Weak Visibility Queries of Line Segments in Simple Polygons

TL;DR

This paper introduces efficient algorithms and data structures for computing the weak visibility polygon from any query line segment within a simple polygon, optimizing for different preprocessing time and space constraints.

Contribution

It presents new data structures with improved query times for weak visibility polygons, balancing preprocessing complexity and query efficiency.

Findings

O(n)-time data structure with O(n) space for O(k log n) query time

O(n^3)-time data structure with O(n^3) space for O(k + log n) query time

Enhanced algorithms for weak visibility queries in simple polygons

Abstract

Given a simple polygon P in the plane, we present new algorithms and data structures for computing the weak visibility polygon from any query line segment in P. We build a data structure in O(n) time and O(n) space that can compute the visibility polygon for any query line segment s in O(k log n) time, where k is the size of the visibility polygon of s and n is the number of vertices of P. Alternatively, we build a data structure in O(n^3) time and O(n^3) space that can compute the visibility polygon for any query line segment in O(k + log n) time.