Discrete and mixed two-center problems for line segments

Given a set of n non-intersecting line segments $L$ and a set Q of m points in $R^2$; we present algorithms of the discrete two-center problem for (i) covering, (ii) stabbing and (iii) hitting the set $L$ in (i) $O\left(m\right(m+n){\log }^{2}n)$, (ii) $O(m^{2}n\log n)$ and (iii) $O\left(m\right(m+n){\log }^{2}n)$ time respectively, where the two disks are centered at two points of Q and radius of the larger disk is minimized. We also study the mixed two-center problems for (i) covering, (ii) stabbing and (iii) hitting the set $L$, where only one of the disks is centered at a point $q_i\in Q$ and the other disk is centered at any point in $R^2$, and these three problems are solved in (i) $O(mn\log n)$, (ii) $O(m{n}^3\log n)$ and (iii) $O(mn{\log }^{2}n)$ time, respectively. The space complexities for all these algorithms are linear.