TY - GEN

T1 - An optimal algorithm to compute the inverse beacon attraction region

AU - Kostitsyna, I.

AU - Kouhestani, Bahram

AU - Langerman, Stefan

AU - Rappaport, David

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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 Ω(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(nlogn) time algorithm to construct it. This improves upon the best previous algorithm which required O(n) time and O(n) space. Furthermore we prove a matching Ω(nlogn) lower bound for this task in the algebraic computation tree model of computation, even if the polygon is monotone.

AB - 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 Ω(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(nlogn) time algorithm to construct it. This improves upon the best previous algorithm which required O(n) time and O(n) space. Furthermore we prove a matching Ω(nlogn) lower bound for this task in the algebraic computation tree model of computation, even if the polygon is monotone.

KW - Algorithm

KW - Beacon attraction

KW - Inverse attraction region

KW - Optimal

UR - http://www.scopus.com/inward/record.url?scp=85048959517&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2018.55

DO - 10.4230/LIPIcs.SoCG.2018.55

M3 - Conference contribution

T3 - Leibniz International Proceedings in Informatics (LIPIcs)

SP - 55:1-55:14

BT - 34th International Symposium on Computational Geometry, SoCG 2018

A2 - Toth, Csaba D.

A2 - Speckmann, Bettina

PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik

CY - Dagstuhl

T2 - 34th International Symposium on Computational Geometry (SoCG 2018)

Y2 - 11 June 2018 through 14 June 2018

ER -