Largest empty circle centered on a query line

The Largest Empty Circle problem seeks the largest circle centered within the convex hull of a set P of n   points in R2R2 and devoid of points from P. In this paper, we introduce a query version of this well-studied problem. In our query version, we are required to preprocess P so that when given a query line Q, we can quickly compute the largest empty circle centered at some point on Q and within the convex hull of P.We present solutions for two special cases and the general case; all our queries run in O(logn) time. We restrict the query line to be horizontal in the first special case, which we preprocess in O(nα(n)logn) time and space, where α(n)α(n) is the slow growing inverse of Ackermann's function. When the query line is restricted to pass through a fixed point, the second special case, our preprocessing takes O(nα(n)O(α(n))logn) time and space. We use insights from the two special cases to solve the general version of the problem with preprocessing time and space in O(n3logn) and O(n3)O(n3) respectively.