Near Optimal Line Segment Weak Visibility Queries in Simple Polygons

TL;DR

This paper introduces an efficient algorithm for computing the weak visibility polygon of any query segment inside a simple polygon, optimizing preprocessing time and space while maintaining fast query response times.

Contribution

It presents a new preprocessing algorithm that reduces time and space complexity for weak visibility polygon queries in simple polygons.

Findings

Preprocessing takes O(n^2 log n) time and O(n^2) space.

Query time is O(log^2 n + |WVP(pq)|).

Improves existing results by optimizing preprocessing at the expense of query time.

Abstract

This paper considers the problem of computing the weak visibility polygon (WVP) of any query line segment pq (or WVP(pq)) inside a given simple polygon P. We present an algorithm that preprocesses P and creates a data structure from which WVP(pq) is efficiently reported in an output sensitive manner. Our algorithm needs O(n^2 log n) time and O(n^2) space in the preprocessing phase to report WVP(pq) of any query line segment pq in time O(log^2 n + |WVP(pq)|). We improve the preprocessing time and space of current results for this problem at the expense of more query time.