On Finding Minimum and Maximum Path Length in Grid-Based Wireless Networks

In this paper, we obtain the minimum and maximum hop counts between any pair of cells in the 3D gridbased wireless networks. We start by determining the minimum path length between any two points in a 2D grid coordinate system. We establish that the minimum path length is the maximum difference between the corresponding coordinates of the two points. We then extend the result to derive the minimum and maximum hop counts for the 3D grid-based wireless networks. We establish that the maximum path length is the sum of the differences between the corresponding coordinates of the two cells. Whilst the minimum path length depends on the positions of the two cells; it does not exceed the maximum difference between the corresponding coordinates of the two cells.