## The Annals of Applied Probability

### The Rate of Convergence of the Mean Length of the Longest Common Subsequence

Kenneth S. Alexander

#### Abstract

Given two i.i.d. sequences of $n$ letters from a finite alphabet, one can consider the length $L_n$ of the longest sequence which is a subsequence of both the given sequences. It is known that $EL_n$ grows like $\gamma n$ for some $\gamma \in \lbrack 0, 1\rbrack$. Here it is shown that $\gamma n \geq EL_n \geq \gamma n - C(n \log n)^{1/2}$ for an explicit numerical constant $C$ which does not depend on the distribution of the letters. In simulations with $n = 100,000, EL_n/n$ can be determined from $k$ such trials with 95% confidence to within $0.0055/\sqrt k$, and the results here show that $\gamma$ can then be determined with 95% confidence to within $0.0225 + 0.0055/\sqrt k$, for an arbitrary letter distribution.

#### Article information

Source
Ann. Appl. Probab., Volume 4, Number 4 (1994), 1074-1082.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177004903

Digital Object Identifier
doi:10.1214/aoap/1177004903

Mathematical Reviews number (MathSciNet)
MR1304773

Zentralblatt MATH identifier
0812.60014

JSTOR