Randomized Algorithms and Probabilistic Methods: Advanced Topics

The stationary distribution is uniform: πi = 1/(kn) for i ∈ V . The Markov chain is connected and therefore irreducible, moreover since it has self loops and is undirected it is also aperiodic. Hence, it converges to π. To apply the flow method, we have to route one unit of flow, for each pair (u, v) of vertices. We do this as follows: if u and v are connected via an edge e, then we route the unit corresponding to (u, v) directly over e. Otherwise, we distribute the flow over all shortes paths from u to v. Now we want to bound the amount of flow on an edge e = {u, v}, with u ∈ Vi and v ∈ Vi+1 for 1 ≤ i < k. We get