On Local Broadcasting Schedules and CONGEST Algorithms in the SINR Model

Abstract

We consider the distributed construction of a deterministic local broadcasting schedule in the SINR model of interference. During the execution of such a schedule each node should be able to transmit one message to its neighbors. Our construction requires only $\mathcal {O}(\varDelta \log n)$ time slots, where $\varDelta$ is the maximum node degree in the network and $n$ the number of nodes. We prove that the length of the constructed schedule is asymptotically optimal, i.e. of length $\mathcal {O}(\varDelta )$. Considering the simulation of $\mathcal {CONGEST}$ algorithms in the SINR model, our deterministic schedule achieves a runtime of $\mathcal {O}(\tau \varDelta ^2 + \varDelta \log n)$ time slots, where $\tau$ is the original runtime in the $\mathcal {CONGEST}$ model. We show that there is a lower bound of $\varOmega (\varDelta ^2)$ for the simulation of each one of the $\tau$ rounds, hence our simulation is optimal apart from the logarithmic factor. If we restrict the knowledge of the nodes and let the maximum node degree $\varDelta$ be unknown, we can prove that at least $\varOmega (\mathrm{D }+ \tau \varDelta ^2)$ time slots are required to simulate synchronized $\mathcal {CONGEST}$ algorithms in the SINR model of interference, where $\mathrm{D }$ is the diameter of the network. For our algorithms we assume location information to be given. Regarding the case without location information we argue that a deterministic algorithm to compute local broadcasting schedules by Derbel and Talbi [ICDCS’10], which requires transmission power adaption, needs messages of size $\mathcal {O}(\log n)$ to simulate $\mathcal {CONGEST}$ algorithms. This is a logarithmic factor less than stated by the authors.