7,227 research outputs found
Gathering Anonymous, Oblivious Robots on a Grid
We consider a swarm of autonomous mobile robots, distributed on a
2-dimensional grid. A basic task for such a swarm is the gathering process: All
robots have to gather at one (not predefined) place. A common local model for
extremely simple robots is the following: The robots do not have a common
compass, only have a constant viewing radius, are autonomous and
indistinguishable, can move at most a constant distance in each step, cannot
communicate, are oblivious and do not have flags or states. The only gathering
algorithm under this robot model, with known runtime bounds, needs
rounds and works in the Euclidean plane. The underlying time
model for the algorithm is the fully synchronous model. On
the other side, in the case of the 2-dimensional grid, the only known gathering
algorithms for the same time and a similar local model additionally require a
constant memory, states and "flags" to communicate these states to neighbors in
viewing range. They gather in time .
In this paper we contribute the (to the best of our knowledge) first
gathering algorithm on the grid that works under the same simple local model as
the above mentioned Euclidean plane strategy, i.e., without memory (oblivious),
"flags" and states. We prove its correctness and an time
bound in the fully synchronous time model. This time bound
matches the time bound of the best known algorithm for the Euclidean plane
mentioned above. We say gathering is done if all robots are located within a
square, because in such configurations cannot be
solved
Rendezvous of Distance-aware Mobile Agents in Unknown Graphs
We study the problem of rendezvous of two mobile agents starting at distinct
locations in an unknown graph. The agents have distinct labels and walk in
synchronous steps. However the graph is unlabelled and the agents have no means
of marking the nodes of the graph and cannot communicate with or see each other
until they meet at a node. When the graph is very large we want the time to
rendezvous to be independent of the graph size and to depend only on the
initial distance between the agents and some local parameters such as the
degree of the vertices, and the size of the agent's label. It is well known
that even for simple graphs of degree , the rendezvous time can be
exponential in in the worst case. In this paper, we introduce a new
version of the rendezvous problem where the agents are equipped with a device
that measures its distance to the other agent after every step. We show that
these \emph{distance-aware} agents are able to rendezvous in any unknown graph,
in time polynomial in all the local parameters such the degree of the nodes,
the initial distance and the size of the smaller of the two agent labels . Our algorithm has a time complexity of
and we show an almost matching lower bound of
on the time complexity of any
rendezvous algorithm in our scenario. Further, this lower bound extends
existing lower bounds for the general rendezvous problem without distance
awareness
Approximate Kalman-Bucy filter for continuous-time semi-Markov jump linear systems
The aim of this paper is to propose a new numerical approximation of the
Kalman-Bucy filter for semi-Markov jump linear systems. This approximation is
based on the selection of typical trajectories of the driving semi-Markov chain
of the process by using an optimal quantization technique. The main advantage
of this approach is that it makes pre-computations possible. We derive a
Lipschitz property for the solution of the Riccati equation and a general
result on the convergence of perturbed solutions of semi-Markov switching
Riccati equations when the perturbation comes from the driving semi-Markov
chain. Based on these results, we prove the convergence of our approximation
scheme in a general infinite countable state space framework and derive an
error bound in terms of the quantization error and time discretization step. We
employ the proposed filter in a magnetic levitation example with markovian
failures and compare its performance with both the Kalman-Bucy filter and the
Markovian linear minimum mean squares estimator
Rendezvous on a Line by Location-Aware Robots Despite the Presence of Byzantine Faults
A set of mobile robots is placed at points of an infinite line. The robots
are equipped with GPS devices and they may communicate their positions on the
line to a central authority. The collection contains an unknown subset of
"spies", i.e., byzantine robots, which are indistinguishable from the
non-faulty ones. The set of the non-faulty robots need to rendezvous in the
shortest possible time in order to perform some task, while the byzantine
robots may try to delay their rendezvous for as long as possible. The problem
facing a central authority is to determine trajectories for all robots so as to
minimize the time until the non-faulty robots have rendezvoused. The
trajectories must be determined without knowledge of which robots are faulty.
Our goal is to minimize the competitive ratio between the time required to
achieve the first rendezvous of the non-faulty robots and the time required for
such a rendezvous to occur under the assumption that the faulty robots are
known at the start. We provide a bounded competitive ratio algorithm, where the
central authority is informed only of the set of initial robot positions,
without knowing which ones or how many of them are faulty. When an upper bound
on the number of byzantine robots is known to the central authority, we provide
algorithms with better competitive ratios. In some instances we are able to
show these algorithms are optimal
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