## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 36, 19 pp.

### Closeness to the diagonal for longest common subsequences in random words

Christian Houdré and Heinrich Matzinger

#### Abstract

The nature of the alignment with gaps corresponding to a longest common subsequence (LCS) of two independent iid random sequences drawn from a finite alphabet is investigated. It is shown that such an optimal alignment typically matches pieces of similar short-length. This is of importance in understanding the structure of optimal alignments of two sequences. Moreover, it is also shown that any property, common to two subsequences, typically holds in most parts of the optimal alignment whenever this same property holds, with high probability, for strings of similar short-length. Our results should, in particular, prove useful for simulations since they imply that the re-scaled two dimensional representation of a LCS gets uniformly close to the diagonal as the length of the sequences grows without bound.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 36, 19 pp.

**Dates**

Received: 29 December 2014

Accepted: 20 April 2016

First available in Project Euclid: 27 April 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1461781966

**Digital Object Identifier**

doi:10.1214/16-ECP4029

**Mathematical Reviews number (MathSciNet)**

MR3492931

**Zentralblatt MATH identifier**

1338.05004

**Subjects**

Primary: 05A05: Permutations, words, matrices 60C05: Combinatorial probability 60F10: Large deviations

**Keywords**

longest common subsequences optimal alignments last passage percolation edit/Levensthein distance

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Houdré, Christian; Matzinger, Heinrich. Closeness to the diagonal for longest common subsequences in random words. Electron. Commun. Probab. 21 (2016), paper no. 36, 19 pp. doi:10.1214/16-ECP4029. https://projecteuclid.org/euclid.ecp/1461781966