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.
Assume that @math sensors of identical range are all initially placed on a line. It was shown in @cite_1 that there is an @math algorithm for minimizing the max displacement of a sensor while the optimization problem becomes NP-complete if there are two separate (non-overlapping) barriers on the line (cf. also @cite_3 for arbitrary sensor ranges). If the optimization cost is the sum of displacements then @cite_9 shows that the problem is NP-complete when arbitrary sensor ranges are allowed, while an @math algorithm is given when all sensing ranges are the same. Similarly, if one is interested in the number of sensors moved then the coverage problem is NP-complete when arbitrary sensor ranges are allowed, and an @math algorithm is given when all sensing ranges are the same @cite_8 . Further, @cite_12 considers the algorithmic complexity of several natural generalizations of the barrier coverage problem with sensors of arbitrary ranges, including when the initial positions of sensors are arbitrary points in the two-dimensional plane, as well as multiple barriers that are parallel or perpendicular to each other.
@cite_8 @cite_9 @cite_1 @cite_3 @cite_12
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We consider several variations of the problems of covering a set of barriers (modeled as line segments) using sensors that can detect any intruder crossing any of the barriers. Sensors are initially located in the plane and they can relocate to the barriers. We assume that each sensor can detect any intruder in a circular area centered at the sensor. Given a set of barriers and a set of sensors located in the plane, we study three problems: the feasibility of barrier coverage, the problem of minimizing the largest relocation distance of a sensor (MinMax), and the problem of minimizing the sum of relocation distances of sensors (MinSum). When sensors are permitted to move to arbitrary positions on the barrier, the problems are shown to be NP-complete. We also study the case when sensors use perpendicular movement to one of the barriers. We show that when the barriers are parallel, both the MinMax and MinSum problems can be solved in polynomial time. In contrast, we show that even the feasibility problem is NP-complete if two perpendicular barriers are to be covered, even if the sensors are located at integer positions, and have only two possible sensing ranges. On the other hand, we give an O(n 3 2) algorithm for a natural special case of this last problem. A set of sensors establishes barrier coverage of a given line segment if every point of the segment is within the sensing range of a sensor. Given a line segment I, n mobile sensors in arbitrary initial positions on the line (not necessarily inside I) and the sensing ranges of the sensors, we are interested in finding final positions of sensors which establish a barrier coverage of I so that the sum of the distances traveled by all sensors from initial to final positions is minimized. It is shown that the problem is NP complete even to approximate up to constant factor when the sensors may have different sensing ranges. When the sensors have an identical sensing range we give several efficient algorithms to calculate the final destinations so that the sensors either establish a barrier coverage or maximize the coverage of the segment if complete coverage is not feasible while at the same time the sum of the distances traveled by all sensors is minimized. Some open problems are also mentioned. 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.