Minimum linear gossip graphs and maximal linear (Δ, k)‐gossip graphs

Abstract

AbstractGossiping is an information dissemination problem in which each node of a communication network has a unique piece of information that must be transmitted to all other nodes using two‐way communications between pairs of nodes along the communication links of the network. In this paper, we study gossiping using a linear‐cost model of communication which includes a start‐up time and a propagation time which is proportional to the amount of information transmitted. A minimum linear gossip graph is a graph (modeling a network), with the minimum possible number of links, in which gossiping can be completed in minimum time under the linear‐cost model. For networks with an even number of nodes, we prove that the structure of minimum linear gossip graphs is independent of the relative values of the start‐up and unit propagation times. We prove that this is not true when the number of nodes is odd. We present four infinite families of minimum linear gossip graphs. We also present minimum linear gossip graphs for all even numbers of nodes n ≤ 32 except n = 22. A linear (Δ, k)‐gossip graph is a graph with maximum degree Δ in which gossiping can be completed in k rounds with minimum propagation time. We present three infinite families of maximal linear (Δ, k)‐gossip graphs, that is, linear (Δ, k)‐gossip graphs with a maximum number of nodes. We show that not all minimum broadcast graphs are maximal linear (Δ, k)‐gossip graphs. © 2001 John Wiley & Sons, Inc.