Standard deviation of the longest common subsequence

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Standard deviation of the longest common subsequence

Jüri Lember and Heinrich Matzinger

Source: Ann. Probab. Volume 37, Number 3 (2009), 1192-1235.

Abstract

Let L_n be the length of the longest common subsequence of two independent i.i.d. sequences of Bernoulli variables of length n. We prove that the order of the standard deviation of L_n is $\sqrt{n}$ , provided the parameter of the Bernoulli variables is small enough. This validates Waterman’s conjecture in this situation [Philos. Trans. R. Soc. Lond. Ser. B 344 (1994) 383–390]. The order conjectured by Chvatal and Sankoff [J. Appl. Probab. 12 (1975) 306–315], however, is different.

Primary Subjects: 60K35, 41A25

Secondary Subjects: 60C05C

Keywords: Longest common subsequence; variance bound; Chvatal–Sankoff conjecture

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1245434031
Digital Object Identifier: doi:10.1214/08-AOP436
Zentralblatt MATH identifier: 05587826

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References

[1] Alexander, K. S. (1994). The rate of convergence of the mean length of the longest common subsequence. Ann. Appl. Probab. 4 1074–1082.

[2] Amsalu, S., Matzinger, H. and Popov, S. (2007). Macroscopic non-uniqueness and transversal fluctuation in optimal random sequence alignment. ESAIM Probab. Stat. 11 281–300.

[3] Arratia, R. and Waterman, M. S. (1994). A phase transition for the score in matching random sequences allowing deletions. Ann. Appl. Probab. 4 200–225.

[4] Baeza-Yates, R. A., Gavaldà, R., Navarro, G. and Scheihing, R. (1999). Bounding the expected length of longest common subsequences and forests. Theory Comput. Syst. 32 435–452.

[5] Bonetto, F. and Matzinger, H. (2006). Fluctuations of the longest common subsequence in the asymmetric case of 2- and 3-letter alphabets. ALEA Lat. Am. J. Probab. Math. Stat. 2 195–216 (electronic).

[6] Boutet de Monvel, J. (1999). Extensive simulations for longest common subsequences. Eur. Phys. J. B 7 293–308.

[7] Chvatal, V. and Sankoff, D. (1975). Longest common subsequences of two random sequences. J. Appl. Probab. 12 306–315.

[8] Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Applications of Mathematics (New York) 31. Springer, New York.

[9] Hauser, R., Martínez, S. and Matzinger, H. (2006). Large deviations-based upper bounds on the expected relative length of longest common subsequences. Adv. in Appl. Probab. 38 827–852.

[10] Houdré, C., Lember, J. and Matzinger, H. (2006). On the longest common increasing binary subsequence. C. R. Math. Acad. Sci. Paris 343 589–594.

[11] Kiwi, M., Loebl, M. and Matoušek, J. (2005). Expected length of the longest common subsequence for large alphabets. Adv. Math. 197 480–498.

[12] Lember, J., Matzinger, H. and Vollmer, A. (2007). Path properties of LCS-optimal alignments. Submitted.

[13] Matzinger, H., Lember, J. and Durringer, C. (2007). Deviation from mean in sequence comparison with a periodic sequence. ALEA Lat. Am. J. Probab. Math. Stat. 3 1–29 (electronic).

[14] Steele, J. M. (1986). An Efron–Stein inequality for nonsymmetric statistics. Ann. Statist. 14 753–758.

[15] Waterman, M. S. (1994). Estimating statistical significance of sequence alignments. Philos. Trans. R. Soc. Lond. Ser. B 344 383–390.

[16] Waterman, M. S. (1995). Introduction to Computational Biology. Chapman & Hall, London.

[17] Waterman, M. S. and Vingron, M. (1994). Sequence comparison significance and Poisson approximation. Statist. Sci. 9 367–381.

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