Complexity of Sequential Pattern Matching Algorithms

We formally define a class of sequential pattern matching algorithms that includes all variations of Morris-Pratt algorithm. For last twenty years it was known that complexity of such algorithms are bounded by a linear function of the text string length. Recenlly, substantial progress has been made in identifying lower bounds. However, it was not known whether really there exists asymptotically a linearity constant. We prove this fact rigorously for the worst case and the average case using Subadditive Ergodic Theorem. We additionally prove an almost sure cOIlvergence. Our results hold for any given pattern and text and for stationary ergodic pattern and text providing the length of the pattern is order of magnitude smaller than the square root of the text length. In the course of the proof, we also establish some structural property of Morris-Pratt-like algorithms. Namely, we prove the exlstence of "unavoidable positions" where the algorithm must stop to compare. This property seems to be uniquely reserved for Morris-Pratt type algorithms since as, we point out in our concluding remarks, a popular pattern matching algorithm proposed by Boyer and Moore does not possess this property.