Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T23:34:35.256Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Speculated Relation Between Chvátal–Sankoff Constants of Several Sequences

Published online by Cambridge University Press:  01 July 2009

M. KIWI  and
J. SOTO
M. KIWI
Affiliation:
Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático (UMI 2807, CNRS), University of Chile (e-mail: mkiwi@dim.uchile.cl)
J. SOTO
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA (e-mail: jsoto@math.mit.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well known that, when normalized by n, the expected length of a longest common subsequence of d sequences of length n over an alphabet of size σ converges to a constant γσ,d. We disprove a speculation by Steele regarding a possible relation between γ2,d and γ2,2. In order to do that we also obtain some new lower bounds for γσ,d, when both σ and d are small integers.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 4 , July 2009 , pp. 517 - 532
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alexander, K. S. (1994) The rate of convergence of the mean of the longest common subsequence. Ann. Appl. Probab. 4 1074–1083.CrossRefGoogle Scholar
[2]Baeza-Yates, R., Navarro, G., Gavaldá, R. and Schehing, R. (1999) Bounding the expected length of the longest common subsequences and forests. Theory Comput. Syst. 32 435–452.CrossRefGoogle Scholar
[3]Bollobás, B. and Winkler, P. (1988) The longest chain among random points in Euclidean space. Proc. Amer. Math. Soc. 103 347–353.CrossRefGoogle Scholar
[4]Chvátal, V. and Sankoff, D. (1975) Longest common subsequences of two random sequences. J. Appl. Probab. 12 306–315.CrossRefGoogle Scholar
[5]Dančík, V. (1994) Expected length of longest common subsequences. PhD thesis, Department of Computer Science, University of Warwick.Google Scholar
[6]Dančík, V. (1998) Common subsequences and supersequences and their expected length. Combin. Probab. Comput. 7 365–373.CrossRefGoogle Scholar
[7]Dančík, V. and Paterson, M. (1995) Upper bounds for the expected length of a longest common subsequence of two binary sequences. Random Struct. Alg. 6 449–458.CrossRefGoogle Scholar
[8]Deken, J. P. (1979) Some limit results for longest common subsequences. Discrete Math. 26 17–31.CrossRefGoogle Scholar
[9]Janson, S., Łuczak, T. and Rucinski, A. (2000) Random Graphs, Wiley.CrossRefGoogle Scholar
[10]Kiwi, M., Loebl, M. and Matoušek, J. (2005) Expected length of the longest common subsequence for large alphabets. Adv. Math. 197 480–498.CrossRefGoogle Scholar
[11]Logan, B. and Shepp, L. (1977) A variational problem of random Young tableaux. Adv. Math. 26 206–222.CrossRefGoogle Scholar
[12]Lueker, G. (2003) Improved bounds on the average length of longest common subsequences. In Proc. 14th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 130–131.Google Scholar
[13]Pevzner, P. (2000) Computational Molecular Biology: An Algorithmic Approach, MIT Press.CrossRefGoogle Scholar
[14]Soto, J. (2006) Variantes aleatorias de la subsecuencia común más grande. Departamento de Ingeniería Matemática, U. Chile. (In Spanish.)Google Scholar
[15]Steele, J. M. (1986) An Efron–Stein inequality for nonsymmetric statistics. Ann. Statist. 14 753–758.CrossRefGoogle Scholar
[16]Steele, J. M. (1996) Probability Theory and Combinatorial Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.Google Scholar
[17]Szpankowski, W. (2000) Average Case Analysis of Algorithms and Sequences, Series in Discrete Mathematics and Optimization, Wiley InterScience.Google Scholar
[18]Vershik, A. and Kerov, S. (1977) Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux. Doklady Akademii Nauk SSSR 233 1024–1028.Google Scholar
[19]Waterman, M. (1995) Introduction to Computational Biology: Average Case Analysis of Algorithms and Sequences. Series in Discrete Mathematics and Optimization, Chapman & Hall/CRC.Google Scholar