The problem of coverage in two-dimensional (2D) wireless sensor networks is challenging and is still open. Precisely, determining the minimum sensor density (i.e, minimum number of sensors per unit area) that is required to cover a 2D field of interest (FoI), where every point in the field is covered by at least one sensor, is still under investigation. The problem of 2D k-coverage, which requires that every point in a 2D FoI be covered by at least k sensors, where $k\ge 1$, is more challenging. In this paper, we attempt to address the 2D connected k-coverage problem, where a 2D FoI is k-covered, while the underlying set of sensors k-covering the field forms a connected network. We propose to solve this problem using an approach based on slicing 2D FoI into convex regular hexagons. Our goal is to achieve k-coverage of a 2D FoI with a minimum number of sensors in order to maximize the network lifetime. First, we compute the minimum sensor density for 2D k-coverage using the regular convex hexagon, which is a 2D paver (i.e., covers a 2D field without gaps or overlaps). Indeed, we found that the regular convex hexagon best assimilates the sensors’ sensing disk with respect to our proposed metric, sensing range usage rate. Second, we derive the ratio of the communication range to the sensing range of the sensors to ensure connected k-coverage. Third, we propose an energy-efficient connected k-coverage protocol based on hexagonal slicing and area stretching. To this end, we formulate a multi-objective optimization problem, which computes an optimum solution to the 2D k-coverage problem that meets two requirements: Maximizing the size of the k-covered area, $C_k$, so as to minimize the sensor density to k-cover a 2D FoI (Requirement 1) and maximizing the area of the sensor locality, $L_k$, i.e., the region where at least k sensors are located to k-cover $C_k$, so as to minimize the interference between sensors (Requirement 2). Fourth, we show various simulation results to substantiate our theoretical analysis. We found a close-to-perfect match between our theoretical and simulation results.

二维（2D）无线传感器网络中的覆盖问题极具挑战性，并且仍然存在。精确地确定覆盖2D感兴趣区域（FoI）所需的最小传感器密度（即，每单位面积的最小传感器数量）的问题仍在研究中，在该2D感兴趣区域中，场中的每个点都被至少一个传感器覆盖。 。2D k覆盖问题，要求2D FoI中的每个点至少要被k个传感器覆盖，其中$ķ\ge \mathrm{1个}$，更具挑战性。在本文中，我们尝试解决2D连通的k覆盖问题，其中二维FoI被k覆盖，而覆盖该场的基础传感器k的基础集形成了一个连通的网络。我们建议使用一种基于将2D FoI切片为凸正六边形的方法来解决此问题。我们的目标是用最少的传感器实现2D FoI的k覆盖率，以最大化网络寿命。首先，我们使用正则凸形六边形（即2D摊铺机，即2D摊铺机）计算2D k覆盖的最小传感器密度（即覆盖2D字段，没有间隙或重叠）。确实，我们发现正则凸六边形相对于我们提出的度量标准（感应范围使用率）可以最佳地同化传感器的感应盘。其次，我们得出通信范围与传感器的感应范围之比，以确保连接的k覆盖率。第三，我们提出了一种基于六边形切片和面积拉伸的高效节能的k-覆盖协议。为此，我们制定了一个多目标优化问题，该问题计算了满足以下两个要求的2D k覆盖问题的最佳解决方案：最大化k覆盖区域的大小，$C_{ķ}$，以便最小化传感器密度以k覆盖2D FoI（要求1）并最大化传感器局部区域，${\mathrm{大号}}_{ķ}$，即至少k个传感器位于k覆盖的区域$C_{ķ}$，以最大程度地减少传感器之间的干扰（要求2）。第四，我们展示了各种模拟结果，以证实我们的理论分析。我们在理论和仿真结果之间找到了接近完美的匹配。