Deterministic coding for interactive communication

Two factors are prominent among those contributing to the increases in speed and storage capacity in current generations of computers. The first is increasing parallelism — whether in actual parallel and distributed computers, or among the steadily more numerous components of a sequential machine. The second is the dramatic miniaturization of logical devices and wires. The first of these factors greatly magnifies the number of interprocessor communications performed during any computation, while the second increases the noise level affecting transmissions. For these reasons, and on the basis that the role of noise should be understood in a model of a physical process, the following concern was recently identified as basic [10]. Consider a problem whose input is split between two processors connect ed by a communication link; and for which an interactive protocol exists which solves the problem in T transmissions on any input, provided the channel is noiseless. If in fact there is some noise on the channel, what is the effect upon the number of transmissions needed in order to solve the communication problem reliably? We describe a deterministic method for simulating noiseless-channel protocols on noisy channels, with only a constant slow-down. This is an analog for general interactive protocols of Shannon’s coding theorem, which dealt only with data transmission, i.e. one-way protocols [11]. This result im*Research supported by an NSF postdoctoral fellowship. Permission to copy without fee all or part of this material is granted provided that the copias are not made or distributed for direct commercial advantage, the ACM copyright notice and tha title of the publication and its date appear, and notioe is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 25th ACM STOC ‘93-51931CA,USA 01993 ACM 0-89791 -591 -71931000510747 . .. S1.50 proves on recent work which provided a randomized simulation method for interactive protocols. The Shannon theorem is thus reproduced for the general interactive case, in all but the constant factor. The randomized method was fundamentally unsuited to further derandomization, and the deterministic solution is entirely different. A key role in the present work is played by tree codes, originally considered by Wozencraft [13] for the sake of comput ationally efficient decoding of noisy data transmissions. In their new setting tree codes are reinterpreted as a way of transforming a highly interactive protocol into one that behaves like a pair of one-way protocols, and which therefore can be implemented at both high rate and reliability y.