19 research outputs found
Fast Collaborative Graph Exploration
International audienceWe study the following scenario of online graph exploration. A team of agents is initially located at a distinguished vertex of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure that every vertex has been visited by some agent. We consider two communication models: one in which all agents have global knowledge of the state of the exploration, and one in which agents may only exchange information when simultaneously located at the same vertex. As our main result, we provide the first strategy which performs exploration of a graph with vertices at a distance of at most from in time , using a team of agents of polynomial size . Our strategy works in the local communication model, without knowledge of global parameters such as or . We also obtain almost-tight bounds on the asymptotic relation between exploration time and team size, for large . For any constant , we show that in the global communication model, a team of agents can always complete exploration in time steps, whereas at least steps are sometimes required. In the local communication model, steps always suffice to complete exploration, and at least steps are sometimes required. This shows a clear separation between the global and local communication models
A general lower bound for collaborative tree exploration
We consider collaborative graph exploration with a set of agents. All
agents start at a common vertex of an initially unknown graph and need to
collectively visit all other vertices. We assume agents are deterministic,
vertices are distinguishable, moves are simultaneous, and we allow agents to
communicate globally. For this setting, we give the first non-trivial lower
bounds that bridge the gap between small () and large () teams of agents. Remarkably, our bounds tightly connect to existing results
in both domains.
First, we significantly extend a lower bound of
by Dynia et al. on the competitive ratio of a collaborative tree exploration
strategy to the range for any . Second,
we provide a tight lower bound on the number of agents needed for any
competitive exploration algorithm. In particular, we show that any
collaborative tree exploration algorithm with agents has a
competitive ratio of , while Dereniowski et al. gave an algorithm
with agents and competitive ratio , for any
and with denoting the diameter of the graph. Lastly, we
show that, for any exploration algorithm using agents, there exist
trees of arbitrarily large height that require rounds, and we
provide a simple algorithm that matches this bound for all trees
Minimizing the Cost of Team Exploration
A group of mobile agents is given a task to explore an edge-weighted graph
, i.e., every vertex of has to be visited by at least one agent. There
is no centralized unit to coordinate their actions, but they can freely
communicate with each other. The goal is to construct a deterministic strategy
which allows agents to complete their task optimally. In this paper we are
interested in a cost-optimal strategy, where the cost is understood as the
total distance traversed by agents coupled with the cost of invoking them. Two
graph classes are analyzed, rings and trees, in the off-line and on-line
setting, i.e., when a structure of a graph is known and not known to agents in
advance. We present algorithms that compute the optimal solutions for a given
ring and tree of order , in time units. For rings in the on-line
setting, we give the -competitive algorithm and prove the lower bound of
for the competitive ratio for any on-line strategy. For every strategy
for trees in the on-line setting, we prove the competitive ratio to be no less
than , which can be achieved by the algorithm.Comment: 25 pages, 4 figures, 5 pseudo-code
Collaborative Delivery with Energy-Constrained Mobile Robots
We consider the problem of collectively delivering some message from a
specified source to a designated target location in a graph, using multiple
mobile agents. Each agent has a limited energy which constrains the distance it
can move. Hence multiple agents need to collaborate to move the message, each
agent handing over the message to the next agent to carry it forward. Given the
positions of the agents in the graph and their respective budgets, the problem
of finding a feasible movement schedule for the agents can be challenging. We
consider two variants of the problem: in non-returning delivery, the agents can
stop anywhere; whereas in returning delivery, each agent needs to return to its
starting location, a variant which has not been studied before.
We first provide a polynomial-time algorithm for returning delivery on trees,
which is in contrast to the known (weak) NP-hardness of the non-returning
version. In addition, we give resource-augmented algorithms for returning
delivery in general graphs. Finally, we give tight lower bounds on the required
resource augmentation for both variants of the problem. In this sense, our
results close the gap left by previous research.Comment: 19 pages. An extended abstract of this paper was published at the
23rd International Colloquium on Structural Information and Communication
Complexity 2016, SIROCCO'1
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
Fault-Tolerant Dispersion of Mobile Robots
We consider the mobile robot dispersion problem in the presence of faulty
robots (crash-fault). Mobile robot dispersion consists of robots in
an -node anonymous graph. The goal is to ensure that regardless of the
initial placement of the robots over the nodes, the final configuration
consists of having at most one robot at each node. In a crash-fault setting, up
to robots may fail by crashing arbitrarily and subsequently lose all
the information stored at the robots, rendering them unable to communicate. In
this paper, we solve the dispersion problem in a crash-fault setting by
considering two different initial configurations: i) the rooted configuration,
and ii) the arbitrary configuration. In the rooted case, all robots are placed
together at a single node at the start. The arbitrary configuration is a
general configuration (a.k.a. arbitrary configuration in the literature) where
the robots are placed in some clusters arbitrarily across the graph. For
the first case, we develop an algorithm solving dispersion in the presence of
faulty robots in rounds, which improves over the previous
-round result by \cite{PS021}. For the
arbitrary configuration, we present an algorithm solving dispersion in
rounds, when the number of edges
and the maximum degree of the graph is known to the robots