Recognizing the Largest Empty Circle and Axis-Parallel Rectangle in a Desired Location

In this paper, we study the query version of the largest empty space recognition problem. Here, a set of n points P is given in a bounded 2D region. The objective is to preprocess P such that given any arbitrary query point q, the largest empty region of some desired shape that contains q but does not contain any point in P can be reported efficiently. We consider only circle and axis-parallel rectangle as the desired shapes, and will refer to these problems as the largest query point enclosing empty circle (QMEC) problem and the largest query point enclosing empty rectangle (QMER) problem, respectively. For the QMEC problem, the preprocessing time and space complexities are O(n log n) and O(n logn) respectively, and the query time is O(log n). For the QMER problem, the preprocessing time and space complexities of our proposed algorithm are both O(n logn), and the query time is O(logn). We also show evidence that improving the above complexities for the QMER problem is unlikely. The techniques employed to solve both problems are quite different from each other.