Tighter Upper Bounds for the Minimum Number of Calls and Rigorous Minimal Time in Fault-Tolerant Gossip Schemes

The gossip problem (telephone problem) is an information dissemination problem in which each of n nodes of a communication network has a unique piece of information that must be transmitted to all the 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) needed to guarantee the k-fault-tolerance. We construct two classes of k-fault-tolerant gossip schemes (sequences of calls) and found two upper bounds of τ (n, k), which improve the previously known results. The first upper bound for general even n is τ (n, k) ≤ 1 2 n ⌈log 2 n⌉ + 1 2 nk. This result is used to obtain the upper bound for general odd n. From the expressions for the second upper bound it follows that τ (n, k) ≤ 2 3 nk + O(n) for large n. Assuming that the calls can take place simultaneously, it is also of interest to find k-fault-tolerant gossip schemes, which can spread the full information in minimal time. For even n we showed that the minimal time is T (n, k) = ⌈log 2 n⌉ + k.