We consider an interval coverage problem. Given @math intervals of the same length on a line @math and a line segment @math on @math , we want to move the intervals along @math such that every point of @math is covered by at least one interval and the sum of the moving distances of all intervals is minimized. As a basic geometry problem, it has applications in mobile sensor barrier coverage in wireless sensor networks. The previous work solved the problem in @math time. In this paper, by discovering many interesting observations and developing new algorithmic techniques, we present an @math time algorithm. We also show an @math time lower bound for this problem, which implies the optimality of our algorithm.
Mehrandish @cite_13 @cite_2 considered another variant of the one-dimensional barrier coverage problem, where the goal is to move the minimum number of sensors to form a barrier coverage. They @cite_13 @cite_2 proved the problem is NP-hard if sensors have different ranges and gave polynomial time algorithms otherwise. In addition, Li @cite_4 considers the linear coverage problem which aims to set an energy for each sensor to form a coverage such that the cost of all sensors is minimized. There @cite_4 , the sensors are not allowed to move, and the more energy a sensor has, the larger the covering range of the sensor and the larger the cost of the sensor. Another problem variation is considered in @cite_12 , where the goal is to maximize the barrier coverage lifetime subject to the limited battery powers.
@cite_13 @cite_12 @cite_4 @cite_2
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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. 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. 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. This paper addresses the coverage breach problem in wireless sensor networks with limited bandwidths. In wireless sensor networks, sensor nodes are powered by batteries. To make efficient use of battery energy is critical to sensor network lifetimes. When targets are redundantly covered by multiple sensors, especially in stochastically deployed sensor networks, it is possible to save battery energy by organizing sensors into mutually exclusive subsets and alternatively activating only one subset at any time. Active nodes are responsible for sensing, computing and communicating. While the coverage of each subset is an important metric for sensor organization, the size of each subset also plays an important role in sensor network performance because when active sensors periodically send data to base stations, contention for channel access must be considered. The number of available channels imposes a limit on the cardinality of each subset. Coverage breach happens when a subset of sensors cannot completely cover all the targets. To make efficient use of both energy and bandwidth with a minimum coverage breach is the goal of sensor network design. This paper presents the minimum breach problem using a mathematical model, studies the computational complexity of the problem, and provides two approximate heuristics. Effects of increasing the number of channels and increasing the number of sensors on sensor network coverage are studied through numerical simulations. Overall, the simulation results reveal that when the number of sensors increases, network lifetimes can be improved without loss of network coverage if there is no bandwidth constraint; with bandwidth constraints, network lifetimes may be improved further at the cost of coverage breach.