Forbidden Factors and Fragment Assembly

¹ University of Palermo, Dipartimento di Matematica ed Applicazioni, Via Archirafi 34, 90123 Palermo, Italy; (mignosi@math.unipa.it)
² University of Palermo, Dipartimento di Matematica ed Applicazioni, Via Archirafi 34, 90123 Palermo, Italy; (restivo@math.unipa.it)
³ University of Palermo, Dipartimento di Matematica ed Applicazioni, Via Archirafi 34, 90123 Palermo, Italy; (mari@math.unipa.it)

Received: 26 June 2001
Revised: 21 February 2002

Abstract

In this paper methods and results related to the notion of minimal forbidden words are applied to the fragment assembly problem. The fragment assembly problem can be formulated, in its simplest form, as follows: reconstruct a word w from a given set I of substrings (fragments) of a word w. We introduce an hypothesis involving the set of fragments I and the maximal length m(w) of the minimal forbidden factors of w. Such hypothesis allows us to reconstruct uniquely the word w from the set I in linear time. We prove also that, if w is a word randomly generated by a memoryless source with identical symbol probabilities, m(w) is logarithmic with respect to the size of w. This result shows that our reconstruction algorithm is suited to approach the above problem in several practical applications e.g. in the case of DNA sequences.