A Simple Convex Layers Algorithm

Given a set of n points P in the plane, the first layer L1 of P is formed by the points that appear on P ’s convex hull. In general, a point belongs to layer Li, if it lies on the convex hull of the set P \ ⋃ j<i{Lj}. The convex layers problem is to compute the convex layers Li. Existing algorithms for this problem either do not achieve the optimal O (n log n) runtime and linear space, or are overly complex and difficult to apply in practice. We propose a new algorithm that is both optimal and simple. The simplicity is achieved by independently computing four sets of monotone convex chains in O (n log n) time and linear space. These are then merged in O (n log n) time.