4,555 research outputs found
Scatter of Weak Robots
In this paper, we first formalize the problem to be solved, i.e., the Scatter
Problem (SP). We then show that SP cannot be deterministically solved. Next, we
propose a randomized algorithm for this problem. The proposed solution is
trivially self-stabilizing. We then show how to design a self-stabilizing
version of any deterministic solution for the Pattern Formation and the
Gathering problems
The Random Bit Complexity of Mobile Robots Scattering
We consider the problem of scattering robots in a two dimensional
continuous space. As this problem is impossible to solve in a deterministic
manner, all solutions must be probabilistic. We investigate the amount of
randomness (that is, the number of random bits used by the robots) that is
required to achieve scattering. We first prove that random bits are
necessary to scatter robots in any setting. Also, we give a sufficient
condition for a scattering algorithm to be random bit optimal. As it turns out
that previous solutions for scattering satisfy our condition, they are hence
proved random bit optimal for the scattering problem. Then, we investigate the
time complexity of scattering when strong multiplicity detection is not
available. We prove that such algorithms cannot converge in constant time in
the general case and in rounds for random bits optimal
scattering algorithms. However, we present a family of scattering algorithms
that converge as fast as needed without using multiplicity detection. Also, we
put forward a specific protocol of this family that is random bit optimal ( random bits are used) and time optimal ( rounds are used).
This improves the time complexity of previous results in the same setting by a
factor. Aside from characterizing the random bit complexity of mobile
robot scattering, our study also closes its time complexity gap with and
without strong multiplicity detection (that is, time complexity is only
achievable when strong multiplicity detection is available, and it is possible
to approach it as needed otherwise)
PT-symmetric deformations of integrable models
We review recent results on new physical models constructed as PT-symmetrical
deformations or extensions of different types of integrable models. We present
non-Hermitian versions of quantum spin chains, multi-particle systems of
Calogero-Moser-Sutherland type and non-linear integrable field equations of
Korteweg-de-Vries type. The quantum spin chain discussed is related to the
first example in the series of the non-unitary models of minimal conformal
field theories. For the Calogero-Moser-Sutherland models we provide three
alternative deformations: A complex extension for models related to all types
of Coxeter/Weyl groups; models describing the evolution of poles in constrained
real valued field equations of non linear integrable systems and genuine
deformations based on antilinearly invariant deformed root systems.
Deformations of complex nonlinear integrable field equations of KdV-type are
studied with regard to different kinds of PT-symmetrical scenarios. A reduction
to simple complex quantum mechanical models currently under discussion is
presented.Comment: 21 pages, 3 figure
On diffeomorphisms over surfaces trivially embedded in the 4-sphere
A surface in the 4-sphere is trivially embedded, if it bounds a 3-dimensional
handle body in the 4-sphere. For a surface trivially embedded in the 4-sphere,
a diffeomorphism over this surface is extensible if and only if this preserves
the Rokhlin quadratic form of this embedded surface.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-33.abs.htm
Rendezvous in Networks in Spite of Delay Faults
Two mobile agents, starting from different nodes of an unknown network, have
to meet at the same node. Agents move in synchronous rounds using a
deterministic algorithm. Each agent has a different label, which it can use in
the execution of the algorithm, but it does not know the label of the other
agent. Agents do not know any bound on the size of the network. In each round
an agent decides if it remains idle or if it wants to move to one of the
adjacent nodes. Agents are subject to delay faults: if an agent incurs a fault
in a given round, it remains in the current node, regardless of its decision.
If it planned to move and the fault happened, the agent is aware of it. We
consider three scenarios of fault distribution: random (independently in each
round and for each agent with constant probability 0 < p < 1), unbounded adver-
sarial (the adversary can delay an agent for an arbitrary finite number of
consecutive rounds) and bounded adversarial (the adversary can delay an agent
for at most c consecutive rounds, where c is unknown to the agents). The
quality measure of a rendezvous algorithm is its cost, which is the total
number of edge traversals. For random faults, we show an algorithm with cost
polynomial in the size n of the network and polylogarithmic in the larger label
L, which achieves rendezvous with very high probability in arbitrary networks.
By contrast, for unbounded adversarial faults we show that rendezvous is not
feasible, even in the class of rings. Under this scenario we give a rendezvous
algorithm with cost O(nl), where l is the smaller label, working in arbitrary
trees, and we show that \Omega(l) is the lower bound on rendezvous cost, even
for the two-node tree. For bounded adversarial faults, we give a rendezvous
algorithm working for arbitrary networks, with cost polynomial in n, and
logarithmic in the bound c and in the larger label L
Deterministic Rendezvous at a Node of Agents with Arbitrary Velocities
We consider the task of rendezvous in networks modeled as undirected graphs.
Two mobile agents with different labels, starting at different nodes of an
anonymous graph, have to meet. This task has been considered in the literature
under two alternative scenarios: weak and strong. Under the weak scenario,
agents may meet either at a node or inside an edge. Under the strong scenario,
they have to meet at a node, and they do not even notice meetings inside an
edge. Rendezvous algorithms under the strong scenario are known for synchronous
agents. For asynchronous agents, rendezvous under the strong scenario is
impossible even in the two-node graph, and hence only algorithms under the weak
scenario were constructed. In this paper we show that rendezvous under the
strong scenario is possible for agents with restricted asynchrony: agents have
the same measure of time but the adversary can arbitrarily impose the speed of
traversing each edge by each of the agents. We construct a deterministic
rendezvous algorithm for such agents, working in time polynomial in the size of
the graph, in the length of the smaller label, and in the largest edge
traversal time.Comment: arXiv admin note: text overlap with arXiv:1704.0888
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