A time–space trade-off for computing the k-visibility region of a point in a polygon

Abstract

Let P be a simple polygon with n vertices, and let q ∈ P be a point in P . Let k ∈ {0, . . . , n − 1}. A point p ∈ P is k-visible from q if and only if the line segment pq crosses the boundary of P at most k times. The k-visibility region of q in P is the set of all points that are k-visible from q. We study the problem of computing the k-visibility region in the limited workspace model, where the input resides in a random-access read-only memory of O(n) words, each with Ω(log n) bits. The algorithm can read and write O(s) additional words of workspace, where s ∈ N is a parameter of the model. The output is written to a write-only stream.Given a simple polygon P with n vertices and a point q ∈ P , we present an algorithm that reports the k-visibility region of q in P in O(cn/s + c log s + min{ k/s n, n log log s n}) expected time using O(s) words of workspace. Here, c ∈ {1, . . . , n} is the number of critical vertices of P for q where the k-visibility region of q may change. We generalize this result for polygons with holes and for sets of non-crossing line segments.

Generation of simple polygons from ordered points using an iterative insertion algorithm

“…Given a polygon and a point outside of the polygon, if a node of the polygon is visible from the point, then a line formed between the node and point does not cross any other polygon line. If and only if a polygon line is visible for the point, the two endpoints of this line are visible for the point [10][11][12].…”