A variant of the tandem duplication - random loss model of genome rearrangement

TL;DR

This paper investigates a variant of the genome rearrangement model focusing on duplication-loss steps with bounded width, analyzing the classes of permutations generated and the number of steps needed for permutation transformation.

Contribution

It introduces a new cost model with bounded width for duplication-loss steps and characterizes the resulting pattern-avoiding permutation classes.

Findings

Permutation classes correspond to pattern-avoiding permutations.

Number of steps needed for permutation transformation is computed.

Analysis includes worst-case and average-case scenarios.

Abstract

In Soda'06, Chaudhuri, Chen, Mihaescu and Rao study algorithmic properties of the tandem duplication - random loss model of genome rearrangement, well-known in evolutionary biology. In their model, the cost of one step of duplication-loss of width k is $\alpha^k$ for $\alpha =1$ or $\alpha >=2 $. In this paper, we study a variant of this model, where the cost of one step of width $k$ is 1 if $k <= K$ and $\infty$ if $k > K$, for any value of the parameter $K in N$. We first show that permutations obtained after $p$ steps of width $K$ define classes of pattern-avoiding permutations. We also compute the numbers of duplication-loss steps of width $K$ necessary and sufficient to obtain any permutation of $S_n$, in the worst case and on average. In this second part, we may also consider the case $K=K(n)$, a function of the size $n$ of the permutation on which the duplication-loss operations are performed.