Visibility Area of a Point in Existence of Circular Obstacles

In this paper, we consider the special class of visibility problems that is when there are a number of circular obstacles in the plane limiting the visibility area of an observer. Here we propose an algorithm to compute the visibility area, in O(nlogn) time. For the query version of the problem, data structure of size O(n) can be built in time O(nlogn) so that the visibility sequence of every point q can be reported in O(logn + |V (q)|) time where |V (q)| is the size of the output visibility sequence. For the streaming version of the problem, where new obstacles appear one after another, the new visibility sequence can be computed in O(log|V(q)|+k) time using the previous visibility sequence, where k is the number of the current visibility sequence elements bounded between two tangent lines of the new circle. When only one new circle is going to appear, a faster algorithm can be proposed which uses O(|V(q)|) space, takes O(|V(q)|) preprocessing time, and computes the new visibility sequence of the point q, in O(log|V(q)|+k’), where k’ is the number of times the new circle appears in the new visibility sequence.