In modern applications of geometric algorithms, it is often unrealistic to assume that the input representation fully captures all relevant aspects of the problem, because the input data is often large and dynamic. To address this challenge, we consider the notion of average sensitivity, which is defined as the average earth mover’s distance between the output distributions of the algorithm when run on an input and the same input with one point removed, where the average is over removed points and the distance between two outputs is measured using the symmetric difference size. We start by showing that a number of classical problems from computational geometry, in particular the convex hull, Delaunay triangulation, and Voronoi diagram problems, are "simple" from the viewpoint of average sensitivity by proving tight bounds for the average sensitivity of any algorithm for these problems. Then, we continue by constructing an algorithm with low average sensitivity that computes, for any ε > 0, a set of (1/3+ε)n guards for the art gallery problem. This is the main technical contribution of this work, which combines algorithms from computational geometry with results from the theory of local computation algorithms (LCAs) and property testing.
Ebbens et al. (Thu,) studied this question.