Moreover, our scheduling is optimal up to a polylogarithmic factor in terms of throughput capacity according to the lower bound of Gupta and Kumar. The main strength of our contribution lies in the fact that the scheduling is set in a fully distributed way and considers non-uniform power ranges, and it can therefore fit the sensor network setting. Clearly, a unit disk graph is a special case of a d-quasi unit disk graph for d 1. Such an edge may be there, but it may not be there. Note that the denition of a quasi unit disk graph does not specify whether there is an edge between two nodes u and v having distance d < dist(u,v) < 1. Precisely, we show that it is possible to emulate a 1/radic(n ln n)-UDG that satisfies the constraints of the SINR over any set of n wireless nodes distributed uniformly in a unit square, with only a O(ln 3 n) time and power stretch factor. is called a d-quasi unit disk graph (d-QUDG). In this paper, we demonstrate how careful scheduling of the nodes enables the two models to be combined to give the benefits of both the algorithmic features of the UDG and the physical validity of the SINR. Nevertheless, due to its complexity, this latter model has been the subject of very few theoretical investigations and lacks of good algorithmic features. The SINR model focuses on radio interferences created over the network depending on the distance to transmitters. For this purpose, the signal interference plus noise ratio model (SINR) is the more commonly used model. However, such a connectivity requirement is basically not compatible with the reality of wireless networks due to the environment of the nodes as well as the constraints of radio transmission. In a rho-UDG, two nodes are connected if and only if their distance is at most rho, for some rho > 0. As a tractable mathematical object, the unit disk graph (UDG) is a popular model that has enabled the development of efficient algorithms for crucial networking problems. Modeling communications in wireless networks is a challenging task, since it requires a simple mathematical object on which efficient algorithms can be designed but which must also reflect the complex physical constraints inherent in wireless networks, such as interferences, the lack of global knowledge, and purely local computations.
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