Optimal odd gossiping

In the gossiping problem, each node in a network starts with a unique piece of information and must acquire the information of all other nodes using two-way communications between pairs of nodes. In this paper we investigate gossiping in n-node networks with n odd. We use a linear cost model in which the cost of communication is proportional to the amount of information transmitted. In synchronous gossiping, the pairwise communications are organized into rounds, and all communications in a round start at the same time. We present optimal synchronous gossip algorithms for all odd values of n, proving the truth of a conjecture by Fertin, Peters, Raabe, and Xu. Central to the construction of the algorithms is a doubly-inductive proof about properties of optimal gossip algorithms.