Résumé | In a continuous diffusive process, a certain amount of a resource, known as "load", is initially placed on the nodes of the graph, possibly in an unfair manner. In successive time steps, each node shares its load evenly among all its neighbors, until the load on the nodes of the graph eventually converges to its limit distribution (e.g., becomes uniform in the case of regular graphs). However, continuous diffusion cannot be applied to "granular load" which is not arbitrarily divisible, i.e., represented by unsplittable unit-load tokens (chips), which are placed on nodes and may be passed around the graph. In such a scenario, one natural way of simulating the continuous diffusion process is to require that each chip follows an independent random walk on the graph.
In this talk, we will instead focus on chip diffusion processes following rules which are both locally fair and deterministic. These rules include the so-called "rotor walk", in which chips are propagated by each node to its neighbors in round-robin fashion and, more broadly, rules in which each node attempts to send out roughly the same number of tokens through each of its outgoing arcs.
We will start by providing a description of the evolution and limit behavior of the rotor walk on the n-node ring with K << n chips. For general graphs, we will show bounds on the "cover time" of a rotor walk system with K chips, proving that in a graph with m edges and diameter D, all nodes will have been visited at least once by some chip during the first O(mD / log K) steps of the process. Finally, we will relate the "blanket time" (the time until all nodes of the graph have been visited by chips a similar number of times) and the "diffusion time" (the time until all nodes all host a similar number of chips, for K >> n) of fair deterministic walks to analogous parameters of the random walk. The latter results allow us to design surprisingly simple and efficient deterministic algorithms for load balancing in the diffusive model.
This talk includes an overview of results presented at PODC'13, STACS'14, and some current work in progress (joint work with: P. Berenbrink, D. Dereniowski, R. Klasing, F. Mallmann-Trent, D. Pajak, T. Sauerwald, P. Uznanski). |