Covering algorithms, continuum percolation and the geometry of wireless networks

Ronald Meester, Massimo Franceschetti, Jehoshua Bruck, Lorna Booth
2003 The Annals of Applied Probability
Continuum percolation models where each point o f a t wo-dimensional Poisson point process is the center of a disc of given or random radius r, h a ve been extensively studied. In this paper, we consider the generalization in which a deterministic algorithm given the points of the point process places the discs on the plane, in such a w ay that AMS 1991 subject classi cations. Primary 60D05, 60K35, 82B26, 82B43, 94C99. 1 each disc covers at least one point of the point process and that each
more » ... s and that each point i s c o vered by at least one disc. This gives a model for wireless communication networks, which w as the original motivation to study this class of problems. We look at the percolation properties of this generalized model, showing the almost sure non-existence of an unbounded connected component of discs for small values of of the density of the Poisson point process, for any c o vering algorithm. In general, it turns out not to be true that unbounded connected components arise when is taken su ciently high. However, we identify some large families of covering algorithms, for which such a n u n bounded component d o e s arise for large values of . We show h o w a simple scaling operation can change the percolation properties of the model, leading to the almost sure existence of an unbounded connected component for large values of , for any c o vering algorithm. Finally, w e show that a large class of covering algorithms, that arise in many practical applications, can get arbitrarily close to achieving a minimal density o f c o vering discs. We also show constructively the existence of algorithms that achieve this minimal density. 6 8 disc centers. We show Theorem 5.1 that: If R=r 1, then, for any grid G, there is a covering algorithm that places discs only at the vertices of G, and a.s. does not form an unbounded connected component, for any v alue of . If 1 R=r 2, then, for some given dense grid G, there is a covering algorithm that places discs only at the vertices of G, and a.s. does not form an unbounded connected component, for any v alue of . If R=r = 2, then, for any grid G, a n y c o vering algorithm that places discs only at the vertices of G forms a.s. an unbounded connected component for large values of . If R=r 2, then any algorithm forms a.s. an unbounded connected component for large values of , e v en if it is not grid-based. Note that the latter case has a high practical value, because it states that if base stations can communicate at a distance larger than twice the maximum communication distance to the clients, an unbounded connected component forms a.s. for large values of the density of the clients, regardless of the covering algorithm used to build the cellular network. Optimality Results. Finally, w e show constructively, in Theorem 6.3 the existence of algorithms that are optimal in achieving a minimal density of covering discs. We also show that a certain class of practical algorithms can achieve densities arbitrarily close to the optimal. 9