The communication complexity of pointer chasing. Applications of entropy and sampling

The following pointer chasing problem plays a central role in the study of bounded round communication complexity. There are two players A and B. There are two sets of vertices V_A and V_B of size n each. Player A is given a function f_A: VA→VB and player B is given a function f_B: VB→VA. In the problem g _k the players have to determine the vertex reached by applying f_A and f_B alternately, k times starting with a fixed vertex v₀∈V_A. That is, in g₁, they must determine f_A(v₀), in g ₂ they must determine f_B(f_A(v₀ )), in g₃ they must determine f_A(f_B (f_A(v₀))), and so on