Hairpin Lengthening

The hairpin completion is a natural operation of formal languages which has been inspired by molecular phenomena in biology and by DNA-computing. We consider here a new variant of the hairpin completion, called hairpin lengthening, which seems more appropriate for practical implementation. The variant considered here concerns the lengthening of the word that forms a hairpin structure, such that this structure is preserved, without necessarily completing the hairpin. Although our motivation is based on biological phenomena, the present paper is more about some algorithmic properties of this operation. We prove that the iterated hairpin lengthening of a language recognizable in O(f(n)) time is recognizable in O(nf(n)) time, while the one-step hairpin lengthening of such a language is recognizable in O(nf(n)) time. Finally, we propose an algorithm for computing the hairpin lengthening distance between two words in quadratic time.