Construction of generalized Čech complex for coverage hole detection in 3-dimensional space

Homology theory has been introduced to address the coverage hole detection problem in the networks covering 2-dimensional regions, whereas the coverage of 3-dimensional space has received limited attention. In this paper, a construction method of generalized Čech complex is proposed for coverage hole detection in 3-dimensional space. The simplices whose dimension are equal or lower than three are built successively to obtain the complex, and then the homology groups of the complex are calculated to see the existence of coverage holes. In particular, the construction criteria of 3-simplices is presented detailedly, which include selection of candidate groups and check of four different situations. The effectiveness of the proposed algorithm is verified on various cases, and the simulation results confirm an accurate detection of coverage holes. Additionally, the complexity of the algorithm is estimated to be in polynomial time, indicating its computational efficiency.