Lower Bounds for Structuring Unreliable Radio Networks

Abstract

In this paper, we study lower bounds for randomized solutions to the maximal independent set (MIS) and connected dominating set (CDS) problems in the dual graph model of radio networks—a generalization of the standard graph-based model that now includes unreliable links controlled by an adversary. We begin by proving that a natural geographic constraint on the network topology is required to solve these problems efficiently (i.e., in time polylogarthmic in the network size). In more detail, we prove that in the absence of this constraint, for a network of size n: every MIS algorithm now requires Ω(n ^1 − ε) rounds to solve the problem, for any constant ε, 0 < ε ≤ 1, and every CDS algorithm that provides a reasonable approximation of a minimum CDS now requires $\Omega(\sqrt{n}/\log{n})$ rounds. We then prove the importance of the assumption that nodes are provided advance knowledge of their reliable neighbors (i.e, neighbors connected by reliable links). In more detail, we prove that in the absence of this assumption, for any CDS algorithm that guarantees a g(n)-approximation of a minimum CDS in f(n) rounds, it follows that g(n) + f(n) = Ω(n). This holds even if we assume the geographic constraint and the weakest possible adversary controlling the unreliable links. Finally, we show that although you can efficiently build an MIS without advance neighborhood knowledge, this omission increases the problem’s dependence on the geographic constraint. When both constraints are missing, every MIS algorithm now requires Ω(n) rounds, even if we assume the weakest possible adversary. Combined, these results answer an open question by proving that the efficient MIS and CDS algorithms from [2] are optimal with respect to their dual graph model assumptions. They also provide insight into what properties of an unreliable network enable efficient local computation.

This work is supported in part by NSF grant number CCF 1320279 and the Ford Motor Company University Research Program.