Distributed control in computer networks and cross-sections of colored multidimensional bodies

The number of messages to match a pair of processes in a multiprocessor network with mobile processes is a measure for the cost of setting up temporary communication between processes. The authors establish lower bounds on the average number of point-to-point transmissions between any pair of nodes in the context. The present analysis allows for the possibility of multiple transmissions (as opposed to a single one) between any two nodes, and also for the possibility of multiple queries (as opposed to the two, i.e., post and a single query considered before). Applications of the results include lower bounds on the number of messages for distributed s-matching, that is, matching a group of s processes, and distributed s-mutual exclusion, that is, s-1 processes may enter a critical section simultaneously, but s processes may not, for s > or = 2. The idea of the proof of the combinatorial result needed for this analysis is further extended to obtain a lower bound on the average number of colors occurring in random cross sections of colored, multidimensional bodies in terms of the total (multidimensional) volume of each color in the whole body.