We consider the model of fully connected networks, where in each round each node can send an O(log n)-bit message to each other node (this is the congest model with diameter 1). It is known that in this model, min-weight spanning trees can be found in O(log log n) rounds. In this paper we show that distributed sorting, where each node has at most n items, can be done in time O(log log n) as well. It is also shown that selection can be done in O(1) time. (Using a concurrent result by Lenzen and Wattenhofer, the complexity of sorting is further reduced to constant.) Our algorithms are randomized, and the stated complexity bounds hold with high probability.

We consider the problem of balancing load items (tokens) on networks. Starting with an arbitrary load distribution, we allow in each round nodes to exchange tokens with their neighbors. Thegoalisto achieveadistribution whereall nodeshavenearlythe samenumber of tokens. For the continuous case where tokens are arbitrarily divisible, most load balancing schemes correspond to Markov chains whose convergence is rather well-understood in terms of their spectral gap. However, since for many applications load items cannot be divided arbitrarily, we focus on the discrete case where the load is composed of indivisible tokens. Unfortunately, this discretization entails a non-linear behavior due to its rounding errors, which makes the analysis much harder than in the continuous case. Therefore, it has been a major open problem to understand the limitations of discrete load balancing and its relation to the continuous case. We investigate several randomized protocols for different communication models in the discrete case. Ourresults demonstratethat there is almost no deviationbetween the discrete