Illumination by floodlights

William Steiger, Rutgers University–New Brunswick
Ileana Streinu, Smith College

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We consider three problems about the illumination of planar regions with floodlights of prescribed angles. Problem 1 is the decision problem: given a wedge W of angle φ ≤ π, n points p1 . . . . . pn in the plane and n angles α1 . . . . . αn such that ∑ni=1 αi ≤ θ, decide whether W can be illuminated by floodlights of angles α1 , . . . , αn placed in some order at the points p1 , . . . , pn and then rotated appropriately. We show that this problem is the exponential time and a specialized version of it (when φ = θ) is in NP. The second problem arises when the n points are in the complementary wedge of W and θ ≥ φ. Boss et al. have shown that a solution exists and gave an O(n log n) algorithm to place the floodlights. Here we give a matching lower bound. Problem 3 involves the illumination of the whole plane. The algorithm of Bose et al. uses an O(n log n) tripartitioning algorithm to reduce problem 3 to problem 2. We give a linear time tripartitioning algorithm of independent interest. © 1998 Elsevier Science B.V.