Approximate pattern matching with k-mismatches in packed text

Given strings P of length m and T of length n over an alphabet of size σ, the string matching with k-mismatches problem is to find the positions of all the substrings in T that are at Hamming distance at most k from P . If T can be read only one character at the time the best known bounds are O(n √ k log k) and O(n+ n √ k/w log k) in the word-RAM model with word length w. In the RAM models (including AC and word-RAM) it is possible to read up to ⌊w/ log σ⌋ characters in constant time if the characters of T are encoded using ⌈log σ⌉ bits. The only solution for k-mismatches in packed text works in O((n log σ/ log n)⌈m log(k+log n/ log σ)/w⌉+n) time, for any ε > 0. We present an algorithm that runs in time O( n ⌊w/(m log σ)⌋ (1 + logmin(k, σ) logm/ log σ)) in the AC model if m = O(w/ log σ) and T is given packed. We also describe a simpler variant that runs in time O( n ⌊w/(m log σ)⌋ logmin(m, logw/ log σ)) in the word-RAMmodel. The algorithms improve the existing bound for w = Ω(log n), for any ǫ > 0. Based on the introduced technique, we present algorithms for several other approximate matching problems.