L₁ pattern matching lower bound

Let a text string T = t₀, ..., t_{n - 1} and a pattern string P = p₀, ..., p_{m - 1}, t_i, p_j ∈ N be given. In the Approximate Pattern Matching in theL₁ 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. ∑_{j = 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 sqrt(m log m)) time. In this paper we show that the latter two problems can be linearly reduced to the problem of L₁-matching.