Minkowski Sum Selection and Finding

Abstract

Let P, Q ⊆ R 2 be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A ∈ R λ×2 , r = [ x y ], and b ∈ R λ . Define the constrained Minkowski sum (P ⊕ Q) Ar≥b as the multiset {(p + q)|p ∈ P, q ∈ Q, A(p + q) ≥ b}. Given P , Q, Ar ≥ b, an objective function f : R 2 → R, and a positive integer k, the Minkowski Sum Selection problem is to find the k th largest objective value among all objective values of points in (P ⊕Q) Ar≥b . Given P , Q, Ar ≥ b, an objective function f : R 2 → R, and a real number δ, the Minkowski Sum Finding problem is to find a point (x * , y * ) in (P ⊕ Q) Ar≥b such that |f (x * , y * ) − δ| is minimized. For the Minkowski Sum Selection problem with linear objective functions, we obtain the following results: (1) optimal O(n log n) time algorithms for λ = 1; (2) O(n log 2 n) time deterministic algorithms and expected O(n log n) time randomized algorithms for any fixed λ > 1. For the Minkowski Sum Finding problem with linear objective functions or objective functions of the form f (x, y) = by ax , we construct optimal O(n log n) time algorithms for any fixed λ ≥ 1. As a byproduct, we obtain improved algorithms for the Length-Constrained Sum Selection problem and the Density Finding problem.