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Suppose that the edges of a complete graph are assigned weights independently at random and we ask for the weight of the minimal-weight spanning tree, or perfect matching, or Hamiltonian cycle. For these and several other common optimisation problems, we establish asymptotically tight bounds when the weights are independent copies of a symmetric random variable (satisfying a mild condition on tail probabilities), in particular when the weights are Gaussian. 

Abstract Beneficial rhizobacteria promote plant growth and protect plants against phytopathogens. Effective colonization on plant roots is critical for the rhizobacteria to exert beneficial activities. How bacteria migrate swiftly in the soil of semisolid or solid nature remains unclear. Here we report that sucrose, a disaccharide ubiquitously deployed by photosynthetic plants for fixed carbon transport and storage, and abundantly secreted from plant roots, promotes solid surface motility (SSM) and root colonization by Bacillus subtilis through a previously uncharacterized mechanism. Sucrose induces robust SSM by triggering a signaling cascade, first through extracellular synthesis of polymeric levan, which in turn stimulates strong production of surfactin and hyper-flagellation of the cells. B. subtilis poorly colonizes the roots of Arabidopsis thaliana mutants deficient in root-exudation of sucrose, while exogenously added sucrose selectively shapes the rhizomicrobiome associated with the tomato plant roots, promoting specifically bacilli and pseudomonad. We propose that sucrose activates a signaling cascade to trigger SSM and promote rhizosphere colonization by B. subtilis. Our findings also suggest a practicable approach to boost prevalence of beneficial Bacillus species in plant protection.