Extremal Problem with Network-Diameter and -Minimum-Degree for Distributed Function Computation

Distributed function computation has a wide spectrum of major applications 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. Distributed computing network-systems are modeled as directed/undirected graphs with vertices representing compute elements and adjacency-edges capturing their uni- or bi-directional communication. To quantify an intuitive tradeoff between two graph-parameters: minimum vertex-degree and diameter of the underlying graph, we formulate an extremal problem with the two parameters: for all positive integers n and d, the extremal value ∇ (n, d) denotes the least minimum vertex-degree among all connected order-n graphs with diameters of at most d. We prove matching upper and lower bounds on the extremal values of ∇ (n, d) for various combinations of n- and d-values. © 2020, Springer Nature Singapore Pte Ltd.