A circle C separates two planar sets if it encloses one of the sets and its
open interior disk does not meet the other set. A separating circle is a l
argest one if it cannot be locally increased while still separating the two
given sets. An Theta(n log n) optimal algorithm is proposed to find all la
rgest circles separating two given sets of line segments when line segments
are allowed to meet only at their endpoints. in the general case, when lin
e segments may intersect Omega(n(2)) times, our algorithm can be adapted to
work in O(n alpha(n) log n) time and O(n alpha(n)) space, where alpha(n) r
epresents the extremely slowly growing inverse of the Ackermann function.