Topology matters in communication

We consider the communication cost of computing functions when inputs are distributed among the vertices of an undirected graph. The communication is assumed to be point-to-point: a processor sends messages only to its neighbors. The processors in the graph act according to a pre-determined protocol, which can be randomized and may err with some small probability. The communication cost of the protocol is the total number of bits exchanged in the worst case. Extending recent work that assumed that the graph was the complete graph (with unit edge lengths), we develop a methodology for showing lower bounds that are sensitive to the graph topology. In particular, for a broad class of graphs, we obtain a lower bound of the form ω(k²n), for computing a function of k inputs, each of which is n-bits long and located at a different vertex. Previous works obtained lower bounds of the form ω(kn). This methodology yields a variety of other results including the following: A tight lower bound (ignoring poly-log factors) for Element Distinctness, settling a question of Phillips, Verbin and Zhang (SODA '12), a distributed XOR lemma, a lower bound for composed functions, settling a question of Phillips et al., new topology-dependent bounds for several natural graph problems considered by Woodruff and Zhang (DISC '13). To obtain these results we use tools from the theory of metric embeddings and represent the topological constraints imposed by the graph as a collection of cuts, each cut providing a setting where our understanding of two-party communication complexity can be effectively deployed.