Visibility Polygon Queries Among Dynamic Polygonal Obstacles in Plane

Abstract

Given a polygonal domain, we devise a fully dynamic algorithm for maintaining the visibility polygon of any query point, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update the visibility polygon of any query point. After preprocessing the initial input polygonal domain to build a few data structures, our dynamic 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 the 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 in $\mathcal P'$, 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.

R. Inkulu’s research is supported in part by SERB MATRICS grant MTR/2017/000474.