Bit-Parallel Witnesses and Their Applications to Approximate String Matching

Abstract

We present a new bit-parallel technique for approximate string matching. We build on two previous techniques. The first one, BPM (Myers, 1999), searches for a pattern of length m in a text of length n permitting k differences in $O(\lceil m/w \rceil n)$ time, where w is the width of the computer word. The second one, ABNDM (Navarro and Raffinot, 2000), extends a sublinear-time exact algorithm to approximate searching. ABNDM relies on another algorithm, BPA (Wu and Manber, 1992), which makes use of an $O(k \lceil m/w \rceil n)$ time algorithm for its internal workings. BPA is slow but flexible enough to support all operations required by ABNDM. We improve previous ABNDM analyses, showing that it is average-optimal in number of inspected characters, although the overall complexity is higher because of the $O(k \lceil m/w \rceil )$ work done per inspected character. We then show that the faster BPM can be adapted to support all the operations required by ABNDM. This involves extending it to compute edit distance, to search for any pattern suffix, and to detect in advance the impossibility of a later match. The solution to those challenges is based on the concept of a witness, which permits sampling some dynamic programming matrix values to bound, deduce or compute others fast. The resulting algorithm is average-optimal for m ≤ w, assuming the alphabet size is constant. In practice, it performs better than the original ABNDM and is the fastest algorithm for several combinations of m, k and alphabet sizes that are useful, for example, in natural language searching and computational biology. To show that the concept of witnesses can be used in further scenarios, we also improve a recent variant of BPM. The use of witnesses greatly improves the running time of this algorithm too.