Problems and Results

Let q1, q2, …, qm be a set of internal points of a polygon P with holes with a total of n vertices. Consider the problem of computing visibility polygons of points q1, q2, …, qm in the polygon P. Since the visibility polygon of P from each point qi can be computed in O(n log n) time by the algorithm of Asano [27] (see Section 2.3), the problem can be solved in O(mn log n) time. Suppose m is quite large compared to n. In that case, it may be a good idea to construct data structures by processing P once so that the visibility polygon from each query point qi can be computed in less than O(n log n) time with the help of these data structures. In fact, it has been shown by Asano et al. [28] that after spending O(n2) time in preprocessing of P, the visibility polygon from each query point qi can be computed in O(n) time. Thus the overall time complexity for solving this problem is reduced from O(mn log n) to O(mn). Such problems, that require a large number of computations of similar type on the same polygonal domain, are known as query problems in computational geometry [291].