Streaming k-mismatch with error correcting and applications

Abstract: We present a new streaming algorithm for the $k$-Mismatch problem, one of the most basic problems in pattern matching. Given a pattern and a text, the task is to find all substrings of the text that are at the Hamming distance at most $k$ from the pattern. Our algorithm is enhanced with an important new feature called Error Correcting, and its complexities for $k=1$ and for a general $k$ are comparable to those of the solutions for the $k$-Mismatch problem by Porat and Porat (FOCS 2009) and Clifford et al. (SODA 2016). In parallel to our research, a yet more efficient algorithm for the $k$-Mismatch problem with the Error Correcting feature was developed by Clifford et al. (SODA 2019). Using the new feature and recent work on streaming Multiple Pattern Matching we develop a series of streaming algorithms for pattern matching on weighted strings, which are a commonly used representation of uncertain sequences in molecular biology. We also show that these algorithms are space-optimal up to polylog factors. A preliminary version of this work was published at DCC 2017 conference.