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Large deviations-based upper bounds on the expected relative length of longest common subsequences

Part of: Combinatorics Parametric inference Chemistry

Published online by Cambridge University Press:  01 July 2016

Raphael Hauser ,
Servet Martínez  and
Heinrich Matzinger
Raphael Hauser*
Affiliation:
University of Oxford
Servet Martínez*
Affiliation:
Universidad de Chile
Heinrich Matzinger*
Affiliation:
Universität Bielefeld and Georgia Institute of Technology
*
∗ Postal address: Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. Email address: hauser@comlab.ox.ac.uk
∗∗ Postal address: CMM-DIM-CNRS 2071, Universidad de Chile, Casilla 170-3 Correo 3, Santiago, Chile. Email address: smartine@dim.uchile.cl
∗∗∗ Postal address: Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany. Email address: matzing@mathematik.uni-bielefeld.de
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Abstract

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Consider the random variable Ln defined as the length of a longest common subsequence of two random strings of length n and whose random characters are independent and identically distributed over a finite alphabet. Chvátal and Sankoff showed that the limit γ=limn→∞E[Ln]/n is well defined. The exact value of this constant is not known, but various methods for the computation of upper and lower bounds have been discussed in the literature. Even so, high-precision bounds are hard to come by. In this paper we discuss how large deviation theory can be used to derive a consistent sequence of upper bounds, (qm)m∈ℕ, on γ, and how Monte Carlo simulation can be used in theory to compute estimates, q̂m, of the qm such that, for given Ξ > 0 and Λ ∈ (0,1), we have P[γ < q̂ < γ + Ξ] ≥ Λ. In other words, with high probability the result is an upper bound that approximates γ to high precision. We establish O((1 − Λ)−1Ξ−(4+ε)) as a theoretical upper bound on the complexity of computing q̂m to the given level of accuracy and confidence. Finally, we discuss a practical heuristic based on our theoretical approach and discuss its empirical behavior.

Keywords

Longest common subsequence problemChvátal-Sankoff constantupper boundlarge deviation theoryMonte Carlo simulation

MSC classification

Primary: 05A16: Asymptotic enumeration 62F10: Point estimation
Secondary: 92E10: Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 827 - 852
Copyright
Copyright © Applied Probability Trust 2006 

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