40 research outputs found
Want to Gather? No Need to Chatter!
A team of mobile agents, starting from different nodes of an unknown network,
possibly at different times, have to meet at the same node and declare that
they have all met. Agents have different labels and move in synchronous rounds
along links of the network. The above task is known as gathering and was
traditionally considered under the assumption that when some agents are at the
same node then they can talk. In this paper we ask the question of whether this
ability of talking is needed for gathering. The answer turns out to be no.
Our main contribution are two deterministic algorithms that always accomplish
gathering in a much weaker model. We only assume that at any time an agent
knows how many agents are at the node that it currently occupies but agents do
not see the labels of other co-located agents and cannot exchange any
information with them. They also do not see other nodes than the current one.
Our first algorithm works under the assumption that agents know a priori some
upper bound N on the network size, and it works in time polynomial in N and in
the length l of the smallest label. Our second algorithm does not assume any a
priori knowledge about the network but its complexity is exponential in the
network size and in the labels of agents. Its purpose is to show feasibility of
gathering under this harsher scenario.
As a by-product of our techniques we obtain, in the same weak model, the
solution of the fundamental problem of leader election among agents. As an
application of our result we also solve, in the same model, the well-known
gossiping problem: if each agent has a message at the beginning, we show how to
make all messages known to all agents, even without any a priori knowledge
about the network. If agents know an upper bound N on the network size then our
gossiping algorithm works in time polynomial in N, in l and in the length of
the largest message
Almost Universal Anonymous Rendezvous in the Plane
Two mobile agents represented by points freely moving in the plane and
starting at two distinct positions, have to meet. The meeting, called
rendezvous, occurs when agents are at distance at most of each other and
never move after this time, where is a positive real unknown to them,
called the visibility radius. Agents are anonymous and execute the same
deterministic algorithm. Each agent has a set of private attributes, some or
all of which can differ between agents. These attributes are: the initial
position of the agent, its system of coordinates (orientation and chirality),
the rate of its clock, its speed when it moves, and the time of its wake-up. If
all attributes (except the initial positions) are identical and agents start at
distance larger than then they can never meet. However, differences between
attributes make it sometimes possible to break the symmetry and accomplish
rendezvous. Such instances of the rendezvous problem (formalized as lists of
attributes), are called feasible.
Our contribution is three-fold. We first give an exact characterization of
feasible instances. Thus it is natural to ask whether there exists a single
algorithm that guarantees rendezvous for all these instances. We give a strong
negative answer to this question: we show two sets and of feasible
instances such that none of them admits a single rendezvous algorithm valid for
all instances of the set. On the other hand, we construct a single algorithm
that guarantees rendezvous for all feasible instances outside of sets and
. We observe that these exception sets and are geometrically
very small, compared to the set of all feasible instances: they are included in
low-dimension subspaces of the latter. Thus, our rendezvous algorithm handling
all feasible instances other than these small sets of exceptions can be justly
called almost universal
Exploring Graphs with Time Constraints by Unreliable Collections of Mobile Robots
A graph environment must be explored by a collection of mobile robots. Some
of the robots, a priori unknown, may turn out to be unreliable. The graph is
weighted and each node is assigned a deadline. The exploration is successful if
each node of the graph is visited before its deadline by a reliable robot. The
edge weight corresponds to the time needed by a robot to traverse the edge.
Given the number of robots which may crash, is it possible to design an
algorithm, which will always guarantee the exploration, independently of the
choice of the subset of unreliable robots by the adversary? We find the optimal
time, during which the graph may be explored. Our approach permits to find the
maximal number of robots, which may turn out to be unreliable, and the graph is
still guaranteed to be explored.
We concentrate on line graphs and rings, for which we give positive results.
We start with the case of the collections involving only reliable robots. We
give algorithms finding optimal times needed for exploration when the robots
are assigned to fixed initial positions as well as when such starting positions
may be determined by the algorithm. We extend our consideration to the case
when some number of robots may be unreliable. Our most surprising result is
that solving the line exploration problem with robots at given positions, which
may involve crash-faulty ones, is NP-hard. The same problem has polynomial
solutions for a ring and for the case when the initial robots' positions on the
line are arbitrary.
The exploration problem is shown to be NP-hard for star graphs, even when the
team consists of only two reliable robots
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
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
Rendezvous of Heterogeneous Mobile Agents in Edge-weighted Networks
We introduce a variant of the deterministic rendezvous problem for a pair of
heterogeneous agents operating in an undirected graph, which differ in the time
they require to traverse particular edges of the graph. Each agent knows the
complete topology of the graph and the initial positions of both agents. The
agent also knows its own traversal times for all of the edges of the graph, but
is unaware of the corresponding traversal times for the other agent. The goal
of the agents is to meet on an edge or a node of the graph. In this scenario,
we study the time required by the agents to meet, compared to the meeting time
in the offline scenario in which the agents have complete knowledge
about each others speed characteristics. When no additional assumptions are
made, we show that rendezvous in our model can be achieved after time in a -node graph, and that such time is essentially in some cases
the best possible. However, we prove that the rendezvous time can be reduced to
when the agents are allowed to exchange bits of
information at the start of the rendezvous process. We then show that under
some natural assumption about the traversal times of edges, the hardness of the
heterogeneous rendezvous problem can be substantially decreased, both in terms
of time required for rendezvous without communication, and the communication
complexity of achieving rendezvous in time
Byzantine Gathering in Networks
This paper investigates an open problem introduced in [14]. Two or more
mobile agents start from different nodes of a network and have to accomplish
the task of gathering which consists in getting all together at the same node
at the same time. An adversary chooses the initial nodes of the agents and
assigns a different positive integer (called label) to each of them. Initially,
each agent knows its label but does not know the labels of the other agents or
their positions relative to its own. Agents move in synchronous rounds and can
communicate with each other only when located at the same node. Up to f of the
agents are Byzantine. A Byzantine agent can choose an arbitrary port when it
moves, can convey arbitrary information to other agents and can change its
label in every round, in particular by forging the label of another agent or by
creating a completely new one.
