A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces

International audienceWe consider the problem of constructing roadmaps of real algebraic sets. The problem was introduced by Canny to answer connectivity questions and solve motion planning problems. Given $s$ polynomial equations with rational coefficients, of degree $D$ in $n$ variables, Canny's algorithm has a Monte Carlo cost of $s^n\log(s) D^{O(n^2)}$ operations in $\mathbb{Q}$; a deterministic version runs in time $s^n \log(s) D^{O(n^4)}$. The next improvement was due to Basu, Pollack and Roy, with an algorithm of deterministic cost $s^{d+1} D^{O(n^2)}$ for the more general problem of computing roadmaps of semi-algebraic sets ($d \le n$ is the dimension of an associated object). We give a Monte Carlo algorithm of complexity $(nD)^{O(n^{1.5})}$ for the problem of computing a roadmap of a compact hypersurface $V$ of degree $D$ in $n$ variables; we also have to assume that $V$ has a finite number of singular points. Even under these extra assumptions, no previous algorithm featured a cost better than $D^{O(n^2)}$

Plasticity of a transcriptional regulation network among alpha-proteobacteria is supported by the identification of CtrA targets in Brucella abortus.

CtrA is a master response regulator found in many alpha-proteobacteria. In Caulobacter crescentus and Sinorhizobium meliloti, this regulator is essential for viability and is transcriptionally autoregulated. In C. crescentus, it is required for the regulation of multiple cell cycle events, such as DNA methylation, DNA replication, flagella and pili biogenesis and septation. Here, we report the characterization of the ctrA gene homologue in the alpha2-proteobacteria Brucella abortus, a facultative intracellular pathogen responsible for brucellosis. We detected CtrA expression in the main Brucella species, and its overproduction led to a phenotype typical of cell division defect, consistent with its expected role. A purified B. abortus CtrA recombinant protein (His6-CtrA) was shown to protect the B. abortus ctrA promoter from DNase I digestion, suggesting transcriptional autoregulation, and this protection was enhanced under CtrA phosphorylation on a conserved Asp residue. Despite the similarities shared by B. abortus and C. crescentus ctrA, the pathway downstream from CtrA may be distinct, at least partially, in both bacteria. Indeed, beside ctrA itself, only one (the ccrM gene) out of four B. abortus homologues of known C. crescentus CtrA targets is bound in vitro by phosphorylated B. abortus CtrA. Moreover, further footprinting experiments support the hypothesis that, in B. abortus, CtrA might directly regulate the expression of the rpoD, pleC, minC and ftsE homologues. Taken together, these results suggest that, in B. abortus and C. crescentus, similar cellular processes are regulated by CtrA through the control of distinct target genes. The plasticity of the regulation network involving CtrA in these two bacteria may be related to their distinct lifestyles