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We consider the following problem for wireless ad hoc networks: assume n nodes, each capable of communicating with nodes within a radius of r, are distributed in a d-dimensional region of side l; how large must the transmitting range r be to ensure that the resulting network is connected? We also consider the mobile version of the problem, in which nodes are allowed to move during a time interval and the value of r ensuring connectedness for a given fraction of the interval must be determined. For the stationary case, we give tight bounds on the relative magnitude of r, n and l yielding a connected graph with high probability in l-dimensional networks, thus solving an open problem. The mobile version of the problem when d=2 is investigated through extensive simulations, which give insight on how mobility affects connectivity and reveal a useful trade-off between communication capability and energy consumption.
Date of Conference: 2002