An optimal solution for dynamic polar diagram

The Polar Diagram [1] of a set of points (i.e. sites) is a partition of the plane. It is a locus approach for problems processing angles. Also, Dynamic Polar Diagram problem is a problem in which some points can be added to or removed from the point set of the Polar Diagram. Sadeghi et al. [4] introduced this problem and solved it using an algorithm which is optimal in the case that some points are deleted from the set. But this algorithm is not optimal when some new points are inserted into the diagram. In this paper, we present an algorithm to solve the Dynamic Polar Diagram in optimal time in which we insert some new points into the diagram one by one. Our approach applies only on the regions that would be changed and solves the problem for each insertion in O(k + log n) time, in which 1 ≤ k ≤ n is the number of the sites which their regions would be changed.