Parameterized matching with mismatches

The problem of approximate parameterized string searching consists of finding, for a given text t=t₁t₂…t_n and pattern p=p₁p₂…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_it_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.