Broadcasting in UDG radio networks with missing and inaccurate information

Abstract

We study broadcasting time in radio networks, modeled as unit disk graphs (UDG). Network stations are represented by points in the plane and a station is connected to all stations at distance at most 1 from it. Stations are unaware of the network topology. Each station can send messages from the beginning of the broadcasting process, even before getting the source message. Emek et al. showed that broadcasting time depends on two parameters of the UDG network, namely, its diameter D (in hops) and its granularity g. The latter is the inverse of the density d of the network which is the minimum Euclidean distance between any two stations. They proved that the minimum broadcasting time is Θ(min {D + g ², D log g}), assuming that each node knows the density of the network and knows exactly its own position in the plane. In many situations these assumptions are unrealistic. Does removing them influence broadcasting time? The aim of this paper is to answer this question, hence we assume that density is unknown and nodes perceive their position with some unknown error margin ${\varepsilon}$. It turns out that this combination of missing and inaccurate information substantially changes the problem: the main new challenge becomes fast broadcasting in sparse networks (with constant density), when optimal time is O(D). Nevertheless, under our very weak scenario, we construct a broadcasting algorithm that maintains optimal time O (min {D + g ², D log g}) for all networks with at least 2 nodes, of diameter D and granularity g (previously obtained with exact positions and known density), if each node perceives its position with error margin ${\varepsilon=\alpha d}$, for any (unknown) constant α < 1/2. Rather surprisingly, the minimum time of an algorithm working correctly for all networks, and hence stopping if the source is alone, turns out to be Θ(D + g ²). Thus, the mere stopping requirement for the special case of the lonely source causes an exponential increase in broadcasting time, for networks of any density and any small diameter. Finally, if ${\varepsilon\geq d/2}$, then broadcasting is impossible.