Visibility polygons and visibility graphs among dynamic polygonal obstacles in the plane

Abstract

We devise an algorithm for maintaining the visibility polygon of any query point in a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update the data structures that store the visibility polygon of the query point. After preprocessing the initial input polygonal domain to build a few data structures, our algorithm takes $O(k(\lg {|VP_{{\mathcal {P}}'}(q)|})+(\lg {n'})^{2}+h)$ (resp. $O(k(\lg n')^2+(\lg |VP_{{\mathcal {P}}'}(q)|)+h)$) worst-case time to update data structures that store visibility polygon $VP_{\mathcal P'}(q)$ of a query point q when any vertex v is inserted to (resp. deleted from) any obstacle of the current polygonal domain ${\mathcal {P}}'$. Here, $n'$ is the number of vertices of ${\mathcal {P}}'$, h is the number of obstacles in ${\mathcal {P}}'$, $VP_{{\mathcal {P}}'}(q)$ is the visibility polygon of q in ${\mathcal {P}}'$ ($|VP_{{\mathcal {P}}'}(q)|$ is the number of vertices of $VP_{{\mathcal {P}}'}(q)$), and k is the number of combinatorial changes in $VP_{{\mathcal {P}}'}(q)$ due to the insertion (resp. deletion) of v. As an application of the above algorithm, we also devise an algorithm for maintaining the visibility graph of a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update data structures that store the visibility graph of the polygonal domain. After preprocessing the initial input polygonal domain, our dynamic algorithm takes $O(k(\lg {n'})^{2}+h)$ (resp. $O(k(\lg {n'})^{2}+h)$) worst-case time to update data structures that store the visibility graph when any vertex v is inserted to (resp. deleted from) any obstacle of the current polygonal domain ${\mathcal {P}}'$. Here, $n'$ is the number of vertices of ${\mathcal {P}}'$, h is the number of obstacles in ${\mathcal {P}}'$, and k is the number of combinatorial changes in the visibility graph of ${\mathcal {P}}'$ due to the insertion (resp. deletion) of v.