L1 pattern matching lower bound

Let a text string T = t(0),..., t(n-1) and a pattern string P = P-0,...,Pm-1, t(i), p(j) is an element of N be given. In the Approximate Pattern Matching in the L-1 metric problem (L-1-matching for short) the output is, for every text location i, the L-1 distance between the pattern and the length m substring of the text starting at i, i.e. Sigma(m-1)(j=0)vertical bar t(i+j) - p(j)vertical bar. The Less Than Matching problem is that of finding all locations i of T where t(i+j) >= p(j), j = 0,..., m - 1. The String Matching with Mismatches problem is that of finding the number of mismatches between the pattern and every length m substring of the text. For the three above problems, the fastest known deterministic solution is O(n root m log m) time. In this paper we show that the latter two problems can be linearly reduced to the problem of L-1-matching. (c) 2007 Elsevier B.V. All rights reserved.