Local Complexity of Polygons

TL;DR

This paper introduces two formal definitions of local complexity in polygons, explores their relationship, and demonstrates bounds and examples illustrating their differences and connections.

Contribution

It provides two formal definitions of local complexity in polygons and analyzes their relationship through bounds and examples.

Findings

Chord visibility width is at most exponential in point visibility width.

Existence of polygons with exponential chord visibility width relative to point visibility width.

Established bounds linking the two complexity measures.

Abstract

Many problems in Discrete and Computational Geometry deal with simple polygons or polygonal regions. Many algorithms and data-structures perform considerably faster, if the underlying polygonal region has low local complexity. One obstacle to make this intuition rigorous, is the lack of a formal definition of local complexity. Here, we give two possible definitions and show how they are related in a combinatorial sense. We say that a polygon $P$ has point visibility width $w=pvw$, if there is no point $q\in P$ that sees more than $w$ reflex vertices. We say that a polygon $P$ has chord visibility width $w=cvw $, if there is no chord $c=\textrm{seg}(a,b)\subset P$ that sees more than w reflex vertices. We show that \[ cvw \leq pvw ^{O( pvw )},\] for any simple polygon. Furthermore, we show that there exists a simple polygon with \[ cvw \geq 2^{\Omega( pvw )}.\]