The Complexity of Data Aggregation in Directed Networks

We study problems of data aggregation, such as approximate counting and computing the minimum input value, in synchronous directed networks with bounded message bandwidth B = Ω(log n). In undirected networks of diameter D, many such problems can easily be solved in O(D) rounds, using O(log n)- size messages. We show that for directed networks this is not the case: when the bandwidth B is small, several classical data aggregation problems have a time complexity that depends polynomially on the size of the network, even when the diameter of the network is constant. We show that computing an e-approximation to the size n of the network requires Ω(min{n, 1/e2}/B) rounds, even in networks of diameter 2. We also show that computing a sensitive function (e.g., minimum and maximum) requires Ω(√n/B) rounds in networks of diameter 2, provided that the diameter is not known in advance to be o(√n/B). Our lower bounds are established by reduction from several well-known problems in communication complexity. On the positive side, we give a nearly optimal O(D+√n/B)-round algorithm for computing simple sensitive functions using messages of size B = Ω(logN), where N is a loose upper bound on the size of the network and D is the diameter.