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arXiv 1603.02853cs.CG
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计算多边形中点的k-可见区域的时空折衷方法

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

Yeganeh Bahoo, Bahareh Banyassady, Prosenjit Bose, Stephane Durocher, Wolfgang Mulzer

备注：
17 pages, 5 figures; a preliminary version appeared in WALCOM 2017; this revision fixes a mistake in the proof of Theorem 4.4 in the published version, the statement of the theorem remains unchanged
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摘要：

令$P$是具有$n$顶点的简单多边形，并且令$q \in P$是$P$中的点。令$k \in \{0, \dots, n - 1\}$。点$p \in P$是$k$ -当且仅当线段$pq$在大多数$k$时间处穿过$P$的边界时，从$q$可见。$P$中$q$的$k$ -可见区域是$q$中$k$ -可见的所有点的集合。我们研究的问题，计算的$k$可见性区域中的有限工作空间模型，其中输入驻留在随机存取只读存储器的$O(n)$字，每个$Ω(\log{n})$位。该算法能够读写$O(s)$工作空间附加字，其中$s \in \mathbb{N}$是模型参数。输出被写入只写流。给定一个简单的多边形$P$，$n$顶点和一个点$q \in P$，我们提出了一个算法，报告$P$中的$q$的$k$ -可见区域在$O(cn/s+c\log{s} + \min\{\lceil k/s \rceil n,n \log{\log_s{n}}\})$预期的时间使用工作空间的$O(s)$字。这里，$c \in \{1, \dots, n\}$是$q$的$P$的关键顶点的数量，其中$q$的$k$可见区域可以改变。我们推广这一结果的多边形孔和非交叉线段集。

英文摘要：

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 $Ω(\log{n})$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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