Sorting With Forbidden Intermediates

TL;DR

This paper introduces a new constraint in permutation sorting problems, requiring the avoidance of certain forbidden intermediates, with a polynomial algorithm provided for involutions under these conditions.

Contribution

It presents the first polynomial-time algorithm for sorting involutions while avoiding forbidden intermediates, expanding the understanding of constrained permutation sorting.

Findings

Polynomial algorithm for sorting involutions with forbidden intermediates.

Framework applicable to evolutionary scenarios in genomics.

Addresses constraints relevant to lethal mutation pathways.

Abstract

A wide range of applications, most notably in comparative genomics, involve the computation of a shortest sorting sequence of operations for a given permutation, where the set of allowed operations is fixed beforehand. Such sequences are useful for instance when reconstructing potential scenarios of evolution between species, or when trying to assess their similarity. We revisit those problems by adding a new constraint on the sequences to be computed: they must \emph{avoid} a given set of \emph{forbidden intermediates}, which correspond to species that cannot exist because the mutations that would be involved in their creation are lethal. We initiate this study by focusing on the case where the only mutations that can occur are exchanges of any two elements in the permutations, and give a polynomial time algorithm for solving that problem when the permutation to sort is an involution.