9,203 research outputs found
Minimum Enclosing Circle of a Set of Fixed Points and a Mobile Point
Given a set S of n static points and a mobile point p in ℝ2, we study the variations of the smallest circle that encloses S ∪ {p} when p moves along a straight line ℓ. In this work, a complete characterization of the locus of the center of the minimum enclosing circle (MEC) of S ∪ {p}, for p ∈ ℓ, is presented. The locus is a continuous and piecewise differentiable linear function, and each of its differentiable pieces lies either on the edges of the farthest-point Voronoi diagram of S, or on a line segment parallel to the line ℓ. Moreover, the locus has differentiable pieces, which can be computed in linear time, given the farthest-point Voronoi diagram of S
Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space
Creating a swarm of mobile computing entities frequently called robots,
agents or sensor nodes, with self-organization ability is a contemporary
challenge in distributed computing. Motivated by this, we investigate the plane
formation problem that requires a swarm of robots moving in the three
dimensional Euclidean space to land on a common plane. The robots are fully
synchronous and endowed with visual perception. But they do not have
identifiers, nor access to the global coordinate system, nor any means of
explicit communication with each other. Though there are plenty of results on
the agreement problem for robots in the two dimensional plane, for example, the
point formation problem, the pattern formation problem, and so on, this is the
first result for robots in the three dimensional space. This paper presents a
necessary and sufficient condition for fully-synchronous robots to solve the
plane formation problem that does not depend on obliviousness i.e., the
availability of local memory at robots. An implication of the result is
somewhat counter-intuitive: The robots cannot form a plane from most of the
semi-regular polyhedra, while they can form a plane from every regular
polyhedron (except a regular icosahedron), whose symmetry is usually considered
to be higher than any semi-regular polyhedrdon
Robots with Lights: Overcoming Obstructed Visibility Without Colliding
Robots with lights is a model of autonomous mobile computational entities
operating in the plane in Look-Compute-Move cycles: each agent has an
externally visible light which can assume colors from a fixed set; the lights
are persistent (i.e., the color is not erased at the end of a cycle), but
otherwise the agents are oblivious. The investigation of computability in this
model, initially suggested by Peleg, is under way, and several results have
been recently established. In these investigations, however, an agent is
assumed to be capable to see through another agent. In this paper we start the
study of computing when visibility is obstructable, and investigate the most
basic problem for this setting, Complete Visibility: The agents must reach
within finite time a configuration where they can all see each other and
terminate. We do not make any assumption on a-priori knowledge of the number of
agents, on rigidity of movements nor on chirality. The local coordinate system
of an agent may change at each activation. Also, by definition of lights, an
agent can communicate and remember only a constant number of bits in each
cycle. In spite of these weak conditions, we prove that Complete Visibility is
always solvable, even in the asynchronous setting, without collisions and using
a small constant number of colors. The proof is constructive. We also show how
to extend our protocol for Complete Visibility so that, with the same number of
colors, the agents solve the (non-uniform) Circle Formation problem with
obstructed visibility
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