Incremental Algorithms to Update Visibility Polygons

Abstract

We consider the problem of updating the visibility polygon of a point located within the given simple polygon as that polygon is modified with the incremental addition of new vertices to it. In particular, we propose the following two semi-dynamic algorithms:

• Given a simple polygon P defined with n vertices and a point $p \in P$, our preprocessing algorithm computes the visibility polygon of p in P and constructs relevant data structures in O(n) time; for every vertex v added to the current simple polygon, our visibility polygon updation algorithm takes $O((k+1)\lg {n})$ time in the worst-case to update the visibility polygon of p in the new simple polygon resulted from adding v. Here, k is the change in combinatorial complexity of visibility polygon of p due to the addition of new vertex v.

• Given a simple polygon P defined with n vertices and an edge pq of P, our preprocessing algorithm computes the weak visibility polygon of pq in P and constructs relevant data structures in O(n) time; for every vertex v added to the current simple polygon, our weak visibility updation algorithm takes $O((k+1)\lg {n})$ time in the worst-case to update the weak visibility polygon of pq in the new simple polygon resulted from adding v. Here, k is the change in combinatorial complexity of shortest path tree rooted at p added to the change in combinatorial complexity of shortest path tree rooted at q, wherein both these changes are due to the addition of new vertex v.

R. Inkulu’s research is supported in part by NBHM grant 248(17)2014-R&D-II/1049.