On Gallager's problem: new lower-bounds for noisy communication
The effect of noise on computation and communication has been studied extensively in recent decades. This field of research can be traced back to Shannon, who proved that transferring information from one party to another over a noisy channel only requires a constant blowup in resource usage, compared to transmission over a noise free channel. Over the years it was shown that a constant blowup is sufficient to overcome noise for many other models.
In this talk I will discuss a model introduced by El Gamal in 1984 for sharing information over a noisy broadcast network: each of N players is given one input bit, and the goal is for all players to learn (with high probability) all the input bits of the other players, using the smallest possible number of broadcasts over a joint communication channel. In each broadcast a player transmits one bit to all other players; however noise flips the bit heard by each recipient with some fixed probability.
Without noise, N broadcasts would trivially suffice for the players to learn all bits. However the best known protocol that deals with noise, discovered by Gallager in 1988, uses $N\log\log N$ broadcasts. Attempts made since to bring the blowup down to a constant have failed. Our main result is that Gallager's protocol is in fact optimal up to a constant factor.
This is joint work with Navin Goyal and Michael Saks.