Deterministic regular expressions in linear time

Abstract

Deterministic regular expressions are widely used in XML processing. For instance, all regular expressions in DTDs and XML Schemas are required to be deterministic. In this paper we show that determinism of a regular expression e can be tested in linear time. The best known algorithms, based on the Glushkov automaton, require O(σ|e|) time, where σ is the number of distinct symbols in e. We further show that matching a word w against an expression e can be achieved in combined linear time O(|e| + |w|), for a wide range of deterministic regular expressions: (i) star-free (for multiple input words), (ii) bounded-occurrence, i.e., expressions in which each symbol appears a bounded number of times, and (iii) bounded plus-depth, i.e., expressions in which the nesting depth of alternating plus (union) and concatenation symbols is bounded. Our algorithms use a new structural decomposition of the parse tree of e. For matching arbitrary deterministic regular expressions we present an O(|e| + |w| log log |e|) time algorithm.

Minimal partial languages and automata

“…Checking all possible configurations of a loop for every loop used in a regular expression for the language is intractable. We could use a standardized configuration of a loop, such as the unambiguous form from [5]. For a regular expression composed of regular expressions x, y, define x(yx) * to be the unambiguous form, as a DFA is easily constructed from it.…”

Minimal Partial Languages and Automata

“…Checking all possible configurations of a loop for every loop used in a regular expression for the language is intractable. We could use a standardized configuration of a loop, such as the unambiguous form from [14]. For a regular expression composed of regular expressions x, y, define x(yx) * to be the unambiguous form, as a DFA is easily constructed from it.…”

Deciding definability by deterministic regular expressions

“…Since then, DREs have been studied in the context of language approximations [15], learning [16], descriptional complexity [17,18] and static analysis [19,20]. Recently, it was shown that testing if a regular expression is deterministic can be done in linear time [21].…”