Hashing and Indexing: Succinct DataStructures and Smoothed Analysis

We consider the problem of indexing a text T (of length n) with a light data structure that supports efficient search of patterns P (of length m) allowing errors under the Hamming distance. We propose a hash-based strategy that employs two classes of hash functions—dubbed Hamming-aware and de Bruijn—to drastically reduce search space and memory footprint of the index, respectively. We use our succinct hash data structure to solve the k-mismatch search problem in 2n log σ+o(n log σ) bits of space with a randomized algorithm having smoothed complexity O((2σ)(logn)(logm+ ξ) + (occ+ 1) ·m), where σ is the alphabet size, occ is the number of occurrences, and ξ is a term depending on m, n, and on the amplitude of the noise perturbing text and pattern. Significantly, we obtain that for any > 0, for m large enough, ξ ∈ O(logm): our results improve upon previous linear-space solutions of the k-mismatch problem.