Vertex Guarding for Dynamic Orthogonal Art Galleries

TL;DR

This paper presents an efficient algorithm for dynamically updating vertex guards in orthogonal polygons as the shape changes, ensuring coverage with minimal guards and optimal update time.

Contribution

It introduces a novel dynamic guard updating algorithm for orthogonal polygons that operates efficiently during modifications, with proven bounds on guard number and update complexity.

Findings

Guard set updates in O(k log(n+n')) time

Guarantees at most (n+2h)/4 guards for any orthogonal polygon

Efficient handling of hole-free polygon updates

Abstract

We devise an algorithm for surveying a dynamic orthogonal polygonal domain by placing one guard at each vertex in a subset of its vertices, i.e., whenever an orthogonal polygonal domain {\cal P'} is modified to result in another orthogonal polygonal domain {\cal P}, our algorithm updates the set of vertex guards surveying {\cal P'} so that the updated guard set surveys {\cal P}. Our algorithm modifies the guard placement in O(k \lg{(n+n')}) amortized time while ensuring the updated orthogonal polygonal domain with h holes and n vertices is guarded using at most \lfloor (n+2h)/4 \rfloor vertex guards. For the special case of the initial orthogonal polygon being hole-free and each update resulting in a hole-free orthogonal polygon, our guard update algorithm takes O(k\lg{(n+n')}) worst-case time. Here, n' and n are the number of vertices of the orthogonal polygon before and after the update, respectively; and, k is the sum of |n - n'| and the number of updates to a few structures maintained by our algorithm. Further, by giving a construction, we show it suffices for the algorithm to consider only the case in which the parity of the number of reflex vertices of both {\cal P'} and {\cal P} are equal.