Linear Time Approximation Algorithms for the Relay Node Placement Problem in Wireless Sensor Networks with Hexagon Tessellation

Algorithm 2

A simple linear time approximation algorithm for the MGDC problem.

Algorithm Hexagon-II

Input: A set of points, and a two-dimensional region .

Output: A set of disks with radius that covers .

Step  1:

  Virtually tessellate the region with hexagons of length . Calculate and store the coordinate of the center and

  the corresponding index of each hexagon in the tessellation. (Each index contains two ranks (_rank, _rank)

  to -axis and -axis, respectively, as shown in Figure 5.)

Step  2:

  For each point with Cartesian coordinate (, ) in do

    (1) ; // find the indices of the basic rectangle that contains

      If _rank is even

      else

    (2) Find the hexagon containing by comparing the distances from to the centers of hexagons with indices

    (_rank, _rank), (_rank + 1, _rank), and (_rank + 1, _rank + 1) if _rank is even or (_rank + 1, _rank − 1)

    if _rank is odd. Ties are broken in favor of the marked hexagon, otherwise broken arbitrary.

    (3) Mark the hexagon chosen in (2).

Step  3:

  Output the circumscribed circles of the chosen hexagons.