Planar lower envelope of monotone polygonal chains

TL;DR

This paper introduces a simple linear search algorithm for constructing the lower envelope of monotone polygonal chains in 2D, achieving optimal output-sensitive performance with improved simplicity and speed over existing methods.

Contribution

It presents a novel, straightforward linear search algorithm for lower envelope construction that is easier to implement and faster than previous output-sensitive algorithms.

Findings

Runs in O(n+mk) time for monotone chains

Achieves optimal O(n log k) time for arbitrary segments

Simpler implementation without complex data structures

Abstract

A simple linear search algorithm running in $O(n+mk)$ time is proposed for constructing the lower envelope of $k$ vertices from $m$ monotone polygonal chains in 2D with $n$ vertices in total. This can be applied to output-sensitive construction of lower envelopes for arbitrary line segments in optimal $O(n\log k)$ time, where $k$ is the output size. Compared to existing output-sensitive algorithms for lower envelopes, this is simpler to implement, does not require complex data structures, and is a constant factor faster.