Clique Here: On the Distributed Complexity in Fully-Connected Networks.

We consider a message. passingwith connected to all other nodes by a link that can deliver a message of B bits in a time unit (typically, B = O(logn)). We assume that each node has an input of size L bits (typically, L = O (a log rt)) and the nodes cooperate in order to compute some function (i.e., perform a distributed task). We are interested in the number of rounds required to compute the function. We give two results regarding this model. First, we show that most boolean functions require FL/B1 1 rounds to compute deterministically, and that even if we consider randomized protocols that are allowed to err, the expected running time remains Omega(L/B) for most boolean function. Second, trying to find explicit functions that require super constant time, we consider the pointer chasing problem. In this problem, each node i is given an array Ai of length cl whose entries are in [IA, and the task is to find, for any j is an element of [n], the value of A 114 2 [. . . . A(O]) [j]. . . ]]. We give a deterministic O(log n/log log n round protocol for this function using message size B = (log n), a slight but non trivial improvement over the O(log n) bound provided by standard "pointer doubling." The question of an explicit function (or functionality) that requires super constant number of rounds in this setting remains, however, open.