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

TL;DR

This paper introduces an efficient algorithm for computing the k-visibility region of a point within a polygon under limited workspace constraints, balancing time and space complexity.

Contribution

It presents a novel algorithm that computes the k-visibility region in polygons with limited workspace, extending to polygons with holes and segment sets.

Findings

Algorithm operates in expected time depending on workspace size

Works for polygons with holes and non-crossing segments

Balances time complexity with limited memory constraints

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 $\Omega(\log{n})$ 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.