Consider @math sensors placed randomly and independently with the uniform distribution in a @math dimensional unit cube ( @math ). The sensors have identical sensing range equal to @math , for some @math . We are interested in moving the sensors from their initial positions to new positions so as to ensure that the @math dimensional unit cube is completely covered, i.e., every point in the @math dimensional cube is within the range of a sensor. If the @math -th sensor is displaced a distance @math , what is a displacement of minimum cost? As cost measure for the displacement of the team of sensors we consider the @math -total movement defined as the sum @math , for some constant @math . We assume that @math and @math are chosen so as to allow full coverage of the @math dimensional unit cube and @math . The main contribution of the paper is to show the existence of a tradeoff between the @math dimensional cube, sensing radius and @math -total movement. The main results can be summarized as follows for the case of the @math dimensional cube. If the @math dimensional cube sensing radius is @math and @math , for some @math , then we present an algorithm that uses @math total expected movement (see Algorithm 2 and Theorem 5). If the @math dimensional cube sensing radius is greater than @math and @math is a natural number then the total expected movement is @math (see Algorithm 3 and Theorem 7). In addition, we simulate Algorithm 2 and discuss the results of our simulations.
An analysis similar to the one for the line segment was provided for the unit square in @cite_14 . Our present paper focuses on the analysis of sensor displacement for a group of sensors placed uniformly at random on the @math dimensional unit cube, thus also generalizing the results of @cite_6 from @math to arbitrary dimension @math . In particular, our approach is the first to generalize the results of @cite_14 to the @math dimensional unit cube using as cost metric the @math -total movement, and also obtain sharper bounds for the case of the unit square.
@cite_14 @cite_6
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Consider @math sensors placed randomly and independently with the uniform distribution in a unit square. The sensors have identical sensing range equal to @math , for some @math . We are interested in moving the sensors from their initial positions to new positions so as to ensure that the unit square is completely covered, i.e., every point in the squarei¾?is within the range of a sensor. If the @math -th sensor is displaced a distance @math , what is a displacement of minimum cost? As cost measure for the displacement of the team of sensors we consider thei¾? @math -total movement defined as the sum @math , for some constant @math . We assume that @math and @math are chosen so as to allow full coverage of the square and @math . The main contribution of the paper is to show the existence of a tradeoff between the square sensing radius and @math -total movement and can be summarized as follows:1.If the square sensing radius is equal to @math and @math is the square of a natural number we present an algorithm and show that in expectation the @math -total movement is in @math .2.If the square sensing radius is greater than @math and @math is natural number then we present an algorithm and show that in expectation the @math -total movement is in @math . Therefore this sharp decrease from @math to @math in the @math -total movement of the sensors to attain complete coverage of the square indicates the presence of an interesting threshold on the square sensing radius when it increases from @math to @math . In addition, we simulate our algorithms above and discuss the results of our simulations. Consider @math mobile sensors placed independently at random with the uniform distribution on a barrier represented as the unit line segment @math . The sensors have identical sensing radius, say @math . When a sensor is displaced on the line a distance equal to @math it consumes energy (in movement) which is proportional to some (fixed) power @math of the distance @math traveled. The energy consumption of a system of @math 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 [10] to any power @math . Assume that n sensors with identical range r = f(n) 2n, for some f(n) > 1 for all n, are thrown randomly and independently with the uniform distribution in the unit interval [0,1]. They are required to move to new positions so as to cover the entire unit interval in the sense that every point in the interval is within the range of a sensor. We obtain tradeoffs between the expected sum and maximum of displacements of the sensors and their range required to accomplish this task. In particular, when f(n) = 1 the expected total displacement is shown to be Θ(√n). For sensors with larger ranges we present two algorithms that prove the upper bound for the sum drops sharply as f(n) increases. The first of these holds for f(n) ≥ 6 and shows the total movement of the sensors is O ( √In n (n)) while the second holds for 12 ≤ f(n) ≤ In n - 2 In In n and gives an upper bound of O(ln n f(n)ef(n) 2 ). Note that the second algorithm improves upon the first for f(n) δ In In n - In In Inn. Further we show a lower bound, for any 1 ω f(n) 0. For the case of the expected maximum displacement of a sensor when f(n) = 1 our bounds are (n -1 2) and for any e > 0, O(n-1 2+e). For larger sensor ranges (up to (1 - e)ln n n e > O) the expected maximum displacement is shown to be Θ(ln n n). We also obtain similar sum and maximum displacement and range tradeoffs for area coverage for sensors thrown at random in a unit square. In this case, for the expected maximum displacement our bounds are tight and for the expected sum they are within a factor of √ln n. Finally, we investigate the related problem of the expected total and maximum displacement for perimeter coverage (whereby only the perimeter of the region need be covered) of a unit square. For example, when n sensors of radius > 2 n are thrown randomly and independently with the uniform distribution in the interior of a unit square, we can show the total expected displacement required to cover the perimeter is n 12 + o(n). © 2013 ACM. In this paper, we study the problem of moving @math n sensors on a line to form a barrier coverage of a specified segment of the line such that the maximum moving distance of the sensors is minimized. Previously, it was an open question whether this problem on sensors with arbitrary sensing ranges is solvable in polynomial time. We settle this open question positively by giving an @math O(n2logn) time algorithm. For the special case when all sensors have the same-size sensing range, the previously best solution takes @math O(n2) time. We present an @math O(nlogn) time algorithm for this case; further, if all sensors are initially located on the coverage segment, our algorithm takes @math O(n) time. Also, we extend our techniques to the cycle version of the problem where the barrier coverage is for a simple cycle and the sensors are allowed to move only along the cycle. For sensors with the same-size sensing range, we solve the cycle version in @math O(n) time, improving the previously best @math O(n2) time solution.