# A New Optimal Algorithm for Computing the Visibility Area of a simple Polygon from a Viewpoint

## TL;DR

This paper introduces an optimal-time algorithm for computing the visibility area of a simple polygon from an internal viewpoint, significantly reducing workspace to sub-linear variables while maintaining linear time complexity.

## Contribution

It presents a novel algorithm that computes visibility areas in linear time using minimal workspace proportional to the number of critical vertices.

## Key findings

- Achieves linear time complexity for visibility computation.
- Reduces workspace to O(c / log n) variables, where c is the number of critical vertices.
- Maintains optimal performance with minimal memory usage.

## Abstract

Given a simple polygon $ \mathcal {P} $ of $ n $ vertices in the Plane. We study the problem of computing the visibility area from a given viewpoint $ q $ inside $ \mathcal {P} $ where only sub-linear variables are allowed for working space. Without any memory-constrained, this problem was previously solved in $ O(n) $-time and $ O(n) $-variables space. In a newer research, the visibility area of a point be computed in $ O(n) $-time, using $ O(\sqrt{n}) $ variables for working space. In this paper, we present an optimal-time algorithm, using $ O(c/\log n) $ variables space for computing visibility area, where $ c<n $ is the number of critical vertices. We keep the algorithm in the linear-time and reduce space as much as possible.