A note on optimal parallel transformations of regular expressions to nondeterministic finite automata

Abstract

In this note we show that the transformation of regular expressions to corresponding nondeterministic finite automata can be done in log n time using n/log n processors on a parallel random-access machine without write conflicts (P-RAM). It is easy to parallelize the classical transformation of Hopcroft and Ullman [5]. The parallelization of the more economical transformation of Sedgewick [8] is more interesting and it is obtained by applying (in two stages) a homomorphism which relates the automaton obtained through the transformation from [5] to the one obtained through the transformation from [8]. We also deal with Sedgewick's transformation which is more economic (compare Fig. 1 and Fig. 4) than the one presented in [5], though it does not change the order of the complexity of algorithms obtained. We develop the following technique for the recursive construction. The tree of the recursion is constructed and some parameters associated with the nodes of this tree (pairs of states) are computed. This is the main step. Then, instead of simulating the recursion bottom-up or top-down, appropriate constructing actions are executed for each node of the recursion tree in parallel. The previously computed parameters (associated with the nodes) are such that the action for a given node is, at this stage, easy and independent of the action for any other node.