154 research outputs found
Tight bounds for undirected graph exploration with pebbles and multiple agents
We study the problem of deterministically exploring an undirected and
initially unknown graph with vertices either by a single agent equipped
with a set of pebbles, or by a set of collaborating agents. The vertices of the
graph are unlabeled and cannot be distinguished by the agents, but the edges
incident to a vertex have locally distinct labels. The graph is explored when
all vertices have been visited by at least one agent. In this setting, it is
known that for a single agent without pebbles bits of memory
are necessary and sufficient to explore any graph with at most vertices. We
are interested in how the memory requirement decreases as the agent may mark
vertices by dropping and retrieving distinguishable pebbles, or when multiple
agents jointly explore the graph. We give tight results for both questions
showing that for a single agent with constant memory
pebbles are necessary and sufficient for exploration. We further prove that the
same bound holds for the number of collaborating agents needed for exploration.
For the upper bound, we devise an algorithm for a single agent with constant
memory that explores any -vertex graph using
pebbles, even when is unknown. The algorithm terminates after polynomial
time and returns to the starting vertex. Since an additional agent is at least
as powerful as a pebble, this implies that agents
with constant memory can explore any -vertex graph. For the lower bound, we
show that the number of agents needed for exploring any graph of size is
already when we allow each agent to have at most
bits of memory for any .
This also implies that a single agent with sublogarithmic memory needs
pebbles to explore any -vertex graph
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
The power of a pebble : exploring and mapping directed graphs
Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 36-39).by Amit Sahai.M.S
Deterministic Graph Exploration with Advice
We consider the task of graph exploration. An -node graph has unlabeled
nodes, and all ports at any node of degree are arbitrarily numbered
. A mobile agent has to visit all nodes and stop. The exploration
time is the number of edge traversals. We consider the problem of how much
knowledge the agent has to have a priori, in order to explore the graph in a
given time, using a deterministic algorithm. This a priori information (advice)
is provided to the agent by an oracle, in the form of a binary string, whose
length is called the size of advice. We consider two types of oracles. The
instance oracle knows the entire instance of the exploration problem, i.e., the
port-numbered map of the graph and the starting node of the agent in this map.
The map oracle knows the port-numbered map of the graph but does not know the
starting node of the agent.
We first consider exploration in polynomial time, and determine the exact
minimum size of advice to achieve it. This size is ,
for both types of oracles.
When advice is large, there are two natural time thresholds:
for a map oracle, and for an instance oracle, that can be achieved
with sufficiently large advice. We show that, with a map oracle, time
cannot be improved in general, regardless of the size of advice.
We also show that the smallest size of advice to achieve this time is larger
than , for any .
For an instance oracle, advice of size is enough to achieve time
. We show that, with any advice of size , the time of
exploration must be at least , for any , and with any
advice of size , the time must be .
We also investigate minimum advice sufficient for fast exploration of
hamiltonian graphs
Efficient Multi-Robot Motion Planning for Unlabeled Discs in Simple Polygons
We consider the following motion-planning problem: we are given unit
discs in a simple polygon with vertices, each at their own start position,
and we want to move the discs to a given set of target positions. Contrary
to the standard (labeled) version of the problem, each disc is allowed to be
moved to any target position, as long as in the end every target position is
occupied. We show that this unlabeled version of the problem can be solved in
time, assuming that the start and target positions are at
least some minimal distance from each other. This is in sharp contrast to the
standard (labeled) and more general multi-robot motion-planning problem for
discs moving in a simple polygon, which is known to be strongly NP-hard
Exploring an Infinite Space with Finite Memory Scouts
Consider a small number of scouts exploring the infinite -dimensional grid
with the aim of hitting a hidden target point. Each scout is controlled by a
probabilistic finite automaton that determines its movement (to a neighboring
grid point) based on its current state. The scouts, that operate under a fully
synchronous schedule, communicate with each other (in a way that affects their
respective states) when they share the same grid point and operate
independently otherwise. Our main research question is: How many scouts are
required to guarantee that the target admits a finite mean hitting time?
Recently, it was shown that is an upper bound on the answer to this
question for any dimension and the main contribution of this paper
comes in the form of proving that this bound is tight for .Comment: Added (forgotten) acknowledgement
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