Compatible Triangulations of Simple Polygons

TL;DR

This paper investigates algorithms for determining and constructing compatible triangulations of simple polygons, providing efficient methods under different input conditions and vertex correspondence scenarios.

Contribution

It introduces algorithms for deciding and finding compatible triangulations with improved time complexities based on input information.

Findings

Decidable in O(n log n + nr) time if a triangulation of P is given.

Find compatible triangulations in O(M(n)) time when vertex correspondence is provided.

Provides theoretical bounds for compatibility decision and construction.

Abstract

Let $P$ and $Q$ be simple polygons with $n$ vertices each. We wish to compute triangulations of $P$ and $Q$ that are combinatorially equivalent, if they exist. We consider two versions of the problem: if a triangulation of $P$ is given, we can decide in $O(n\log n + nr)$ time if $Q$ has a compatible triangulation, where $r$ is the number of reflex vertices of $Q$. If we are already given the correspondence between vertices of $P$ and $Q$ (but no triangulation), we can find compatible triangulations of $P$ and $Q$ in time $O(M(n))$, where $M(n)$ is the running time for multiplying two $n\times n$ matrices.