Before revision:  We also study the parameterized space complexity of estimating \alpha and \beta, where the parameter is the ratio of maximum to minimum interval length . For unit-radius disks, we obtain a \frac{8\sqrt{3}{\pi}} -approximation to \alpha and \beta in\textrm{poly}%DIFAUXCMD  \frac{\log(n){\epsilon}} space, which is closely related to the hexagonal circle packing constant .
After revision:  We also study the parameterized space complexity of estimating \alpha and \beta, where the parameter is the ratio of maximum to minimum interval length  \frac{8\sqrt{3}{\pi}} -approximation to \alpha and \beta in\textrm{poly}%DIFAUXCMD  \frac{\log(n){\epsilon}} space, which is closely related to the hexagonal circle packing constant .