Abstract
The median problem seeks a permutation whose total distance to a given set of permutations (the base set) is minimal. This is an important problem in comparative genomics and has been studied for several distance measures such as reversals. The transposition distance is less relevant biologically, but it has been shown that it behaves similarly to the most important biological distances, and can thus give important information on their properties. We have derived an algorithm which solves the transposition median problem, giving all transposition medians (the median cloud). We show that our algorithm can be modified to accept median clouds as elements in the base set and briefly discuss the new concept of median iterates (medians of medians) and limit medians, that is the limit of this iterate. Le problème de la médiane est de trouver une permutation dont la distance totale à un ensemble donné de permutations (l´ensemble de base) est minimale. C'est un problème important en génomique comparative et il a été étudié pour certaines mesures de distance. La distance de transposition n'est pas directement liée à la biologie, mais il a été démontré que son comportement est similaire à celui des distances biologiques essentielles, et elle peut donc donner des indications sur leurs propriétés. Nous construisons un algorithme qui résout le problème de la médiane pour la transposition, et donne toutes les transpositions médianes (le nuage des médianes). Nous démontrons que notre algorithme peut être modifié pour admettre des nuages de médianes comme éléments de l´ensemble de base et introduisons le concept de médianes itérées (médianes de médianes) et de médianes limites, c-à-d de limites de ces itérations.
Highlights
The median problem in comparative genomics calls for a permutation such that the total distance to a given set S of permutations is minimised
The gene order typically changes in a species by reversals, where a segment is taken out and inserted backwards at the same place, block transpositions, where a segment is taken out and inserted, possibly backwards, at another place (changing 1234567 to 1456237, 1365–8050 c 2009 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France for instance), or Double Cut and Join (DCJ), which generalise reversals to genomes with several chromosomes, noting that a reversal can be seen as cutting the genome in two places and putting it together again)
We conjecture that the transposition median problem is NP-hard as well and expect that this can be proved by using the same techniques as Caprara, but it does not seem trivial to change his proof for undirected graphs into a similar proof for directed graphs
Summary
The median problem in comparative genomics calls for a permutation such that the total distance to a given set S of permutations is minimised. We consider the median problem under the usual transposition distance (exchanging positions of any two elements) While this operation has no relevance in genomic development, the distance function behaves very to the reversal and the DCJ distances for signed genomes [6], which both take the number of genes, subtract the number of cycles and add some more terms which for most permutations are zero [3]. We conjecture that the transposition median problem is NP-hard as well and expect that this can be proved by using the same techniques as Caprara, but it does not seem trivial to change his proof for undirected graphs into a similar proof for directed graphs. There are good reasons to believe that median solvers for other distances (breakpoints, reversals, DCJ) can be extended to compute median clouds and accepting them in their base sets. We are confident that our results will improve on biologically relevant median computations
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