Computing largest empty circles with location constraints

LetQ = {q1, q2,..., qn} be a set ofn points on the plane. The largest empty circle (LEG) problem consists in finding the largest circleC with center in the convex hull ofQ such that no pointqi εQ lies in the interior ofC. Shamos recently outlined anO(n logn) 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 anO(n logn) algorithm. The new approach has the advantage that it can also solve more general problems. In particular, it is shown that if the center ofC is constrained to lie in an arbitrary convexn-gon, an0(n logn) algorithm can still be obtained. Finally, an0(n logn +k logn) algorithm is given for solving this problem when the center ofC is constrained to lie in an arbitrary simplen-gonP. wherek denotes the number of intersections occurring between edges ofP and edges of the Voronoi diagram ofQ andk ⩽O(n2).