Detecting false matches in string matching algorithms

Consider a text string of length n, a pattern string of length m, and a match vector of length n which declares each location in the text to be either a mismatch (the pattern does not occur beginning at that location in the text) or a potential match (the pattern may occur beginning at that location in the text). Some of the potential matches could be false, i.e., the pattern may not occur beginning at some location in the text declared to be a potential match. We investigate the complexity of two problems in this context, namely, checking if there is any false match, and identifying all the false matches in the match vector. We present an algorithm on the CRCW PRAM that checks if there exists any false match in O(1) time using O(n) processors. Since string matching takes Ω(log log m) time on the CRCW PRAM, checking for false matches is provably simpler than string matching. As an important application, we use this simple algorithm to convert the Karp-Rabin Monte Carlo type string matching algorithm into a Las Vegas type algorithm without asymptotic loss in complexity. We also present an efficient algorithm for identifying all the false matches and as a consequence, show that string matching algorithms take Ω(log log m) time even given the flexibility to output a few false matches. In addition, we give a sequential algorithm for checking using three heads on a 2-way deterministic finite slate automaton (DFA) in linear time and another on a 1-way DFA with a fixed number of heads.