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.
A Wireless Sensor Network (WSN) uses a large number of sensors to monitor some surrounding environmental phenomena @cite_0 . Each sensor is equipped with a sensing device with limited battery-supplied energy. The sensors process data obtained and forward the data to a base station. Intrusion detection and border surveillance constitute a major application category for WSNs. A main goal of these applications is to detect intruders as they cross the boundary of a region or domain. For example, research efforts were made to extend the scalability of WSNs to the monitoring of international borders @cite_16 @cite_1 . Unlike the traditional full coverage @cite_15 @cite_7 @cite_3 which requires an entire target region to be covered by the sensors, the barrier coverage @cite_8 @cite_9 @cite_10 @cite_14 @cite_1 only seeks to cover the perimeter of the region to ensure that any intruders are detected as they cross the region border. Since barrier coverage requires fewer sensors, it is often preferable to full coverage. Because sensors have limited battery-supplied energy, it is desired to minimize their movements.
@cite_14 @cite_7 @cite_8 @cite_9 @cite_1 @cite_3 @cite_0 @cite_15 @cite_16 @cite_10
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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. The efficiency of sensor networks depends on the coverage of the monitoring area. Although, in general, a sufficient number of sensors are used to ensure a certain degree of redundancy in coverage, a good sensor deployment is still necessary to balance the workload of sensors. In a sensor network with locomotion facilities, sensors can move around to self-deploy. The movement-assisted sensor deployment deals with moving sensors from an initial unbalanced state to a balanced state. Therefore, various optimization problems can be defined to minimize different parameters, including total moving distance, total number of moves, communication computation cost, and convergence rate. In this paper, we first propose a Hungarian-algorithm-based optimal solution, which is centralized. Then, a localized scan-based movement-assisted sensor deployment method (SMART) and several variations of it that use scan and dimension exchange to achieve a balanced state are proposed. An extended SMART is developed to address a unique problem called communication holes in sensor networks. Extensive simulations have been done to verify the effectiveness of the proposed scheme. Intrusion detection, area coverage and border surveillance are important applications of wireless sensor networks today. They can be (and are being) used to monitor large unprotected areas so as to detect intruders as they cross a border or as they penetrate a protected area. We consider the problem of how to optimally move mobile sensors to the fence (perimeter) of a region delimited by a simple polygon in order to detect intruders from either entering its interior or exiting from it. We discuss several related issues and problems, propose two models, provide algorithms and analyze their optimal mobility behavior. Global barrier coverage that requires much fewer sensors than full coverage, is known to be an appropriate model of coverage for movement detection applications such as intrusion detection. However, it has been proved that given a sensor deployment, sensors can not locally determine whether the deployment provides global barrier coverage, making it impossible to develop localized algorithms, thus limiting its use in practice. In this paper, we introduce the concept of local barrier coverage to address this limitation. Motivated by the observation that movements are likely to follow a shorter path in crossing a belt region, local barrier coverage guarantees the detection of all movements whose trajectory is confined to a slice of the belt region of deployment. We prove that it is possible for individual sensors to locally determine the existence of local barrier coverage, even when the region of deployment is arbitrarily curved. Although local barrier coverage does not always guarantee global barrier coverage, we show that for thin belt regions, local barrier coverage almost always provides global barrier coverage. To demonstrate that local barrier coverage can be used to design localized algorithms, we develop a novel sleep-wakeup algorithm for maximizing the network lifetime, called Localized Barrier Coverage Protocol (LBCP). We show that LBCP provides close to optimalenhancement in network lifetime, while providing global barrier coverage most of the time. It outperforms an existing algorithm called Randomized Independent Sleeping (RIS) by up to 6 times. When a sensor network is deployed to detect objects penetrating a protected region, it is not necessary to have every point in the deployment region covered by a sensor. It is enough if the penetrating objects are detected at some point in their trajectory. If a sensor network guarantees that every penetrating object will be detected by at least k distinct sensors before it crosses the barrier of wireless sensors, we say the network provides k-barrier coverage. In this paper, we develop theoretical foundations for k-barrier coverage. We propose efficient algorithms using which one can quickly determine, after deploying the sensors, whether the deployment region is k-barrier covered. Next, we establish the optimal deployment pattern to achieve k-barrier coverage when deploying sensors deterministically. Finally, we consider barrier coverage with high probability when sensors are deployed randomly. The major challenge, when dealing with probabilistic barrier coverage, is to derive critical conditions using which one can compute the minimum number of sensors needed to ensure barrier coverage with high probability. Deriving critical conditions for k-barrier coverage is, however, still an open problem. We derive critical conditions for a weaker notion of barrier coverage, called weak k-barrier coverage. Due to their low cost and small form factors, a large number of sensor nodes can be deployed in redundant fashion in dense sensor networks. The availability of redundant nodes increases network lifetime as well as network fault tolerance. It is, however, undesirable to keep all the sensor nodes active at all times for sensing and communication. An excessive number of active nodes lead to higher energy consumption and it places more demand on the limited network bandwidth. We present an efficient technique for the selection of active sensor nodes in dense sensor networks. The active node selection procedure is aimed at providing the highest possible coverage of the sensor field, i.e., the surveillance area. It also assures network connectivity for routing and information dissemination. We first show that the coverage-centric active nodes selection problem is NP-complete. We then present a distributed approach based on the concept of a connected dominating set (CDS). We prove that the set of active nodes selected by our approach provides full coverage and connectivity. We also describe an optimal coverage-centric centralized approach based on integer linear programming. We present simulation results obtained using an ns2 implementation of the proposed technique. This paper describes the concept of sensor networks which has been made viable by the convergence of micro-electro-mechanical systems technology, wireless communications and digital electronics. First, the sensing tasks and the potential sensor networks applications are explored, and a review of factors influencing the design of sensor networks is provided. Then, the communication architecture for sensor networks is outlined, and the algorithms and protocols developed for each layer in the literature are explored. Open research issues for the realization of sensor networks are also discussed. We pinpoint a new sensor self-deployment problem, achieving focused coverage around a Point of Interest (POI), and introduce an evaluation metric, coverage radius. We propose two purely localized solution protocols Greedy Advance (GA) and Greedy-Rotation-Greedy (GRG), both of which are resilient to node failures and work regardless of network partition. The two algorithms drive sensors to move along a locally-computed triangle tessellation (TT) to surround the POI. In GA, nodes greedily proceed as close to the POI as they can; in GRG, when their greedy advance is blocked, nodes rotate around the POI to a TT vertex where greedy advance can resume. They both yield a connected network of TT layout with hole-free coverage. Further, GRG ensures a hexagon coverage shape centered at the POI.