Randomized algorithms have become a central part of the algorithms curriculum, based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high probability estimates on the performance of randomized algorithms. It covers the basic toolkit from the Chernoff–Hoeffding bounds to more sophisticated techniques like martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as Chernoff–Hoeffding bounds in dependent settings. The authors emphasise comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus making the book accessible to computer scientists as well as probabilists and discrete mathematicians.

For this in turn, it suffices to prove that for any nondecreasing f, and any integer t ,

Plasmids are important members of the bacterial mobile gene pool, and are among the most important contributors to horizontal gene transfer between bacteria. They typically harbour a wide spectrum of host beneficial traits, such as antibiotic resistance, inserted into their backbones. Although these inserted elements have drawn considerable interest, evolutionary information about the plasmid backbones, which encode plasmid related traits, is sparse. Here we analyse 25 complete backbone genomes from the broad-host-range IncP-1 plasmid family. Phylogenetic analysis reveals seven clades, in which two plasmids that we isolated from a marine biofilm represent a novel clade. We also found that homologous recombination is a prominent feature of the plasmid backbone evolution. Analysis of genomic signatures indicates that the plasmids have adapted to different host bacterial species. Globally circulating IncP-1 plasmids hence contain mosaic structures of segments derived from several parental plasmids that have evolved in, and adapted to, different, phylogenetically very distant host bacterial species.

COVID-19 pandemic represents an unprecedented global health crisis in the last 100 years. Its economic, social and health impact continues to grow and is likely to end up as one of the worst global disasters since the 1918 pandemic and the World Wars. Mathematical models have played an important role in the ongoing crisis; they have been used to inform public policies and have been instrumental in many of the social distancing measures that were instituted worldwide. In this article, we review some of the important mathematical models used to support the ongoing planning and response efforts. These models differ in their use, their mathematical form and their scope.