Approximate matching in the L_∞ metric

Let a text T = t₀, ..., t_{n - 1} and a pattern P = p₀, ..., p_{m - 1}, strings of natural numbers, be given. In the Approximate Matching in theL_∞ metric problem the output is, for every text location i, the L_∞ distance between the pattern and the length m substring of the text starting at i, i.e., Max_{j = 0}^{m - 1} | t_{i + j} - p_j |. We consider the Approximatek - L_∞ distance problem. Given text T and pattern P as before, and a natural number k the output of the problem is the L_∞ distance of the pattern from the text only at locations i in the text where the distance is bounded by k. For the locations where the distance exceeds k the output is φ{symbol}. We show an algorithm that solves this problem in O (n (k + log (min (m, | Σ |))) log m) time.