Lower Bounds for Noisy Wireless Networks using Sampling Algorithms

We show a tight lower bound of Omega(N\log\log N) on the number of transmissions required to compute several functions (including the parity function and the majority function) in a network of N randomly placed sensors, communicating using local transmissions, and operating with power near the connectivity threshold. This result considerably simplifies and strengthens an earlier result of Dutta, Kanoria Manjunath and Radhakrishnan (SODA 08) that such networks cannot compute the parity function reliably with significantly fewer than N\log \log N transmissions, thereby showing that the protocol with O(N\log \log N) transmissions due to Ying, Srikant and Dullerud (WiOpt 06) is optimal. We also observe that all the lower bounds shown by Evans and Pippenger (SIAM J. on Computing, 1999) on the average noisy decision tree complexity for several functions can be derived using our technique simply and in a unified way.