Vertex labeling and routing in expanded Apollonian networks

We present a family of networks, expanded deterministic Apollonian networks, which are a generalization of the Apollonian networks and are simultaneously scale-free, small-world, and highly clustered. We introduce a labeling of their vertices that allows to determine a shortest path routing between any two vertices of the network based only on the labels.Comment: 16 pages, 2 figure

Sorting signed circular permutations by super short reversals

We consider the problem of sorting a circular permutation by reversals of length at most 2, a problem that finds application in comparative genomics. Polynomial-time solutions for the unsigned version of this problem are known, but the signed version remained open. In this paper, we present the first polynomial-time solution for the signed version of this problem. Moreover, we perform an experiment for inferring distances and phylogenies for published Yersinia genomes and compare the results with the phylogenies presented in previous works.We consider the problem of sorting a circular permutation by reversals of length at most 2, a problem that finds application in comparative genomics. Polynomial-time solutions for the unsigned version of this problem are known, but the signed version rema9096272283FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIORCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO2013/08293-72014/04718-6306730/2012-0; 477692/2012-5; 483370/2013-411th International Symposium on Bioinformatics Research and Application

Minimal chordal sense of direction and circulant graphs

A sense of direction is an edge labeling on graphs that follows a globally consistent scheme and is known to considerably reduce the complexity of several distributed problems. In this paper, we study a particular instance of sense of direction, called a chordal sense of direction (CSD). In special, we identify the class of k-regular graphs that admit a CSD with exactly k labels (a minimal CSD). We prove that connected graphs in this class are Hamiltonian and that the class is equivalent to that of circulant graphs, presenting an efficient (polynomial-time) way of recognizing it when the graphs' degree k is fixed

Upward Three-Dimensional Grid Drawings of Graphs

A \emph{three-dimensional grid drawing} of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. We prove that every $n$-vertex graph with bounded degeneracy has a three-dimensional grid drawing with $O(n^{3/2})$ volume. This is the broadest class of graphs admiting such drawings. A three-dimensional grid drawing of a directed graph is \emph{upward} if every arc points up in the z-direction. We prove that every directed acyclic graph has an upward three-dimensional grid drawing with $(n^3)$ volume, which is tight for the complete dag. The previous best upper bound was $O(n^4)$. Our main result is that every $c$-colourable directed acyclic graph ($c$ constant) has an upward three-dimensional grid drawing with $O(n^2)$ volume. This result matches the bound in the undirected case, and improves the best known bound from $O(n^3)$ for many classes of directed acyclic graphs, including planar, series parallel, and outerplanar

Sorting Signed Circular Permutations by Super Short Reversals

International audienceWe consider the problem of sorting a circular permutation by reversals of length at most 2, a problem that finds application in comparative genomics. Polynomial-time solutions for the unsigned version of this problem are known, but the signed version remained open. In this paper, we present the first polynomial-time solution for the signed version of this problem. Moreover, we perform an experiment for inferring distances and phylogenies for published Yersinia genomes and compare the results with the phylogenies presented in previous works

On the PATHGROUPS approach to rapid small phylogeny

We present a data structure enabling rapid heuristic solution to the ancestral genome reconstruction problem for given phylogenies under genomic rearrangement metrics. The efficiency of the greedy algorithm is due to fast updating of the structure during run time and a simple priority scheme for choosing the next step. Since accuracy deteriorates for sets of highly divergent genomes, we investigate strategies for improving accuracy and expanding the range of data sets where accurate reconstructions can be expected. This includes a more refined priority system, and a two-step look-ahead, as well as iterative local improvements based on a the median version of the problem, incorporating simulated annealing. We apply this to a set of yeast genomes to corroborate a recent gene sequence-based phylogeny

Limited Lifespan of Fragile Regions in Mammalian Evolution

An important question in genome evolution is whether there exist fragile regions (rearrangement hotspots) where chromosomal rearrangements are happening over and over again. Although nearly all recent studies supported the existence of fragile regions in mammalian genomes, the most comprehensive phylogenomic study of mammals (Ma et al. (2006) Genome Research 16, 1557-1565) raised some doubts about their existence. We demonstrate that fragile regions are subject to a "birth and death" process, implying that fragility has limited evolutionary lifespan. This finding implies that fragile regions migrate to different locations in different mammals, explaining why there exist only a few chromosomal breakpoints shared between different lineages. The birth and death of fragile regions phenomenon reinforces the hypothesis that rearrangements are promoted by matching segmental duplications and suggests putative locations of the currently active fragile regions in the human genome

Estimating true evolutionary distances under rearrangements, duplications, and losses

Background: The rapidly increasing availability of whole-genome sequences has enabled the study of whole-genome evolution. Evolutionary mechanisms based on genome rearrangements have attracted much attention and given rise to many models; somewhat independently, the mechanisms of gene duplication and loss have seen much work. However, the two are not independent and thus require a unified treatment, which remains missing to date. Moreover, existing rearrangement models do not fit the dichotomy between most prokaryotic genomes (one circular chromosome) and most eukaryotic genomes (multiple linear chromosomes). Results: To handle rearrangements, gene duplications and losses, we propose a new evolutionary model and the corresponding method for estimating true evolutionary distance. Our model, inspired from the DCJ model, is simple and the first to respect the prokaryotic/eukaryotic structural dichotomy. Experimental results on a wide variety of genome structures demonstrate the very high accuracy and robustness of our distance estimator. Conclusions: We give the first robust, statistically based, estimate of genomic pairwise distances based on rearrangements, duplications and losses, under a model that respects the structural dichotomy between prokaryotic and eukaryotic genomes. Accurate and robust estimates in true evolutionary distances should translate into much better phylogenetic reconstructions as well as more accurate genomic alignments, while our new model of genome rearrangements provides another refinement in simplicity and verisimilitude

Heuristics for the inversion median problem

Background: The study of genome rearrangements has become a mainstay of phylogenetics and comparative genomics. Fundamental in such a study is the median problem: given three gene arrangements, find a fourth that minimizes the sum of the evolutionary distances between itself and the given three. Many exact algorithms and heuristics have been developped for the inversion median problem, of which the best known is MGR. Results: We present a unifying framework for median heuristics, which enables us to clarify existing strategies and to place them in a partial ordering. Analysis of this framework leads to a new insight: the best strategies continue to refer to the input data rather than just to updated estimates. Using this insight, we develop a new heuristic for inversion medians that uses input data to the end of its computation and leverages our previous work with DCJ medians. Finally, we present the results of extensive experimentation showing that our new heuristic outperforms all others in accuracy and, especially, in running time: the heuristic typically returns solutions within 1 % of optimal and runs in seconds to minutes even on genomes with 25’000 genes—in contrast, MGR can take days on instances of 200 genes and cannot be used beyond 1’000 genes. Conclusions: Finding good rearrangement medians, in particular inversion medians, had long been regarded as the computational bottleneck in whole-genome studies. Our new heuristic for inversion medians, ASM, which dominates all others in our framework, puts that issue to rest by providing near-optimal solutions within seconds to minutes on even the largest genomes