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/sub A/ and V/sub B/ of size n each. Player A is given a function f/sub A/: VA/spl rarr/VB and player B is given a function f/sub B/: VB/spl rarr/VA. In the problem g/sub k/ the players have to determine the vertex reached by applying f/sub A/ and f/sub B/ alternately, k times starting with a fixed vertex v/sub 0//spl isin/V/sub A/. That is, in g/sub 1/, they must determine f/sub A/(v/sub 0/), in g/sub 2/ they must determine f/sub B/(f/sub A/(v/sub 0/)), in g/sub 3/ they must determine f/sub A/(f/sub B/(f/sub A/(v/sub 0/))), and so on.