Approximation Algorithms for Sorting <i>λ</i>-Permutations by <i>λ</i>-Operations

• Main Authors: Guilherme Henrique Santos Miranda, Alexsandro Oliveira Alexandrino, Carla Negri Lintzmayer, Zanoni Dias
• Format: Article
• Language: English
• Published: MDPI AG 2021-06-01
• Series: Algorithms
• Subjects: genome rearrangements; approximation algorithms; sorting permutations
• Online Access: https://www.mdpi.com/1999-4893/14/6/175

Understanding how different two organisms are is one question addressed by the comparative genomics field. A well-accepted way to estimate the evolutionary distance between genomes of two organisms is finding the rearrangement distance, which is the smallest number of rearrangements needed to transform one genome into another. By representing genomes as permutations, one of them can be represented as the identity permutation, and, so, we reduce the problem of transforming one permutation into another to the problem of sorting a permutation using the minimum number of rearrangements. This work investigates the problems of sorting permutations using reversals and/or transpositions, with some additional restrictions of biological relevance. Given a value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>, the problem now is how to sort a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-permutation, which is a permutation whose elements are less than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> positions away from their correct places (regarding the identity), by applying the minimum number of rearrangements. Each <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-rearrangement must have size, at most, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>, and, when applied to a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-permutation, the result should also be a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-permutation. We present algorithms with approximation factors of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>λ</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> for the problems of Sorting <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-Permutations by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-Reversals, by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-Transpositions, and by both operations.