper bound of $O\big(\frac{n}{\log_\sigma^{0.19} n}\big)$ (which was very recently improved to $O\big(\frac{n \cdot \log \log_\sigma n}{\log_\sigma n}\big)$ by H\"ubschle-Schneider and Raman) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size $n$, in which only $m \leq n$ different variables occur, can be transformed (in linear time as well as in logspace) into an arithmetical circuit of size $O\big(\frac{n \cdot \log m}{\log n}\big)$ and depth $O(\log n)$. This refines a classical result of Brent from 1974, according to which an arithmetical formula of size $n$ can be transformed into a logarithmic depth circuit of size $O(n)$.

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BibTeX RIS

Moses Ganardi, Danny Hucke, Artur Jez, Markus Lohrey, Eric Noeth. 2014-07-16. Constructing small tree grammars and small circuits for formulas. https://arxiv.org/abs/1407.4286

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