Computing in fault tolerant broadcast networks and noisy decision trees

We consider a fault tolerant 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 broadcast 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. We present here the first linear complexity protocols for several classes of Boolean functions. This answer an open question of Yao (Invited talk in the 5th ISTCS Conf., 1997), considering this fault tolerant model that was introduced by El Gamal (Open problems presented at the 1984 workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research) and studied also by Gallager [IEEE Tran Info Theory 34 (1988) 176-180].