Achieving Dilution without Knowledge of Coordinates in the SINR Model

When multiple radio nodes, situated in an arena, simultaneously transmit some messages, whether a node receives a message or not is determined by a mathematical formula involving the signal strengths of the transmitting nodes and their respective physical coordinates and sometimes even the topology in which they are located. Considerable literature has been developed for various fundamental distributed problems in this SINR (Signal-to-Interference-plus-NoiseRatio) model for radio transmission. A setting typically studied is when all radio nodes transmit a signal of the same strength, and each device only has access to knowledge about the total number of nodes in the network n, the range from which each node’s label is taken [1, . . . , N ], and the label of the device itself. In addition, an assumption is made that each node also knows its coordinates in the Euclidean plane. In this paper, we create a mechanism which allows algorithm designers to remove that last assumption in the given setting. The assumption about the unavailability of the knowledge of the physical coordinates of the nodes truly captures the ‘ad-hoc’ nature of wireless networks. From a practical and theoretical perspective, devising solutions for this setting is a very important challenge to undertake. Previous work in this area uses a flavor of a beautiful technique called dilution, in which the nodes transmit in a (predetermined) round-robin fashion, and are able to reach all their neighbors. However, without knowing the physical coordinates, it’s not possible to know the coordinates of their containing (pivotal) grid box and seemingly not possible to use dilution (to coordinate their transmissions). We propose a new technique to achieve dilution without using the knowledge of physical coordinates. This technique exploits the understanding that the transmitting nodes lie in 2-D space, segmented by an appropriate pivotal grid, without explicitly referring to the actual physical coordinates of these nodes. Using this technique, it is possible for every weak device to successfully transmit its message to all of its neighbors in Θ(lgN) rounds, as long as the density of transmitting nodes in any physical grid box is bounded by a known constant. This technique, we feel, is an important generic tool for devising practical protocols when physical coordinates of the nodes are not known.