A Novel Approximation Algorithm for Minimum Geometric Disk Cover Problem with Hexagon Tessellation

Given a set P ofn points in the Euclidean plane, the minimum geometric disk cover (MGDC) problem is to identify a minimally sized set of congruent disks with prescribed radiusr that cover all the points in P. It is known that the MGDC problem is NP-complete. Solutions to the MGDC problem can be used to solve the relay node placement problems of wireless sensor networks. In this study, we proposed an approximation algorithm for the MGDC problem that identifies covering disks via the regular hexagon tessellation of the plane. We show that the approximation ratio of the proposed algorithm is 5. Furthermore, we show that if the set of points in P is uniformly distributed, then there is 41.7% probability for the proposed algorithm to use less than or equal to 5 times the optimal number of disks, and 58.3% probability of using no more than 4 times the optimal number of disks.