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Considerable literature has been developed for various fundamentaldistributed problems in the SINR (Signal-to-Interference-plus-Noise-Ratio)model for radio transmission. A setting typically studied is when all nodestransmit a signal of the same strength, and each device only has access toknowledge about the total number of nodes in the network $n$, the range fromwhich each node's label is taken $[1,\dots,N]$, and the label of the deviceitself. In addition, an assumption is made that each node also knows itscoordinates in the Euclidean plane. In this paper, we create a technique whichallows algorithm designers to remove that last assumption. The assumption aboutthe unavailability of the knowledge of the physical coordinates of the nodestruly captures the `ad-hoc' nature of wireless networks.  Previous work in this area uses a flavor of a technique called dilution, inwhich nodes transmit in a (predetermined) round-robin fashion, and are able toreach all their neighbors. However, without knowing the physical coordinates,it's not possible to know the coordinates of their containing (pivotal) gridbox and seemingly not possible to use dilution (to coordinate theirtransmissions). We propose a new technique to achieve dilution without usingthe knowledge of physical coordinates. This technique exploits theunderstanding that the transmitting nodes lie in 2-D space, segmented by anappropriate pivotal grid, without explicitly referring to the actual physicalcoordinates of these nodes. Using this technique, it is possible for every weakdevice to successfully transmit its message to all of its neighbors in$\Theta(\lg N)$ rounds, as long as the density of transmitting nodes in anyphysical grid box is bounded by a known constant. This technique, we feel, isan important generic tool for devising practical protocols when physicalcoordinates of the nodes are not known.

Achieving Dilution without Knowledge of Coordinates in the SINR Model