Vertex Guarding for Dynamic Orthogonal Art Galleries

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 [Formula: see text] is modified to result in another orthogonal polygonal domain [Formula: see text], our algorithm updates the set of vertex guards surveying [Formula: see text] so that the updated guard set surveys [Formula: see text]. Our algorithm modifies the guard placement in [Formula: see text] amortized time, while ensuring the updated orthogonal polygonal domain with [Formula: see text] holes and [Formula: see text] vertices is guarded using at most [Formula: see text] 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 [Formula: see text] worst-case time. Here, [Formula: see text] and [Formula: see text] are the number of vertices of the orthogonal polygon before and after the update, respectively; and, [Formula: see text] is the sum of [Formula: see text] 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 [Formula: see text] and [Formula: see text] are equal.