What is the minimum number M of good agents that guarantees deterministic
gathering of all of them, with termination?
We provide exact answers to this open problem by considering the case when
the agents initially know the size of the network and the case when they do
not. In the former case, we prove M=f+1 while in the latter, we prove M=f+2.
More precisely, for networks of known size, we design a deterministic algorithm
gathering all good agents in any network provided that the number of good
agents is at least f+1. For networks of unknown size, we also design a
deterministic algorithm ensuring the gathering of all good agents in any
network but provided that the number of good agents is at least f+2. Both of
our algorithms are optimal in terms of required number of good agents, as each
of them perfectly matches the respective lower bound on M shown in [14], which
is of f+1 when the size of the network is known and of f+2 when it is unknown
Asynchronous rendezvous with different maps
© Springer Nature Switzerland AG 2019. This paper provides a study on the rendezvous problem in which two anonymous mobile entities referred to as robots rA and rB are asked to meet at an arbitrary node of a graph G = (V,E). As opposed to more standard assumptions robots may not be able to visit the entire graph G. Namely, each robot has its own map which is a connected subgraph of G. Such mobility restrictions may be dictated by the topological properties combined with the intrinsic characteristics of robots preventing them from visiting certain edges in E. We consider four different variants of the rendezvous problem introduced in [Farrugia et al. SOFSEM’15] which reflect on restricted maneuverability and navigation ability of rA and rB in G. In the latter, the focus is on models in which robots’ actions are synchronised. The authors prove that one of the maps must be a subgraph of the other. I.e., without this assumption (or some extra knowledge) the rendezvous problem does not have a feasible solution. In this paper, while we keep the containment assumption, we focus on asynchronous robots and the relevant bounds in the four considered variants. We provide some impossibility results and almost tight lower and upper bounds when the solutions are possible
Gathering of Robots on Anonymous Grids without multiplicity detection
International audienceThe paper studies the gathering problem on grid networks. A team of robots placed at different nodes of a grid, have to meet at some node and remain there. Robots operate in Look-Compute-Move cycles; in one cycle, a robot perceives the current configuration in terms of occupied nodes (Look), decides whether to move towards one of its neighbors (Compute), and in the positive case makes the computed move instantaneously (Move). Cycles are performed asynchronously for each robot. The problem has been deeply studied for the case of ring networks. However, the known techniques used on rings cannot be directly extended to grids. Moreover, on rings, another assumption concerning the so-called multiplicity detection capability was required in order to accomplish the gathering task. That is, a robot is able to detect during its Look operation whether a node is empty, or occupied by one robot, or occupied by an undefined number of robots greater than one. In this paper, we provide a full characterization about gatherable configurations for grids. In particular, we show that in this case, the multiplicity detection is not required. Very interestingly, sometimes the problem appears trivial, as it is for the case of grids with both odd sides, while sometimes the involved techniques require new insights with respect to the well-studied ring case. Moreover, our results reveal the importance of a structure like the grid that allows to overcome the multiplicity detection with respect to the ring case
How to gather asynchronous oblivious robots on anonymous rings
A set of robots arbitrarily placed on the nodes of an anonymous graph have to meet at one common node and remain in there. This problem is known in the literature as the \emph{gathering}. Robots operate in Look-Compute-Move cycles; in one cycle, a robot takes a snapshot of the current configuration (Look), decides whether to stay idle or to move to one of its neighbors (Compute), and in the latter case makes the computed move instantaneously (Move). Cycles are performed asynchronously for each robot. Moreover, each robot is empowered by the so called \emph{multiplicity detection} capability, that is, a robot is able to detect during its Look operation whether a node is empty, or occupied by one robot, or occupied by an undefined number of robots greater than one. The described problem has been extensively studied during the last years. However, the known solutions work only for specific initial configurations and leave some open cases. In this paper, we provide an algorithm which solves the general problem, and is able to detect all the non-gatherable configurations. It is worth noting that our new algorithm makes use of a unified and general strategy for any initial configuration.Un ensemble de robots placés arbitrairement sur les sommets d'un graphe anonyme doivent se rencontrer sur un sommet commun. Ce problème est connu dans la littérature comme le \emph{gathering}. Les robots utilisent des cycles Look-Compute-Move; dans un cycle, un robot prend un instantané de la configuration actuelle (Look), décide de rester inactif ou de se déplacer sur l'un de ses voisins (Compute), et dans ce cas, fait le déplacement (Move). Les cycles sont exécutés de manière asynchrone pour chaque robot. Chaque robot possède la capacité de \emph{multiplicity detection}: un robot est capable de détecter au cours de son opération Look si un sommet est vide, occupé par un robot, ou occupé par un nombre indéfini de robots. Le problème décrit a été largement étudié au cours des dernières années. Toutefois, les solutions connues ne sont valides que pour des configurations initiales spécifiques. Nous fournissons un algorithme qui résout le problème général, et est capable de détecter toutes les configurations initiales pour lesquelles le problème est impossible. Il est intéressant de noter que notre nouvel algorithme utilise une stratégie unifiée et générale pour chaque configuration initiale