The paper addresses the fundamental issue of coverage in wireless sensor networks, focusing on determining whether every point in the service area is covered by at least \( k \) sensors, where \( k \) is a given parameter. The authors propose polynomial-time algorithms that can be easily translated into distributed protocols, generalizing earlier results where \( k = 1 \). The algorithms are designed to handle both unit and non-unit disk sensing ranges. Key contributions include: 1. **Problem Formulation**: The coverage problem is formulated as a decision problem to determine if the entire area is \( k \)-covered. 2. **Algorithms**: Efficient algorithms are presented to check the perimeter coverage of sensors, which can be used to determine the overall coverage level of the network. 3. **Complexity Analysis**: The complexity of the algorithms is analyzed, showing that they run in \( O(nd \log d) \) time, where \( d \) is the maximum number of sensors intersecting a given sensor's range. 4. **Applications**: The solutions have applications in identifying insufficiently covered areas, enhancing fault tolerance, and conserving energy in redundant sensors. 5. **Extensions**: The paper discusses extensions to irregular sensing regions and scenarios with hot spots. 6. **Software Tool**: A software tool is developed to implement the proposed algorithms and visualize the coverage levels. The authors conclude by highlighting the practical implications of their work and the availability of the software tool for further research and application.The paper addresses the fundamental issue of coverage in wireless sensor networks, focusing on determining whether every point in the service area is covered by at least \( k \) sensors, where \( k \) is a given parameter. The authors propose polynomial-time algorithms that can be easily translated into distributed protocols, generalizing earlier results where \( k = 1 \). The algorithms are designed to handle both unit and non-unit disk sensing ranges. Key contributions include: 1. **Problem Formulation**: The coverage problem is formulated as a decision problem to determine if the entire area is \( k \)-covered. 2. **Algorithms**: Efficient algorithms are presented to check the perimeter coverage of sensors, which can be used to determine the overall coverage level of the network. 3. **Complexity Analysis**: The complexity of the algorithms is analyzed, showing that they run in \( O(nd \log d) \) time, where \( d \) is the maximum number of sensors intersecting a given sensor's range. 4. **Applications**: The solutions have applications in identifying insufficiently covered areas, enhancing fault tolerance, and conserving energy in redundant sensors. 5. **Extensions**: The paper discusses extensions to irregular sensing regions and scenarios with hot spots. 6. **Software Tool**: A software tool is developed to implement the proposed algorithms and visualize the coverage levels. The authors conclude by highlighting the practical implications of their work and the availability of the software tool for further research and application.