Lower Bound on Network Diameter for Distributed Function Computation

Abstract

Parallel and distributed computing network-systems are modeled as graphs with vertices representing compute elements and adjacency-edges capturing their uni- or bi-directional communication. Distributed function computation covers a wide spectrum of major applications, such as quantized consensus and collaborative hypothesis testing, in distributed systems. Distributed computation over a network-system proceeds in a sequence of time-steps in which vertices update and/or exchange their values based on the underlying algorithm constrained by the time-(in)variant network-topology. For finite convergence of distributed information dissemination and function computation in the model, we study lower bounds on the number of time-steps for vertices to receive (initial) vertex-values of all vertices regardless of underlying protocol or algorithmics in time-invariant networks via the notion of vertex-eccentricity in a graph-theoretic framework. We prove a lower bound on the maximum vertex-eccentricity in terms of graph-order and -size in a strongly connected directed graph, and demonstrate its optimality via an explicitly constructed family of strongly connected directed graphs.