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 Mathematics , 2012, Abstract: This work considers the identification of the available whitespace, i.e., the regions that are not covered by any of the existing transmitters, within a given geographical area. To this end, $n$ sensors are deployed at random locations within the area. These sensors detect for the presence of a transmitter within their radio range $r_s$, and their individual decisions are combined to estimate the available whitespace. The limiting behavior of the recovered whitespace as a function of $n$ and $r_s$ is analyzed. It is shown that both the fraction of the available whitespace that the nodes fail to recover as well as their radio range both optimally scale as $\log(n)/n$ as $n$ gets large. The analysis is extended to the case of unreliable sensors, and it is shown that, surprisingly, the optimal scaling is still $\log(n)/n$ even in this case. A related problem of estimating the number of transmitters and their locations is also analyzed, with the sum absolute error in localization as performance metric. The optimal scaling of the radio range and the necessary minimum transmitter separation is determined, that ensure that the sum absolute error in transmitter localization is minimized, with high probability, as $n$ gets large. Finally, the optimal distribution of sensor deployment is determined, given the distribution of the transmitters, and the resulting performance benefit is characterized.
 Computer Science , 2015, Abstract: Consider $n$ mobile sensors placed independently at random with the uniform distribution on a barrier represented as the unit line segment $[0,1]$. The sensors have identical sensing radius, say $r$. When a sensor is displaced on the line a distance equal to $d$ it consumes energy (in movement) which is proportional to some (fixed) power $a > 0$ of the distance $d$ traveled. The energy consumption of a system of $n$ sensors thus displaced is defined as the sum of the energy consumptions for the displacement of the individual sensors. We focus on the problem of energy efficient displacement of the sensors so that in their final placement the sensor system ensures coverage of the barrier and the energy consumed for the displacement of the sensors to these final positions is minimized in expectation. In particular, we analyze the problem of displacing the sensors from their initial positions so as to attain coverage of the unit interval and derive trade-offs for this displacement as a function of the sensor range. We obtain several tight bounds in this setting thus generalizing several of the results of [12] to any power $a >0$.
 Computer Science , 2015, Abstract: Consider $n$ sensors placed randomly and independently with the uniform distribution in a $d-$dimensional unit cube ($d\ge 2$). The sensors have identical sensing range equal to $r$, for some $r >0$. We are interested in moving the sensors from their initial positions to new positions so as to ensure that the $d-$dimensional unit cube is completely covered, i.e., every point in the $d-$dimensional cube is within the range of a sensor. If the $i$-th sensor is displaced a distance $d_i$, what is a displacement of minimum cost? As cost measure for the displacement of the team of sensors we consider the $a$-total movement defined as the sum $M_a:= \sum_{i=1}^n d_i^a$, for some constant $a>0$. We assume that $r$ and $n$ are chosen so as to allow full coverage of the $d-$dimensional unit cube and $a > 0$. The main contribution of the paper is to show the existence of a tradeoff between the $d-$dimensional cube, sensing radius and $a$-total movement. The main results can be summarized as follows for the case of the $d-$dimensional cube. If the $d-$dimensional cube sensing radius is $\frac{1}{2n^{1/d}}$ and $n=m^d$, for some $m\in N$, then we present an algorithm that uses $O\left(n^{1-\frac{a}{2d}}\right)$ total expected movement (see Algorithm 2 and Theorem 5). If the $d-$dimensional cube sensing radius is greater than $\frac{3^{3/d}}{(3^{1/d}-1)(3^{1/d}-1)}\frac{1}{2n^{1/d}}$ and $n$ is a natural number then the total expected movement is $O\left(n^{1-\frac{a}{2d}}\left(\frac{\ln n}{n}\right)^{\frac{a}{2d}}\right)$ (see Algorithm 3 and Theorem 7). In addition, we simulate Algorithm 2 and discuss the results of our simulations.