Sliding Window String Indexing in Streams
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Given a string S over an alphabet Σ, the 'string indexing problem' is to preprocess S to subsequently support efficient pattern matching queries, i.e., given a pattern string P report all the occurrences of P in S. In this paper we study the 'streaming sliding window string indexing problem'. Here the string S arrives as a stream, one character at a time, and the goal is to maintain an index of the last w characters, called the 'window', for a specified parameter w. At any point in time a pattern matching query for a pattern P may arrive, also streamed one character at a time, and all occurrences of P within the current window must be returned. The streaming sliding window string indexing problem naturally captures scenarios where we want to index the most recent data (i.e. the window) of a stream while supporting efficient pattern matching. Our main result is a simple O(w) space data structure that uses O(log w) time with high probability to process each character from both the input string S and the pattern string P. Reporting each occurrence from P uses additional constant time per reported occurrence. Compared to previous work in similar scenarios this result is the first to achieve an efficient worst-case time per character from the input stream. We also consider a delayed variant of the problem, where a query may be answered at any point within the next δ characters that arrive from either stream. We present an O(w + δ) space data structure for this problem that improves the above time bounds to O(log(w/δ)). In particular, for a delay of δ = ϵ w we obtain an O(w) space data structure with constant time processing per character. The key idea to achieve our result is a novel and simple hierarchical structure of suffix trees of independent interest, inspired by the classic log-structured merge trees.
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