Balancing Straight-Line Programs

02/10/2019

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by   Moses Ganardi, et al.

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It is shown that a context-free grammar of size m that produces a single string w (such a grammar is also called a string straight-line program) can be transformed in linear time into a context-free grammar for w of size O(m), whose unique derivation tree has depth O( |w|). This solves an open problem in the area of grammar-based compression. Similar results are shown for two formalism for grammar-based tree compression: top dags and forest straight-line programs. These balancing results are all deduced from a single meta theorem stating that the depth of an algebraic circuit over an algebra with a certain finite base property can be reduced to O( n) with the cost of a constant multiplicative size increase. Here, n refers to the size of the unfolding (or unravelling) of the circuit.