An Improved Algorithm for Reconstructing a Simple Polygon from the Visibility Angles

TL;DR

This paper introduces an improved algorithm for reconstructing simple polygons from visibility angles, reducing the computational complexity from O(n^3 log n) to O(n^2), which is optimal given the input size.

Contribution

The paper presents a new algorithm that significantly improves the efficiency of polygon reconstruction from visibility angles, achieving worst-case optimal performance.

Findings

The new algorithm operates in O(n^2) time.

It is optimal given the input size of O(n^2).

The approach is based on novel geometric observations.

Abstract

In this paper, we study the following problem of reconstructing a simple polygon: Given a cyclically ordered vertex sequence of an unknown simple polygon P of n vertices and, for each vertex v of P, the sequence of angles defined by all the visible vertices of v in P, reconstruct the polygon P (up to similarity). An O(n^3 log n) time algorithm has been proposed for this problem. We present an improved algorithm with running time O(n^2), based on new observations on the geometric structures of the problem. Since the input size (i.e., the total number of input visibility angles) is O(n^2) in the worst case, our algorithm is worst-case optimal.