Compatible Triangulations of Simple Polygons
🤖 AI Summary
This study addresses the existence and construction of compatible triangulations between two simple polygons with the same number of vertices and a given vertex correspondence. The work proposes efficient algorithms for two scenarios: when a triangulation of one polygon is provided, it determines in $O(n \log n + nr)$ time whether a compatible triangulation exists for the other polygon; when only the vertex correspondence is given, it constructs a compatible triangulation—if one exists—in $O(M(n))$ time, where $M(n)$ denotes the complexity of matrix multiplication. By integrating techniques from computational geometry, including triangulation methods, reflex vertex analysis, and fast matrix multiplication, the approach achieves near-optimal theoretical complexity, substantially improving the efficiency of both decision and construction procedures for compatible triangulations.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
compatible triangulations
simple polygons
combinatorial equivalence
triangulation
Innovation
Methods, ideas, or system contributions that make the work stand out.
compatible triangulation
simple polygons
combinatorial equivalence
matrix multiplication
reflex vertices
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