Computing largest empty circles with location constraints

Let Q = {q₁, q₂,..., q_n} be a set of n points on the plane. The largest empty circle (LEG) problem consists in finding the largest circle C with center in the convex hull of Q such that no point q_i εQ lies in the interior of C. Shamos recently outlined an O(n log n) algorithm for solving this problem.⁽⁹⁾ In this paper it is shown that this algorithm does not always work correctly. A different approach is proposed here and shown to also result in an O(n log n) algorithm. The new approach has the advantage that it can also solve more general problems. In particular, it is shown that if the center of C is constrained to lie in an arbitrary convex n-gon, an 0(n log n) algorithm can still be obtained. Finally, an 0(n log n +k log n) algorithm is given for solving this problem when the center of C is constrained to lie in an arbitrary simple n-gon P. where k denotes the number of intersections occurring between edges of P and edges of the Voronoi diagram of Q and k ≤O(n²).