Feasibility and complexity of broadcasting with random transmission failures

We consider fault-tolerant broadcasting in the message passing and radio models under a probabilistic failure model. At each step, the transmitter of each node may fail independently with fixed probability p < 1. We study both omission and Byzantine transmission failures. Our goal is to establish conditions on feasibility and to estimate the complexity of almost-safe broadcasting (i.e., broadcasting which is correct with probability at least 1 - 1/n on n-node graphs for sufficiently large n) under these scenarios. If only omission failures are assumed, almost-safe broadcasting is feasible for any p < 1, in both communication models. For Byzantine faults, almost-safe broadcasting is feasible in the message passing model iff p < 1/2 and in the radio model iff p < (1-p)^Δ+1, where Δ is the maximum degree of the network. For the time complexity of almost-safe broadcasting, we give the following upper and lower bounds. Consider an n-node graph G with a given source s, and denote by D the radius of G w.r.t. s (namely, the largest distance from s to any node in G). Then for the message passing model we show that assuming omission faults, the optimal almost-safe broadcasting time is ⊖(D + log n). Assuming Byzantine faults, almost-safe broadcasting is possible in time O(D + log^α n), for any constant α > 1. For the radio model we show that almost-safe broadcasting in time O(opt + log n) (where opt is the optimal fault-free broadcasting time) is impossible for some graphs, even with omission failures, and we give an almost-safe broadcasting algorithm of time O(opt · log n) for any graph, for both types of failures.