The Average Transition Complexity of Glushkov and Partial Derivative Automata

In this paper, the relation between the Glushkov automaton (

$\mathcal{A}_{\mathrm{pos}}$

) and the partial derivative automaton (

$\mathcal{A}_{\mathrm{pd}}$

) of a given regular expression, in terms of transition complexity, is studied. The average transition complexity of

$\mathcal{A}_{\mathrm{pos}}$

was proved by Nicaud to be linear in the size of the corresponding expression. This result was obtained using an upper bound of the number of transitions of

$\mathcal{A}_{\mathrm{pos}}$

. Here we present a new quadratic construction of

$\mathcal{A}_{\mathrm{pos}}$

that leads to a more elegant and straightforward implementation, and that allows the exact counting of the number of transitions. Based on that, a better estimation of the average size is presented. Asymptotically, and as the alphabet size grows, the number of transitions per state is on average 2.

Broda

et al.

computed an upper bound for the ratio of the number of states of

$\mathcal{A}_{\mathrm{pd}}$

to the number of states of

$\mathcal{A}_{\mathrm{pos}}$

, which is about

$\frac{1}{2}$

for large alphabet sizes. Here we show how to obtain an upper bound for the number of transitions in

$\mathcal{A}_{\mathrm{pd}}$

, which we then use to get an average case approximation. Some experimental results are presented that illustrate the quality of our estimate.