On Regular Expression Matching and Deterministic Finite Automata

Given a regular expression R and a string T the regular expression matching problem is to determine if T matches any string in the language generated by R. The best known solution to the problem uses linear space and O ( nm log logn log3/2 n +n+m ) time in the worst-case [2], where m and n are the lengths of R and T , respectively. A common misconception is that we can solve the problem efficiently by building a deterministic finite automaton (DFA) for R using 2O(m) space and then run it on T in O(n) time [1]. However, this analysis completely ignores issues of addressing into exponential sized data structures. An address in a DFA of size 2Ω(m) requires Ω(m) bits. Hence, on a standard unit-cost word RAM with word length Θ(logn) [3], we need at least Ω(m/ logn) time to simply write an address in the DFA. It follows that traversing the DFA for R uses at least Ω(nm/ logn+n+m) worst-case time (note that we do not even include DFA construction time). This bound can only be O(n) when m = O(logn) and is never better than the above best known bound. BODY Even ignoring construction time, deterministic finite automata do not solve regular expression matching in worst-case linear time.