L₁ pattern matching lower bound

Let a text string T = t₀, . . ., t_n-1 and a pattern string P = Po, . . ., p_m-1 t_i, p_j ∈ IN 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. ∑_i=0^m-1|t_i+j - p_j|. 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√m log m) time. In this paper we show that the latter two problems can be linearly reduced to the problem of L₁-matching.