Approximate matching in the L₁ 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 L₁ metric - the k-L₁-distance problem. Given text T = to, ..., t_n-1 and pattern P = p_o, ..., p_m-1 strings of natural number, and a natural number k, we seek all text locations i where the L₁ distance of the pattern from the length m substring of text starting at i is not greater than k, i.e. ∑_j=0^m-1 |t_i+j-p_j| ≤ k. We provide an algorithm that solves the k-L₁ -distance problem in time O(n √k log k). The algorithm applies a bounded divide-and-conquer approach and makes noveluses of non-boolean convolutions.