GUARDING A POLYGON FROM TWO NEARLY-OPPOSITE DIRECTIONS

초록

In this paper, we study a variant of the classical art-gallery problem, in which we require that each point of the art gallery is observed by two guards from nearly-opposite directions, ideally from front and back. We call a polygon P alpha-guarded by a set G of guards if every point p in P is visible from two guards g(1),g(2) in G satisfying alpha <= angle g(1)pg(2) <= pi. We show that any simple n-gon is 2/3 pi-guarded by its n vertex guards, and there are simple n-gons that require atleast n guards. Any simple n-gon can be (pi - delta)-guarded by (4 pi/delta + 1)n guards, and there are simple n-gons that require at least pi/4 delta(n - 1) guards in any guard placement. We prove that the region of a simple n-gon that is alpha-guarded by k guards can have Theta(nk + k(4)) complexity in the worst case. Given simple n-gon P and k guard positions, we can decide for a fixed 0 < alpha < pi whether P is alpha-guarded in O ((nk(2) + k(4)) log(2) nk) time and find the largest alpha < pi for which P is alpha-guarded in O((nk(2) + k(4)) log(6) nk) time.

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