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Distributed computation of symmetric functions with binary inputs

This paper considers the following network computation problem: n nodes are placed on a √n×√n grid, each node in the network is connected to every other node within distance r(n) of itself, and is given an arbitrary input bit. Connected nodes communicate with each other over independent binary symmetric channels of a given transition probability ε ≥ 0, and an arbitrarily designated node computes a symmetric target function ƒ of the input bits. We characterize up to order the minimum number of transmissions required to compute ƒ with a probability of error less than any given positive constant δ. As a side result, we answer an open question posed by El Gamal in 1987 [1] regarding the number of transmissions required to compute the parity function over ring and tree networks.