Dynamic algorithms for visibility polygons in simple polygons

Abstract

We devise the following dynamic algorithms for both maintaining as well as querying for the visibility and weak visibility polygons amid vertex insertions and/or deletions to the simple polygon. * A fully-dynamic algorithm for maintaining the visibility polygon of a fixed point located interior to the simple polygon amid vertex insertions and deletions to the simple polygon. The time complexity to update the visibility polygon of a point $q$ due to the insertion (resp. deletion) of vertex $v$ to (resp. from) the current simple polygon is expressed in terms of the number of combinatorial changes needed to the visibility polygon of $q$ due to the insertion (resp. deletion) of $v$. * An output-sensitive query algorithm to answer the visibility polygon query corresponding to any point $p$ in $\mathbb{R}^2$ amid vertex insertions and deletions to the simple polygon. If $p$ is not exterior to the current simple polygon, then the visibility polygon of $p$ is computed. Otherwise, our algorithm outputs the visibility polygon corresponding to the exterior visibility of $p$. * An incremental algorithm to maintain the weak visibility polygon of a fixed-line segment located interior to the simple polygon amid vertex insertions to the simple polygon. The time complexity to update the weak visibility polygon of a line segment $pq$ due to the insertion of vertex $v$ to the current simple polygon is expressed in terms of the sum of the number of combinatorial updates needed to the geodesic shortest path trees rooted at $p$ and $q$ due to the insertion of $v$. * An output-sensitive algorithm to compute the weak visibility polygon corresponding to any query line segment located interior to the simple polygon amid both the vertex insertions and deletions to the simple polygon. Each of these algorithms requires preprocessing the initial simple polygon.