Rapid Sampling for Visualizations with Ordering Guarantees.

Visualizations are frequently used as a means to understand trends and gather insights from datasets, but often take a long time to generate. In this paper, we focus on the problem of rapidly generating approximate visualizations while preserving crucial visual properties of interest to analysts. Our primary focus will be on sampling algorithms that preserve the visual property of ordering; our techniques will also apply to some other visual properties. For instance, our algorithms can be used to generate an approximate visualization of a bar chart very rapidly, where the comparisons between any two bars are correct. We formally show that our sampling algorithms are generally applicable and provably optimal in theory, in that they do not take more samples than necessary to generate the visualizations with ordering guarantees. They also work well in practice, correctly ordering output groups while taking orders of magnitude fewer samples and much less time than conventional sampling schemes.

Gene maps linearization using genomic rearrangement distances.

A preliminary step to most comparative genomics studies is the annotation of chromosomes as ordered sequences of genes. Different genetic mapping techniques often give rise to different maps with unequal gene content and sets of unordered neighboring genes. Only partial orders can thus be obtained from combining such maps. However, once a total order O is known for a given genome, it can be used as a reference to order genes of a closely related species characterized by a partial order P. Our goal is to find a linearization of P that is as close as possible to O, in term of a given genomic distance. We first prove NP-completeness complexity results considering the breakpoint and the common interval distances. We then focus on the breakpoint distance and give a dynamic programming algorithm whose running time is exponential for general partial orders, but polynomial when the partial order is derived from a bounded number of genetic maps. A time-efficient greedy heuristic is then given for the general case and is empirically shown to produce solutions within 10% of the optimal solution, on simulated data. Applications to the analysis of grass genomes are presented.

On the inference of parsimonious indel evolutionary scenarios.

Given a multiple alignment of orthologous DNA sequences and a phylogenetic tree for these sequences, we investigate the problem of reconstructing a most parsimonious scenario of insertions and deletions capable of explaining the gaps observed in the alignment. This problem, called the Indel Parsimony Problem, is a crucial component of the problem of ancestral genome reconstruction, and its solution provides valuable information to many genome functional annotation approaches. We first show that the problem is NP-complete. Second, we provide an algorithm, based on the fractional relaxation of an integer linear programming formulation. The algorithm is fast in practice, and the solutions it produces are, in most cases, provably optimal. We describe a divide-and-conquer approach that makes it possible to solve very large instances on a simple desktop machine, while retaining guaranteed optimality. Our algorithms are tested and shown efficient and accurate on a set of 1.8 Mb mammalian orthologous sequences in the CFTR region.