Partial Derivative Automaton by Compressing Regular Expressions

Abstract

The partial derivative automaton ($\mathcal {A}_{{{\,\mathrm{\mathsf {PD}}\,}}}$) is an elegant simulation of a regular expression. Although it is, in general, smaller than the position automaton ($\mathcal {A}_{{{\,\mathrm{\mathsf {POS}}\,}}}$), the algorithms that build $\mathcal {A}_{{{\,\mathrm{\mathsf {PD}}\,}}}$ in quadratic worst-case time, first compute $\mathcal {A}_{{{\,\mathrm{\mathsf {POS}}\,}}}$. Asymptotically, and on average for the uniform distribution, the size of $\mathcal {A}_{{{\,\mathrm{\mathsf {PD}}\,}}}$ is half the size of $\mathcal {A}_{{{\,\mathrm{\mathsf {POS}}\,}}}$, being both linear on the size of the expression. We address the construction of $\mathcal {A}_{{{\,\mathrm{\mathsf {PD}}\,}}}$ efficiently, on average, avoiding the computation of $\mathcal {A}_{{{\,\mathrm{\mathsf {POS}}\,}}}$. 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 $\mathcal {A}_{{{\,\mathrm{\mathsf {PD}}\,}}}$’s from expressions in strong star normal form of size n that runs in time $O\big (n^{3/2}\,\root 4 \of {\log (n)}\,\big )$ and space $O\big (n^{3/2}/(\log n)^{3/4}\,\big )$, on average. Empirical results corroborate its good practical performance.

Research supported by NSERC (Canada) and by CMUP through FCT project UIDB/00144/2020.