Graph-Theoretic Solutions to Computational Geometry Problems

TL;DR

This paper reviews how graph-theoretic approaches can efficiently solve various computational geometry problems by constructing auxiliary graphs and leveraging their properties.

Contribution

It provides a survey of multiple computational geometry problems solved through graph-theoretic methods, highlighting the importance of graph properties for algorithm efficiency.

Findings

Graph-theoretic methods unify solutions to diverse geometry problems.

Auxiliary graph construction is key to efficient algorithms.

Special graph properties influence computational performance.

Abstract

Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm depends on the special properties of the graph constructed in this way. We survey the art gallery problem, partition into rectangles, minimum-diameter clustering, rectilinear cartogram construction, mesh stripification, angle optimization in tilings, and metric embedding from this perspective.