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 R2; we present algorithms of the discrete two-center problem for (i) covering, (ii) stabbing and (iii) hitting the set L in (i) O(m(m+n)log2⁡n), (ii) O(m2nlog⁡n) and (iii) O(m(m+n)log2⁡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 qi∈Q and the other disk is centered at any point in R2, and these three problems are solved in (i) O(mnlog⁡n), (ii) O(mn3log⁡n) and (iii) O(mnlog2⁡n) time, respectively. The space complexities for all these algorithms are linear. •Covering, hitting and stabbing a set of line segments by two circular disks.•The discrete- and mixed- two-center problem for the line segments are computed.•The line segments are arbitrarily oriented and non-intersecting.•Voronoi diagram of the point set is used to solve the problem.