The Dynamic k-Mismatch Problem
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The text-to-pattern Hamming distances problem asks to compute the Hamming distances between a given pattern of length m and all length-m substrings of a given text of length n≥ m. We focus on the k-mismatch version of the problem, where a distance needs to be returned only if it does not exceed a threshold k. We assume n≤ 2m (in general, one can partition the text into overlapping blocks). In this work, we show data structures for the dynamic version of this problem supporting two operations: An update performs a single-letter substitution in the pattern or the text, and a query, given an index i, returns the Hamming distance between the pattern and the text substring starting at position i, or reports that it exceeds k. First, we show a data structure with Õ(1) update and Õ(k) query time. Then we show that Õ(k) update and Õ(1) query time is also possible. These two provide an optimal trade-off for the dynamic k-mismatch problem with k ≤√(n): we prove that, conditioned on the strong 3SUM conjecture, one cannot simultaneously achieve k^1-Ω(1) time for all operations. For k≥√(n), we give another lower bound, conditioned on the Online Matrix-Vector conjecture, that excludes algorithms taking n^1/2-Ω(1) time per operation. This is tight for constant-sized alphabets: Clifford et al. (STACS 2018) achieved Õ(√(n)) time per operation in that case, but with Õ(n^3/4) time per operation for large alphabets. We improve and extend this result with an algorithm that, given 1≤ x≤ k, achieves update time Õ(n/k +√(nk/x)) and query time Õ(x). In particular, for k≥√(n), an appropriate choice of x yields Õ(√(nk)) time per operation, which is Õ(n^2/3) when no threshold k is provided.
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