MINKOWSKI SUM SELECTION AND FINDING

Abstract

Let P,Q ⊆ ℝ2 be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A ∈ ℝλ×2, [Formula: see text], and b ∈ ℝλ. 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 : ℝ2 → ℝ, and a positive integer k, the MINKOWSKI SUM SELECTION problem is to find the kth largest objective value among all objective values of points in (P ⊕ Q)Ar≥ b . Given P, Q, Ar ≥ b , an objective function f : ℝ2 → 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 [Formula: see text], 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.