Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

TL;DR

This paper presents a linear-time algorithm for drawing graphs of line or pseudoline arrangements on small grids, advancing understanding of planarity and related geometric problems.

Contribution

It introduces a new efficient algorithm for planar graph drawing within small grids, linking graph drawing complexity to the k-set problem in discrete geometry.

Findings

Algorithm runs in linear time

Drawings fit within O(n^{7/6}) grid area

No known input requires larger area, linking to k-set problem

Abstract

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.