License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2014.457
URN: urn:nbn:de:0030-drops-48639
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2014/4863/
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Hucke, Danny ; Lohrey, Markus ; Noeth, Eric

Constructing Small Tree Grammars and Small Circuits for Formulas

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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 into O(n/log_sigma(n)) = O((n * log(sigma))/ log(n)) many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size O(n/log_sigma(n)), 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(n/log^{0.19}(n)) 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 <= n different variables occur, can be transformed (in time O(n * log(n)) into an arithmetical circuit of size O((n * log(m))/log(n)) and depth O(log(n)). This refines a classical result of Brent, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O(n).

BibTeX - Entry

@InProceedings{hucke_et_al:LIPIcs:2014:4863,
  author =	{Danny Hucke and Markus Lohrey and Eric Noeth},
  title =	{{Constructing Small Tree Grammars and Small Circuits for Formulas}},
  booktitle =	{34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)},
  pages =	{457--468},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-77-4},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{29},
  editor =	{Venkatesh Raman and S. P. Suresh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4863},
  URN =		{urn:nbn:de:0030-drops-48639},
  doi =		{10.4230/LIPIcs.FSTTCS.2014.457},
  annote =	{Keywords: grammar-based compression, tree compression, arithmetical circuits}
}

Keywords: grammar-based compression, tree compression, arithmetical circuits
Collection: 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)
Issue Date: 2014
Date of publication: 12.12.2014

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