Fast Computation of Small Cuts via Cycle Space Sampling

We describe a new sampling-based method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to simple new linear-time sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph. In the model of distributed computing in a graph G=(V, E) with O(log V)-bit messages, our approach yields faster algorithms for several problems. The diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain simple O(Diam)-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving upon previous time bounds. Under natural conditions these new algorithms are universally optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A simple extension of our work yields the first distributed algorithm with sub-linear time for 3-edge-connected components. The basic distributed algorithms are Monte Carlo, but they can be made Las Vegas without increasing the asymptotic complexity. In the model of parallel computing on the EREW PRAM our approach yields a simple algorithm with optimal time complexity O(log V) for finding cut pairs and 3-edge-connected components.Comment: Previous version appeared in Proc. 35th ICALP, pages 145--160, 200

CS Circles: An In-Browser Python Course for Beginners

Computer Science Circles is a free programming website for beginners that is designed to be fun, easy to use, and accessible to the broadest possible audience. We teach Python since it is simple yet powerful, and the course content is well-structured but written in plain language. The website has over one hundred exercises in thirty lesson pages, plus special features to help teachers support their students. It is available in both English and French. We discuss the philosophy behind the course and its design, we describe how it was implemented, and we give statistics on its use.Comment: To appear in SIGCSE 201

Natural convection and the evolution of a reactive porous medium

We describe a mathematical model of buoyancy-driven flow and solute transport in a saturated porous medium, the porosity and permeability of which evolve through precipitation and dissolution as a mineral is lost or gained from the pore fluid. Imposing a vertically varying equilibrium solubility creates a density gradient which can drive convective circulation. We characterise the onset of convection using linear stability analysis, and explore the further development of the coupled reaction–convection system numerically. At low Rayleigh numbers, the effect of the reaction–permeability feedback is shown to be destabilising through a novel reaction–diffusion mechanism; at higher Rayleigh numbers, the precipitation and dissolution have a stabilising effect. Over longer time scales, reaction–permeability feedback triggers secondary instabilities in quasi-steady convective circulation, leading to rapid reversals in the direction of circulation. Over very long time scales, characteristic patterns of porosity emerge, including horizontal layering as well as the development of vertical chimneys of enhanced porosity. We discuss the implications of these findings for more comprehensive models of reactive convection in porous media

Diameter Bounds for Planar Graphs

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(V-1)-E)/3 and 4V^2/3E on the diameter (for connected planar graphs), which are also tight