Efficient special cases of Pattern Matching with Swaps

Let a text string T of n symbols and a pattern string P of m symbols from alphabet Σ be given. A swapped version T′ of T is a length n string derived from T by a series of local swaps (i.e., t′_ℓ ← t_ℓ+1 and t′_ℓ+1 ← tℓ) where each element can participate in no more than one swap. The Pattern Matching with Swaps problem is that of finding all locations i for which there exists a swapped version T′ of T where there is an exact matching of P at location i of T′. It was recently shown that the Pattern Matching with Swaps problem has a solution in time Q(nm^1/3 log m log² σ), where σ = min(|Σ|, m). We consider some interesting special cases of patterns, namely, patterns where there is no length-one run, i.e., there are no a, b, c ∈ Σ where b ≠ a and b ≠ c and where the substring abc appears in the pattern. We show that for such patterns the Pattern Matching with Swaps problem can be solved in time O(n log² m).