On the Complexity of Fixed-Schedule Neighbourhood Learning in Wireless Ad Hoc Radio Networks

Abstract

Consider a synchronous static radio network of $n$ nodes represented by an undirected graph with maximum degree $\varDelta$. Suppose that each node has a unique ID from $\{1,\ldots ,N\}$, where $N \gg n$. In the complete neighbourhood learning task, each node $p$ must produce a set $L_p$ of IDs such that ID $i \in L_p$ if and only if $p$ has a neighbour with ID $i$. We study the complexity of this task when it is assumed that each node fixes its entire transmission schedule at the start of the algorithm. We prove a $\varOmega (\frac{\varDelta ^2}{\log \varDelta }\log {N})$-slot lower bound on schedule length that holds in very general models, e.g., when nodes possess collision detectors, messages can be of arbitrary size, and nodes know the schedules being followed by all other nodes. We also prove a similar result for the SINR model of radio networks. To prove these results, we introduce a new generalization of cover-free families of sets, which may be of independent interest. We also show a separation between the class of fixed-schedule algorithms and the class of algorithms where nodes can choose to leave out some transmissions from their schedule.