Lower Bounds on Broadcasting Time in UDG Radio Networks with Unknown Topology

The paper considers broadcasting in radio networks, modeled as unit disk graphs (UDG). Network stations are modeled as points in the plane, where a station is connected to all stations at Euclidean distance at most 1 from it. A message transmitted by a station reaches all its neighbors, but a station hears a message (receives the message correctly) only if exactly one of its neighbors transmits at a given time step. One station of the network, called the source, has a message which has to be disseminated to all other stations. Stations are unaware of the network topology. Two broadcasting models are considered. In the conditional wake up model, the stations other than the source are initially idle and cannot transmit until they hear a message for the first time. In the spontaneous wake up model, all stations are awake (and may transmit messages) from the beginning. It turns out that broadcasting time depends on two parameters of the UDG network, namely, its diameter D and its granularity g, which is the inverse of the minimum Euclidean distance between any two stations. For the conditional wake up model, we prove that any deterministic algorithm requires Ω(D √ g) time to accomplish broadcasting. This narrows the gap left by the best known broadcasting algorithm in this model which works in time O(Dg). For the spontaneous wake up model, we establish a tight lower bound of Ω ( min { D + g, D log g }) on deterministic broadcasting time. Thus our results yield a provable separation between the two models: for some parameter values, the lower bound in the first model is significantly larger than the upper bound in the second. keywords: radio network, unit disk graph, ad hoc network, algorithm, broadcasting ∗Department of Computer Science, The University of Liverpool, Ashton Building, Ashton Street, Liverpool L69 3BX, UK. E-mail:{leszek,suc}@csc.liv.ac.uk. †Département d’informatique, Université du Québec en Outaouais, Gatineau, Québec J8X 3X7, Canada. E-mail: [email protected]. Research partially supported by NSERC discovery grant and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais. ‡Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: {yuval.emek,erez.kantor,david.peleg}@weizmann.ac.il. Supported in part by grants from the Minerva Foundation and the Israel Ministry of Science.