Optimal Line Bipartitions of Point Sets

Let S be a set of n points in the plane. We study the following problem: Partition S by a line into two subsets Sa and Sb such that max {f(Sa), f(Sb)} is minimal, where f is any monotone function defined over 2S. We first present a solution to the case where the points in S are the vertices of some convex polygon and apply it to some common cases — f(S′) is the perimeter, area, or width of the convex hull of S′ ⊆ S — to obtain linear solutions (or O(n log n) solutions if the convex hull of S is not given) to the corresponding problems. This solution is based on an efficient procedure for finding a minimal entry in matrices of some special type, which we believe is of independent interest. For the general case we present a linear space solution which is in some sense output sensitive. It yields solutions to the perimeter and area cases that are never slower and often faster than the best previous solutions.