Computing in fault tolerance broadcast networks

We consider a fault tolerance broadcast network of n processors each holding one bit of information. The goal is to compute a given Boolean function on the n bits. In each step, a processor may broadcast one bit of information. Each listening processor receives the bit that was broadcasted with error probability bounded by a fixed constant ε. The errors in different steps, as well as for different receiving processors in the same step, are mutually independent. The protocols that are considered in this model are oblivious protocols: At each step, the processors that broadcast are fixed in advanced and independent of the input and the outcome of previous steps. The primal complexity measure in this model is the total number of broadcasts that is performed by the protocol. We present here the first linear complexity protocols for several classes of Boolean functions, including the OR function, functions that have O(l)-minterm (maxterm) size, functions that have linear size AC₀ formulae and some other functions. This answer an open question of Yao (1997), considering this fault tolerance model of El-Gamal (1984) and Gallager (1988).