Finding Largest Empty Circles with Location Constraints

Let S be a set of n points in the plane and let CH(S) represent the convex hull of S. The Largest Empty Circle (LEC) problem is the problem of finding the largest circle centered with CH(S) such that no point of S lies within the circle. Shamos and Hoey (SH75) outlined an algorithm for solving this problem in time O(n log n) by first computing the Voronoi diagram, V(S), in time O(n log n), then using V(S) and CH(S) to compute the largest empty circle in time O(n). In a recent paper [Tou83], Toussaint pointed out some problems with the algorithm as outlined by Shamos and presented an algorithm which, given V(S) and CH(S), solves the LEC problem in time O(n log n). In this note we show that Shamos\u27 original claim was correct: given V(S) and CH(S), the LEC problem can be solved in time O(n). More generally, given V(S) and a convex k-gon P, the LEC centered within P can be found in time O(k+n). We also improve on an algorithm given by Toussaint for computing the LEC when the center is constrained to lie within an arbitrary simple polygon. Given a set S of n points and an arbitrary simple k-gon P, the largest empty circle centered within P can be found in time O(kn + n log n). This becomes O(kn) if the Voronoi diagram of S is already given