Compressed Structures for Partial Derivative Automata Constructions

Abstract

The partial derivative automaton (A(PD)) is an elegant simulation of a regular expression. Although it is, in general, smaller than the position automaton (A(POS)), the algorithms that build A(PD) in quadratic worst-case time first compute A(POS). Asymptotically, and on average for the uniform distribution, the size of A(PD) is half the size of A(POS), being both linear on the size of the expression. We address the construction of A(PD) efficiently, on average, avoiding the computation of A(POS). The expression and the set of its partial derivatives are represented by a directed acyclic graph with shared common subexpressions. We develop an algorithm for building A(PDS) from expressions in strong star normal form of size n that runs, on average, in time that asymptotically behaves as n(3/2) (4)root log(n), and space as n(3/2)/(log n)(3/4). Empirical results corroborate its good practical performance.