Broadcasting through a noisy one-dimensional network

We study one bit broadcast in a one-dimensional network with nodes .No, ... ,.Nn , in which each .Ni.-l sends information to .Ni.. We suppose that the broadcasting is synchronous, and at each step each atomic transmission .Ni.-l --+ .Ni. could be temporarily incorrect with probability equal to a constant 0 < P < 1/2. The probabilities of failure for different steps and different nodes are supposed to be independent. For each constant c there is a "classical" algorithm with O( n log n) broadcast time and error prob ability O(n). The paper studies the possibility of a reliable broadcasting in o( n log n) time. We first show that one natural generalization of the dassical algorithm, which was believed to behave weil, has very bad properties (the prob ability of an error dose to 1/2). The second part of the paper presents the ultimate solution of the problem of the broadcast time in a one-dimensional nework with faults. Our algorithms have linear broadcast time, good (though not optimal) delay time, and they are extremely reliable. For example we can transmit a bit through a network of N = 1000000 of nodes with p = 0.1 in 8999774 < 9N steps with prob ability of error less than 10-436 •