## The Annals of Applied Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60C05: Combinatorial probability

**Keywords**

Longest common subsequence subadditivity first-passage percolation

#### Citation

Alexander, Kenneth S. The Rate of Convergence of the Mean Length of the Longest Common Subsequence. Ann. Appl. Probab. 4 (1994), no. 4, 1074--1082. doi:10.1214/aoap/1177004903. https://projecteuclid.org/euclid.aoap/1177004903

#### Corrections

- See Correction: Kenneth S. Alexander. Correction: The Rate of Convergence of the Mean Length of the Longest Common Subsequence. Ann. Appl. Probab., Vol. 5, Iss. 1 (1995), 327.Project Euclid: euclid.aoap/1177004844