Parameterized matching with mismatches

The problem of approximate parameterized string searching consists of finding, for a given text t = t₁ t₂ ... t_n and pattern p = p₁ p₂ ... p_m over respective alphabets Σ_t and Σ_p, the injection π_i from Σ_p to Σ_t maximizing the number of matches between π_i (p) and t_i t_{i + 1} ... t_{i + m - 1}(i = 1, 2, ..., n - m + 1) . We examine the special case where both strings are run-length encoded, and further restrict to the case where one of the alphabets is binary. For this case, we give a construction working in time O (n + (r_p × r_t) α (r_t) log (r_t)), where r_p and r_t denote the number of runs in the corresponding encodings for y and x, respectively, and α is the inverse of the Ackermann's function.