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.