Hucke, Danny ;
Lohrey, Markus ;
Noeth, Eric
Constructing Small Tree Grammars and Small Circuits for Formulas
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 straightline linear contextfree 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 grammarbased tree compressors were not analyzed for the worstcase 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 = {457468},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897774},
ISSN = {18688969},
year = {2014},
volume = {29},
editor = {Venkatesh Raman and S. P. Suresh},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2014/4863},
URN = {urn:nbn:de:0030drops48639},
doi = {10.4230/LIPIcs.FSTTCS.2014.457},
annote = {Keywords: grammarbased compression, tree compression, arithmetical circuits}
}
2014
Keywords: 

grammarbased compression, tree compression, arithmetical circuits 
Seminar: 

34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

Issue date: 

2014 
Date of publication: 

2014 