Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals
Journal of the ACM (JACM)
On the complexity and approximation of syntenic distance
Discrete Applied Mathematics - Special volume on computational molecular biology DAM-CMB series volume 2
CPM '96 Proceedings of the 7th Annual Symposium on Combinatorial Pattern Matching
Reversal distance for partially ordered genomes
Bioinformatics
Journal of Computer and System Sciences
Inferring gene orders from gene maps using the breakpoint distance
RCG'06 Proceedings of the RECOMB 2006 international conference on Comparative Genomics
A linear kernel for co-path/cycle packing
AAIM'10 Proceedings of the 6th international conference on Algorithmic aspects in information and management
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The PQ-tree is a fundamental data structure that can encode large sets of permutations. It has recently been used in comparative genomics to model ancestral genomes with some uncertainty: given a phylogeny for some species, extant genomes are represented by permutations on the leaves of the tree, and each internal node in the phylogenetic tree represents an extinct ancestral genome, represented by a PQ-tree. An open problem related to this approach is then to quantify the evolution between genomes represented by PQ-trees. In this paper we present results for two problems of PQ-tree comparison motivated by this application. First, we show that the problem of comparing two PQ-trees by computing the minimum breakpoint distance among all pairs of permutations generated respectively by the two considered PQ-trees is NP-complete for unsigned permutations. Next, we consider a generalization of the classical Breakpoint Median problem, where an ancestral genome is represented by a PQ-tree and p permutations are given, with p ≥ 1, and we want to compute a permutation generated by the PQ-tree that minimizes the sum of the breakpoint distances to the p permutations. We show that this problem is Fixed-Parameter Tractable with respect to the breakpoint distance value. This last result applies both on signed and unsigned permutations, and to uni-chromosomal and multi-chromosomal permutations.