32 research outputs found
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
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
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
Gathering in Dynamic Rings
The gathering problem requires a set of mobile agents, arbitrarily positioned
at different nodes of a network to group within finite time at the same
location, not fixed in advanced.
The extensive existing literature on this problem shares the same fundamental
assumption: the topological structure does not change during the rendezvous or
the gathering; this is true also for those investigations that consider faulty
nodes. In other words, they only consider static graphs. In this paper we start
the investigation of gathering in dynamic graphs, that is networks where the
topology changes continuously and at unpredictable locations.
We study the feasibility of gathering mobile agents, identical and without
explicit communication capabilities, in a dynamic ring of anonymous nodes; the
class of dynamics we consider is the classic 1-interval-connectivity.
We focus on the impact that factors such as chirality (i.e., a common sense
of orientation) and cross detection (i.e., the ability to detect, when
traversing an edge, whether some agent is traversing it in the other
direction), have on the solvability of the problem. We provide a complete
characterization of the classes of initial configurations from which the
gathering problem is solvable in presence and in absence of cross detection and
of chirality. The feasibility results of the characterization are all
constructive: we provide distributed algorithms that allow the agents to
gather. In particular, the protocols for gathering with cross detection are
time optimal. We also show that cross detection is a powerful computational
element.
We prove that, without chirality, knowledge of the ring size is strictly more
powerful than knowledge of the number of agents; on the other hand, with
chirality, knowledge of n can be substituted by knowledge of k, yielding the
same classes of feasible initial configurations
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