743 research outputs found
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
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
Time Versus Cost Tradeoffs for Deterministic Rendezvous in Networks
Two mobile agents, starting from different nodes of a network at possibly
different times, have to meet at the same node. This problem is known as
. Agents move in synchronous rounds. Each agent has a
distinct integer label from the set . Two main efficiency
measures of rendezvous are its (the number of rounds until the
meeting) and its (the total number of edge traversals). We
investigate tradeoffs between these two measures. A natural benchmark for both
time and cost of rendezvous in a network is the number of edge traversals
needed for visiting all nodes of the network, called the exploration time.
Hence we express the time and cost of rendezvous as functions of an upper bound
on the time of exploration (where and a corresponding exploration
procedure are known to both agents) and of the size of the label space. We
present two natural rendezvous algorithms. Algorithm has cost
(and, in fact, a version of this algorithm for the model where the
agents start simultaneously has cost exactly ) and time . Algorithm
has both time and cost . Our main contributions are
lower bounds showing that, perhaps surprisingly, these two algorithms capture
the tradeoffs between time and cost of rendezvous almost tightly. We show that
any deterministic rendezvous algorithm of cost asymptotically (i.e., of
cost ) must have time . On the other hand, we show that any
deterministic rendezvous algorithm with time complexity must have
cost
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
Asynchronous approach in the plane: A deterministic polynomial algorithm
In this paper we study the task of approach of two mobile agents having the
same limited range of vision and moving asynchronously in the plane. This task
consists in getting them in finite time within each other's range of vision.
The agents execute the same deterministic algorithm and are assumed to have a
compass showing the cardinal directions as well as a unit measure. On the other
hand, they do not share any global coordinates system (like GPS), cannot
communicate and have distinct labels. Each agent knows its label but does not
know the label of the other agent or the initial position of the other agent
relative to its own. The route of an agent is a sequence of segments that are
subsequently traversed in order to achieve approach. For each agent, the
computation of its route depends only on its algorithm and its label. An
adversary chooses the initial positions of both agents in the plane and
controls the way each of them moves along every segment of the routes, in
particular by arbitrarily varying the speeds of the agents. A deterministic
approach algorithm is a deterministic algorithm that always allows two agents
with any distinct labels to solve the task of approach regardless of the
choices and the behavior of the adversary. The cost of a complete execution of
an approach algorithm is the length of both parts of route travelled by the
agents until approach is completed. Let and be the initial
distance separating the agents and the length of the shortest label,
respectively. Assuming that and are unknown to both agents, does
there exist a deterministic approach algorithm always working at a cost that is
polynomial in and ? In this paper, we provide a positive answer to
the above question by designing such an algorithm
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