This seems like a fairly complicated problem...
We could temporarily define the following variables:
refers to the number of lines across the grid
refers to the TOTAL amount of positions in the grid
refers to the amount of circles or sensors in the grid
refers to the radius of a sensor
So going through the grid and covering all positions requires
covering checks, at the mininum. In the worst case
, this could mean
compares for each "covering check".
However, going through the grid also requires
sensor placements which each require
refers to the geometric average of all sensor radii. This gives us approximately
So far, we're still left with two running times and an initial estimate of:
I believe that we still should know more about the algorithm. Is this definitely the algorithm that you want to stick with? I ask this because we should know exactly how the sensor placement works to get a very exact estimate of the running time. The sensor placement could really change this.