Oblivious Set-maxima for Intersection of Convex Polygons

Abstract: In this paper we revisit the well known set-maxima problem in the oblivious setting. Let $X={x_1,\ldots, x_n}$ be a set of $n$ elements with an underlying total order. Let $\mathcal{S}={S_1,\ldots,S_m}$ be a collection of $m$ distinct subsets of $X$. The set-maxima problem asks to determine the maxima of all the sets in the collection. In the comparison tree model we are interested in determining the number of comparisons necessary and sufficient to solve the problem. We present an oblivious algorithm based on the lattice structure of the input set system. Our algorithm is simple and yet for many set systems gives a non-trivial improvement over known deterministic algorithms. We apply our algorithm to a special $\cal S$ which is determined by an intersection structure of convex polygons and show that $O(n)$ comparisons suffice.