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 $q\in P$ be a point in P. Let $k\in \{0,\dots ,n-1\}$. A point $p\in P$ 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 $\mathrm{\Omega }(\log n)$ bits. The algorithm can read and write $O(s)$ additional words of workspace, where $s\in \mathbb{N}$ 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 $q\in P$, we present an algorithm that reports the k-visibility region of q in P in $O(cn/s+c\log s+\min \{\lceil k/s\rceil n,n\log {\log }_{s}n\})$ expected time using $O(s)$ words of workspace. Here, $c\in \{1,\dots ,n\}$ 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.