Abstract
The beacon model is a recent paradigm for guiding the trajectory of messages or small robotic agents in complex environments. A beacon is a fixed point with an attraction pull that can move points within a given polygon. Points move greedily towards a beacon: if unobstructed, they move along a straight line to the beacon, and otherwise they slide on the edges of the polygon. The Euclidean distance from a moving point to a beacon is monotonically decreasing. A given beacon attracts a point if the point eventually reaches the beacon.
The problem of attracting all points within a polygon with a set of beacons can be viewed as a variation of the art gallery problem. Unlike most variations, the beacon attraction has the intriguing property of being asymmetric, leading to separate definitions of attraction region and inverse attraction region. The attraction region of a beacon is the set of points that it attracts. It is connected and can be computed in linear time for simple polygons. By contrast, it is known that the inverse attraction region of a point  the set of beacon positions that attract it  could have Omega(n) disjoint connected components.
In this paper, we prove that, in spite of this, the total complexity of the inverse attraction region of a point in a simple polygon is linear, and present a O(n log n) time algorithm to construct it. This improves upon the best previous algorithm which required O(n^3) time and O(n^2) space. Furthermore we prove a matching Omega(n log n) lower bound for this task in the algebraic computation tree model of computation, even if the polygon is monotone.
BibTeX  Entry
@InProceedings{kostitsyna_et_al:LIPIcs:2018:8768,
author = {Irina Kostitsyna and Bahram Kouhestani and Stefan Langerman and David Rappaport},
title = {{An Optimal Algorithm to Compute the Inverse Beacon Attraction Region}},
booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)},
pages = {55:155:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770668},
ISSN = {18688969},
year = {2018},
volume = {99},
editor = {Bettina Speckmann and Csaba D. T{\'o}th},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8768},
URN = {urn:nbn:de:0030drops87686},
doi = {10.4230/LIPIcs.SoCG.2018.55},
annote = {Keywords: beacon attraction, inverse attraction region, algorithm, optimal}
}
Keywords: 

beacon attraction, inverse attraction region, algorithm, optimal 
Collection: 

34th International Symposium on Computational Geometry (SoCG 2018) 
Issue Date: 

2018 
Date of publication: 

08.06.2018 