Fast Deterministic Selection

The Median of Medians (also known as BFPRT) algorithm, although a landmark theoretical achievement, is seldom used in practice because it and its variants are slower than simple approaches based on sampling. The main contribution of this paper is a fast linear-time deterministic selection algorithm QuickselectAdaptive based on a refined definition of MedianOfMedians. The algorithm's performance brings deterministic selection---along with its desirable properties of reproducible runs, predictable run times, and immunity to pathological inputs---in the range of practicality. We demonstrate results on independent and identically distributed random inputs and on normally-distributed inputs. Measurements show that QuickselectAdaptive is faster than state-of-the-art baselines.Comment: Pre-publication draf

Statistical analysis of simple repeats in the human genome

The human genome contains repetitive DNA at different level of sequence length, number and dispersion. Highly repetitive DNA is particularly rich in homo-- and di--nucleotide repeats, while middle repetitive DNA is rich of families of interspersed, mobile elements hundreds of base pairs (bp) long, among which the Alu families. A link between homo- and di-polymeric tracts and mobile elements has been recently highlighted. In particular, the mobility of Alu repeats, which form 10% of the human genome, has been correlated with the length of poly(A) tracts located at one end of the Alu. These tracts have a rigid and non-bendable structure and have an inhibitory effect on nucleosomes, which normally compact the DNA. We performed a statistical analysis of the genome-wide distribution of lengths and inter--tract separations of poly(X) and poly(XY) tracts in the human genome. Our study shows that in humans the length distributions of these sequences reflect the dynamics of their expansion and DNA replication. By means of general tools from linguistics, we show that the latter play the role of highly-significant content-bearing terms in the DNA text. Furthermore, we find that such tracts are positioned in a non-random fashion, with an apparent periodicity of 150 bases. This allows us to extend the link between repetitive, highly mobile elements such as Alus and low-complexity words in human DNA. More precisely, we show that Alus are sources of poly(X) tracts, which in turn affect in a subtle way the combination and diversification of gene expression and the fixation of multigene families

Algorithmic construction of Chevalley bases

We present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic at least 5, our algorithm has complexity involving the 7th power of the Lie rank, which is likely to be close to best possible