Constructing Small Tree Grammars and Small Circuits for Formulas

It is shown that every tree of size n over a fixed set of σ different ranked symbols can be decomposed into O( n logσ n ) = O(n log σ 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 logσ 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. [4], for which only the weaker upper bound of O( n log0.19 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( m log n ) 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).