Forbidden factors and fragment assembly
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 6, pp. 565-577.

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.

Classification : 68Q45, 68R15
Mots-clés : factor automaton, minimal forbidden factor, fragment assembly 
@article{ITA_2001__35_6_565_0,
     author = {Mignosi, F. and Restivo, A. and Sciortino, M.},
     title = {Forbidden factors and fragment assembly},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {565--577},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {6},
     year = {2001},
     mrnumber = {1922296},
     zbl = {1005.68122},
     language = {en},
     url = {http://www.numdam.org/item/ITA_2001__35_6_565_0/}
}
TY  - JOUR
AU  - Mignosi, F.
AU  - Restivo, A.
AU  - Sciortino, M.
TI  - Forbidden factors and fragment assembly
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2001
SP  - 565
EP  - 577
VL  - 35
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/ITA_2001__35_6_565_0/
LA  - en
ID  - ITA_2001__35_6_565_0
ER  - 
%0 Journal Article
%A Mignosi, F.
%A Restivo, A.
%A Sciortino, M.
%T Forbidden factors and fragment assembly
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2001
%P 565-577
%V 35
%N 6
%I EDP-Sciences
%U http://www.numdam.org/item/ITA_2001__35_6_565_0/
%G en
%F ITA_2001__35_6_565_0
Mignosi, F.; Restivo, A.; Sciortino, M. Forbidden factors and fragment assembly. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 6, pp. 565-577. http://www.numdam.org/item/ITA_2001__35_6_565_0/

[1] A.V. Aho, J.E. Hopcroft and J.D. Ullman, Data Structures and Algorithms. Addison Wesley, Reading, Mass (1983). | MR | Zbl

[2] M.-P. Béal, F. Mignosi, A. Restivo and M. Sciortino, Forbidden Words in Symbolic Dynamics. Adv. in Appl. Math. 25 (2000) 163-193. | MR | Zbl

[3] A. Blumer, J. Blumer, A. Ehrenfeucht, D. Haussler, M.T. Chen and J. Seiferas, The smallest automaton recognizing the subwords of a text. Theoret. Comput. Sci. 40 (1985) 31-55. | MR | Zbl

[4] A. Carpi, A. De Luca and S. Varricchio, Words, univalent factors, and boxes. Acta Inform. (to appear). | Zbl

[5] M. Crochemore and C. Hancart, Automata for matching patterns, in Handbook of Formal Languages, Vol. 2, Chap. 9, edited by G. Rosenberg and A. Salomaan. Springer (1997) 399-462. | MR

[6] M. Crochemore, F. Mignosi and A. Restivo, Automata and forbidden words. Inform. Process. Lett. 67 (1998) 111-117. | MR

[7] M. Crochemore, F. Mignosi, A. Restivo and S. Salemi, Data compression using antidictionaries, in Proc. of the IEEE, Special Issue on Lossless Data Compression, Vol. 88, edited by J.A. Storer (2000) 1756-1768.

[8] A. Frieze and B.V. Halldórsson, Optimal Sequencing by Hybridization in Rounds, in Proc. of RECOMB 2001, edited by T. Lengauer, D. Sankoff, S. Istrail, P. Pevzner and M. Waterman. ACM Press (2001) 141-148

[9] D. Gusfield, Algorithms on strings, trees, and sequences: Computer science and computational biology. Cambridge University Press (1997). | MR | Zbl

[10] R. Idury and M. Waterman, A new algorithm for DNA sequence assembly. J. Comput. Biol. 2 (1995) 291-306.

[11] F. Mignosi and A. Restivo, Periodicity, in M. Lothaire, Algebraic Combinatorics on Words, Chap. 8. Cambridge University Press (to appear) 237-274. Also available at url: http://www-igm.univ-mlv.fr/~berstel/Lothaire/index.html | MR

[12] F. Mignosi, A. Restivo and M. Sciortino, Forbidden Factors and Fragment Assembly. Lecture Notes in Comput. Sci. (2001). Proceedings of DLT'01. | Zbl

[13] F. Mignosi, A. Restivo and M. Sciortino, Words and Forbidden Factors. Theoret. Comput. Sci. 273 (2002) 99-117. | MR | Zbl

[14] F. Mignosi, A. Restivo, M. Sciortino and J. Storer, On Sequence Assembly, Technical Report cs-00-210. Brandeis University (2000).

[15] S. Muthukrishnan and S.C. Sahinalp, Approximate nearest neighbors and sequence comparison with block operations. ACM Press (2000). Proceedings of STOC 2000. | MR | Zbl

[16] G. Myers, Whole-Genome DNA Sequencing, IEEE Comput. Engrg. Sci. 3 (1999) 33-43.

[17] H. Peltola, H. Soderlund and E. Ukkonen, SEQAID: A DNA Sequence Assembly Program Based on a Mathematical Model. Nucl. Acids Res. 12 (1984) 307-321.

[18] M. Peltola, H. Soderlund, J. Tarhio and E. Ukkonen, Algorithms for some string matching problems arising in molecular genetics, in Proc. of the 9th IFIP World Computer Congress (1983) 59-64.

[19] P.A. Pevzner, H. Tang and M. Waterman, A New Approach Fragment Assembly in DNA Sequencing, in Proc. of RECOMB 2001, edited by T. Lengauer, D. Sankoff, S. Istrail, P. Pevzner and M. Waterman. ACM Press (2001) 141-148.

[20] R. Shamir and D. Tsur, Large Scale Sequencing by Hybridization, in Proc. of RECOMB 2001, edited by T. Lengauer, D. Sankoff, S. Istrail, P. Pevzner and M. Waterman. ACM Press (2001) 269-278.

[21] M.S. Waterman, Introduction to computational biology: Maps, sequences and genomes. Chapman & Hall (1995). | Zbl