How Hard is Computing Parity with Noisy Communications?

Abstract

We show a tight lower bound of Ω(N log log N ) on the number of transmissions required to compute the parity of N input bits with constant error in a noisy communication network of N randomly placed sensors, each having one input bit and communicating with others using local transmissions with power near the connectivity threshold. This result settles the lower bound question left open by Ying, Srikant and Dullerud (WiOpt 06), who showed how the sum of all the N bits can be computed using O(N log log N ) transmissions. The same lower bound has been shown to hold for a host of other functions including majority by Dutta and Radhakrishnan (FOCS 2008).Most works on lower bounds for communication networks considered mostly the full broadcast model without using the fact that the communication in real networks is local, determined by the power of the transmitters. In fact, in full broadcast networks computing parity needs θ(N ) transmissions. To obtain our lower bound we employ techniques developed by Goyal, Kindler and Saks (FOCS 05), who showed lower bounds in the full broadcast model by reducing the problem to a model of noisy decision trees. However, in order to capture the limited range of transmissions in real sensor networks, we adapt their definition of noisy decision trees and allow each node of the tree access to only a limited part of the input. Our lower bound is obtained by exploiting special properties of parity computations in such noisy decision trees.

A Sampling Technique of Proving Lower Bounds for Noisy Computations

“…We then use our technique to derive the main result of this paper -a lower bound for wireless sensor networks. This simplifies and extends a lower bound of [DKmR08,DKmR15] for this model. A wireless sensor network consists of sensors that collect and cooperatively process data in order to compute some global function.…”

“…It was shown by Dutta, Kanoria, Manjunath and Radhakrishnan [DKmR08,DKmR15], that computing the parity of the inputs requires Ω(n log log n) transmissions. This result showed that the protocol presented by Ying, Srikant and Dullerud [YSD06] for computing the sum of all the bits (and hence any symmetric functions of these bits) is optimal.…”

“…This result showed that the protocol presented by Ying, Srikant and Dullerud [YSD06] for computing the sum of all the bits (and hence any symmetric functions of these bits) is optimal. The lower bound argument in [DKmR08,DKmR15] depended strongly on the properties of the parity function. In particular, the argument was not applicable for showing superlinear lower bounds for the majority and other symmetric functions in this model.…”

“…The starting point of our method is the connection between noisy wireless network protocols, and certain randomized decision trees. This connection was derived in [DKmR08,DKmR15], and it in turn made crucial use of a result in [GKS05]. The lower bound for parity in [DKmR08,DKmR15] was derived by rearranging the randomized decision trees obtained from wireless protocols computing parity.…”

“…This connection was derived in [DKmR08,DKmR15], and it in turn made crucial use of a result in [GKS05]. The lower bound for parity in [DKmR08,DKmR15] was derived by rearranging the randomized decision trees obtained from wireless protocols computing parity. In this work, we show that computations on decision trees that arise in our context can be simulated by randomized algorithms that leave a non-trivial fraction of their inputs unread.…”