A Time-Space Trade-off for Computing the k-Visibility Region of a Point in a Polygon

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 $O(cn\mathrm{/}s+c\log s+\min \{\lceil k\mathrm{/}s\rceil n,n\log {\log }_{s}n\})$ 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.