Querying languages over sliding windows

We study the space complexity of querying languages over data streams in the sliding window model. The algorithm has to answer at any point of time whether the content of the sliding window belongs to a fixed regular language. For regular languages, a trichotomy is shown: For every regular language the optimal space requirement is asymptotically either constant, logarithmic, or linear in the size of the sliding window. Moreover, given a DFA for the language, one can check in nondeterministic logspace (and hence polynomial time), which of the three cases holds. For context-free languages the trichotomy no longer holds: For every c ≥ 1 there exists a context-free language Lc for which the optimal space required to query Lc in the sliding window model is asymptotically n. Finally, it is shown that all results also hold for randomized (Monte-Carlo) streaming algorithms. This paper is an extended version of the conference version [17].