80 research outputs found
An Optimal Algorithm to Compute the Inverse Beacon Attraction Region
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
Chasing Puppies: Mobile Beacon Routing on Closed Curves
We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired
by his and others' work on beacon-based routing. Consider a human and a puppy
on a simple closed curve in the plane. The human can walk along the curve at
bounded speed and change direction as desired. The puppy runs with unbounded
speed along the curve as long as the Euclidean straight-line distance to the
human is decreasing, so that it is always at a point on the curve where the
distance is locally minimal. Assuming that the curve is smooth (with some mild
genericity constraints) or a simple polygon, we prove that the human can always
catch the puppy in finite time.Comment: Full version of a SOCG 2021 paper, 28 pages, 27 figure
Metric Dimension for Gabriel Unit Disk Graphs is NP-Complete
We show that finding a minimal number of landmark nodes for a unique virtual
addressing by hop-distances in wireless ad-hoc sensor networks is NP-complete
even if the networks are unit disk graphs that contain only Gabriel edges. This
problem is equivalent to Metric Dimension for Gabriel unit disk graphs. The
Gabriel edges of a unit disc graph induce a planar O(\sqrt{n}) distance and an
optimal energy spanner. This is one of the most interesting restrictions of
Metric Dimension in the context of wireless multi-hop networks.Comment: A brief announcement of this result has been published in the
proceedings of ALGOSENSORS 201
Control for Localization and Visibility Maintenance of an Independent Agent using Robotic Teams
Given a non-cooperative agent, we seek to formulate a control strategy to enable a team of robots to localize and track the agent in a complex but known environment while maintaining a continuously optimized line-of-sight communication chain to a fixed base station. We focus on two aspects of the problem. First, we investigate the estimation of the agent\u27s location by using nonlinear sensing modalities, in particular that of range-only sensing, and formulate a control strategy based on improving this estimation using one or more robots working to independently gather information. Second, we develop methods to plan and sequence robot deployments that will establish and maintain line-of-sight chains for communication between the independent agent and the fixed base station using a minimum number of robots. These methods will lead to feedback control laws that can realize this plan and ensure proper navigation and collision avoidance
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