A Simple Convex Layers Algorithm

TL;DR

This paper introduces a new, simple, and optimal algorithm for computing convex layers in a set of points, achieving the best theoretical runtime with practical simplicity.

Contribution

The paper presents a novel algorithm that computes convex layers efficiently using monotone convex chains, simplifying implementation while maintaining optimal runtime.

Findings

Achieves $ ext{O}(n ext{log} n)$ runtime and linear space complexity.

Uses four sets of monotone convex chains for computation.

Merges chains efficiently to compute convex layers.

Abstract

Given a set of $n$ points $P$ in the plane, the first layer $L_1$ of $P$ is formed by the points that appear on $P$'s convex hull. In general, a point belongs to layer $L_i$, if it lies on the convex hull of the set $P \setminus \bigcup_{j<i}\{L_j\}$. The \emph{convex layers problem} is to compute the convex layers $L_i$. Existing algorithms for this problem either do not achieve the optimal $\mathcal{O}\left(n\log n\right)$ 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 $\mathcal{O}\left(n\log n\right)$ time and linear space. These are then merged in $\mathcal{O}\left(n\log n\right)$ time.