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# A Linear Time Algorithm for an Extended Version of the Breakpoint Double Distance

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## Acknowledgements

We would like to thank Cedric Chauve for bringing our attention to the class of σ_k distances as a means for studying the hardness bound between the breakpoint distance and the DCJ distance in combinatorial problems related to genome evolution. Thanks also to Eloi Araujo, Daniel Doerr and Fábio H. V. Martinez for helping us studying the median problem under this class.

## Cite As

Marília D. V. Braga, Leonie R. Brockmann, Katharina Klerx, and Jens Stoye. A Linear Time Algorithm for an Extended Version of the Breakpoint Double Distance. In 22nd International Workshop on Algorithms in Bioinformatics (WABI 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 242, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.WABI.2022.13

## Abstract

Two genomes over the same set of gene families form a canonical pair when each of them has exactly one gene from each family. A genome is circular when it contains only circular chromosomes. Different distances of canonical circular genomes can be derived from a structure called breakpoint graph, which represents the relation between the two given genomes as a collection of cycles of even length. Then, the breakpoint distance is equal to n-c_2, where n is the number of genes and c_2 is the number of cycles of length 2. Similarly, when the considered rearrangements are those modeled by the double-cut-and-join (DCJ) operation, the rearrangement distance is n-c, where c is the total number of cycles. The distance problem is a basic unit for several other combinatorial problems related to genome evolution and ancestral reconstruction, such as median or double distance. Interestingly, both median and double distance problems can be solved in polynomial time for the breakpoint distance, while they are NP-hard for the rearrangement distance. One way of exploring the complexity space between these two extremes is to consider a σ_k distance, defined to be n-(c_2+c_4+…+c_k), and increasingly investigate the complexities of median and double distance for the σ₄ distance, then the σ₆ distance, and so on. While for the median much effort was done in our and in other research groups but no progress was obtained even for the σ₄ distance, for solving the double distance under σ₄ and σ₆ distances we could devise linear time algorithms, which we present here.

## Subject Classification

##### ACM Subject Classification
• Applied computing → Bioinformatics
##### Keywords
• Comparative genomics
• genome rearrangement
• breakpoint distance
• double-cut-and-join (DCJ) distance
• double distance

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