L1 pattern matching lower bound

Let a text string T=t0,…,tn−1 and a pattern string P=p0,…,pm−1, ti,pj∈N be given. In the Approximate Pattern Matching in the L1 metric problem (L1-matching for short) the output is, for every text location i, the L1 distance between the pattern and the length m substring of the text starting at i, i.e. . The Less Than Matching problem is that of finding all locations i of T where ti+j⩾pj, 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 time. In this paper we show that the latter two problems can be linearly reduced to the problem of L1-matching.