Before revision:  Using quadtrees, we provide an alternative algorithm that runs in near linear time , although this second algorithm has a logarithmic dependence on either\big(\big(\big)\big) the ratio of the fastest speed to the slowest speed of disks or the spread of disk centers (the ratio of the maximum to the minimum distance between them).
After revision:  Using quadtrees, we provide an alternative algorithm that runs in near linear time , although this second algorithm has a logarithmic dependence on either\big(\big(\big)\big) the ratio of the fastest and the slowest growth rate and \Phi is the ratio of the maximum to the minimum distance between them).