A Sweep-Line Algorithm for the Inclusion Hierarchy among Circles

Abstract

Suppose that there are a number of circles in a plane and some of them may contain several smaller circles. In this case, it is necessary to find the inclusion hierarchy among circles for the various applications such as the simulation of emulsion and diameter estimation for wire bundles. In this paper, we present a plane-sweep algorithm that can identify the inclusion hierarchy among the circles in $O(n\log n)$ time in the worst-case. Also, the proposed algorithm uses the sweep-line method and a red-black tree for the efficient computation.