Regular expression constrained sequence alignment revisited

We introduce regular expression constrained sequence alignment as the problem of finding the maximum alignment score between given strings S₁ and S₂ over all alignments such that in these alignments there exists a segment where some substring s₁ of S₁ is aligned to some substring s₂ of S₂, and both s₁ and s₂ match a given regular expression R, i.e. s₁, s₂ ∈ L (R) where L (R) is the regular language described by R. For complexity results we assume, without loss of generality, that n = | S₁ | ≥ | m | = | S₂ |. A motivation for the problem is that protein sequences can be aligned in a way that known motifs guide the alignments. We present an O (n m r) time algorithm for the regular expression constrained sequence alignment problem where r = O (t⁴), and t is the number of states of a nondeterministic finite automaton N that accepts L (R). We use in our algorithm a nondeterministic weighted finite automaton M that we construct from N. M has O (t²) states where the transition-weights are obtained from the given costs of edit operations, and state-weights correspond to optimum alignment scores we compute using the underlying dynamic programming solution for sequence alignment. If we are given a deterministic finite automaton D accepting L (R) with t_d states then our construction creates a deterministic finite automaton M_d with t_d ² states. In this case, our algorithm takes O (t_d ² n m) time. Using M_d results in faster computation than using M when t_d < t². If we only want to compute the optimum score, the space required by our algorithm is O (t² n) (O (t_d ² m) if we use a given M_d). If we also want to compute an optimal alignment then our algorithm uses O (t² m + t² | s₁ | | s₂ |) space (O (t_d ² m + t_d ² | s₁ | | s₂ |) space if we use a given M_d) where s₁ and s₂ are substrings of S₁ and S₂, respectively, s₁, s₂ ∈ L (R), and s₁ and s₂ are aligned together in the optimal alignment that we construct. We also show that our method generalizes for the case of the problem with affine gap penalties, and for finding optimal regular expression constrained local sequence alignments.