Computing Largest Circles Separating Two Sets of Segments

Abstract

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 largest 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 largest circles separating two given sets of line segments when line segments are allow ed to meet only at their endpoints. This settles an open problem from a previous paper\cite{bcdy-csp-95}. In the general case, when line 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 $\alpha(n)$ represents the extremely slowly growing inverse of Ackermann function.