New methods of construction of fault-tolerant gossip graphs

The gossip problem (telephone problem) is an information dissemination problem in which each the of n nodes of communication network has a unique piece of information that should be transmitted to all other nodes using two-way communications (telephone calls) between the pairs of nodes. During a call between the given two nodes, they exchange the whole information known to them at that moment. In this paper we investigate the k-fault-tolerant gossip problem, which is a generalization of the gossip problem, where at most k arbitrary faults of calls are allowed. The problem is to find the minimal number of calls τ(n, k) required to ensure the k-fault-tolerance. We construct two classes of k-fault-tolerant gossip schemes (sequences of calls) and improve the previously known results on upper bounds of τ(n, k). Assuming that calls can be made simultaneously, it is also of interest to find k-fault-tolerant gossip schemes, which can spread the full information during a minimal time. For even n we showed that the minimal time is T(n, k) = ⌈log
₂
n⌉ + k.