Localized Geometric Query Problems

Largest empty space recognition is a classical problem in computational geometry, and has applications in several disciplines like database management, operations research, wireless sensor network, VLSI, to name a few. Here the problem is to identify an empty space of a desired shape and of maximum size in a region containing a set of obstacles. Given a set P of n points in R 2 , an empty circle, is a circle that does not contain any member of P . An empty circle is said to be a maximal empty circle (MEC) if it is not fully contained in any other empty circle. Among the MECs, the one having the maximum radius is the largest empty circle. The largest empty circle among a point set P can easily be located by using the Voronoi diagram of P in O(n log n) time [32].

Although a lot of study has been made on the empty space recognition problem, surprisingly, the query version of the problem has not received much attention. The problem of finding the largest empty circle centered on a given query line segment has been considered in [3]. The preprocessing time, space and query time complexities of the algorithm in [3] are O(n 3 log n), O(n 3 ) and O(log n), respectively. In practical applications, one may need to locate the largest empty circle in a desired location. For example, in the VLSI physical design, one may need to place a large circuit component in the vicinity of some already placed components. Such problems arise in mining large data sets as well, where one of the objectives is to search for empty spaces in data sets [23]. In [12], Edmonds et al. formalized the problem of finding large empty spaces in geometric data sets. In particular, they studied the problem of finding large empty rectangles in data sets.

An important problem in this context is the circular separability problem. Two planar sets P 1 and P 2 are circularly separable if there is a circle that encloses P 1 but excludes P 2 . O’Rourke et al. [27] showed that the decision version of the circularly separability of two sets can be solved in O(n) time using linear programming. Furthermore, they show that a smallest separating circle can be found in O(n) time while the computation of the largest separating circle needs O(n log n) time. Detailed study on circular separability problem can be found in [7,8,13,27]. Boissonnat et al. [8] proposed a linear-time algorithm for solving the decision version of circular separability problem where the sets P 1 and P 2 are simple polygons, and the algorithm outputs the smallest separating circle. They also consider the query version of this problem where the objective is to preprocess a convex polygon P such that given a query point q and a query line , report the largest circle inside P that contains q and does not intersect . The preprocessing time and space complexities of their proposed algorithm are both O(n log n), and the query can be answered in O(log n) time. They also showed that a convex polygon P can be preprocessed in O(n) time and space such that for a query set S of k points, the largest circle inside P that encloses S can be computed in O(k log n) time.

In addition to empty circles, empty rectangles have also been studied. We introduced the query version of the maximal empty rectangle in [1]. The problem entails preprocessing a set of n points such that, given a query point q, the largest empty rectangle containing q can be reported efficiently. We gave a solution with query time O(log n) with preprocessing time and space being O(n 2 log n) and O(n 2 ), respectively. Recently, Kaplan et al. [19] improved the preprocessing time and space complexities to O(nα(n) log 4 n) and O(nα(n) log 3 n), respectively, while the query time has increased to O(log 4 n). Here α(n) is the inverse Ackermann function.

In this paper, we study the query versions of the maximum empty circle problem (QMEC). The following variations are considered.

-Given a simple polygon P , preprocess it such that given a query point q, the largest circle inside P that contains the query point q can be identified efficiently. -Given a set of points P , preprocess it such that given a query point q, the largest circle that does not contain any member of P , but contains the query point q can be identified efficiently.

Our results are summarized in Table 1.

We believe that our work will motivate the study of new types of geometric query problems and may lead to a very active research area. The main theme of our work is to achieve subquadratic preprocessing time and space, while ensuring polylogarithmic query times. The results in this paper, improve upon the results in our previous work [1]. Very recently, Kaplan and Sharir [20] provided a solution to the QMEC problem for point sets that only requires O(n log 2 n) time and O(n log n) space for preprocessing. Their query times, however, are O(log 2 n). 1). Our first algorithm uses the concept of planar separators [22] on the underlying planar graph corresponding to the Voronoi

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