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.
We study the problem of achieving maximum barrier coverage by sensors on a barrier modeled by a line segment, by moving the minimum possible number of sensors, initially placed at arbitrary positions on the line containing the barrier. We consider several cases based on whether or not complete coverage is possible, and whether non-contiguous coverage is allowed in the case when complete coverage is impossible. When the sensors have unequal transmission ranges, we show that the problem of finding a minimum-sized subset of sensors to move in order to achieve maximum contiguous or non-contiguous coverage on a finite line segment barrier is NP-complete. In contrast, if the sensors all have the same range, we give efficient algorithms to achieve maximum contiguous as well as non-contiguous coverage. For some cases, we reduce the problem to finding a maximum-hop path of a certain minimum (maximum) weight on a related graph, and solve it using dynamic programming. 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.  In this paper, we study the problem of moving 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 O(n2lognloglogn) time algorithm. Further, if all sensors have the same-size sensing range, we give an O(nlogn) time algorithm, which improves the previous best O(n2) time solution. 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.