Minkowski Sum Selection and Finding

Abstract

For the \textsc{Minkowski Sum Selection} problem with linear objective functions, we obtain the following results: (1) optimal $O(n\log n)$ time algorithms for $\lambda =1$; (2) $O(n{\log }^{2}n)$ time deterministic algorithms and expected $O(n\log n)$ time randomized algorithms for any fixed $\lambda >1$. For the \textsc{Minkowski Sum Finding} problem with linear objective functions or objective functions of the form $f(x,y)=\frac{by}{ax}$, we construct optimal $O(n\log n)$ time algorithms for any fixed $\lambda \ge 1$.

23 pages, 10 figures, accepted by ISAAC 2008