Approximate matching in the L 1 metric

Approximate matching is one of the fundamental problems in pattern matching, and a ubiquitous problem in real applications. The Hamming distance is a simple and well studied example of approximate matching, motivated by typing, or noisy channels. Biological and image processing applications assign a different value to mismatches of different symbols. We consider the problem of approximate matching in the L 1 metric – the k- L 1 -distance problem. Given text T=t 0,...,t n − 1 and pattern P=p 0,...,p m − 1 strings of natural number, and a natural number k, we seek all text locations i where the L 1 distance of the pattern from the length m substring of text starting at i is not greater than k, i.e. ∑m−1j=0|ti+j−pj|≤k∑j=0m−1|ti+j−pj|≤k. We provide an algorithm that solves the k-L 1-distance problem in time O(nklogk−−−−−√)O(nklog⁡k). The algorithm applies a bounded divide-and-conquer approach and makes novel uses of non-boolean convolutions.