33 research outputs found
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
Ants: Mobile Finite State Machines
Consider the Ants Nearby Treasure Search (ANTS) problem introduced by
Feinerman, Korman, Lotker, and Sereni (PODC 2012), where mobile agents,
initially placed at the origin of an infinite grid, collaboratively search for
an adversarially hidden treasure. In this paper, the model of Feinerman et al.
is adapted such that the agents are controlled by a (randomized) finite state
machine: they possess a constant-size memory and are able to communicate with
each other through constant-size messages. Despite the restriction to
constant-size memory, we show that their collaborative performance remains the
same by presenting a distributed algorithm that matches a lower bound
established by Feinerman et al. on the run-time of any ANTS algorithm
Collaborative search on the plane without communication
We generalize the classical cow-path problem [7, 14, 38, 39] into a question
that is relevant for collective foraging in animal groups. Specifically, we
consider a setting in which k identical (probabilistic) agents, initially
placed at some central location, collectively search for a treasure in the
two-dimensional plane. The treasure is placed at a target location by an
adversary and the goal is to find it as fast as possible as a function of both
k and D, where D is the distance between the central location and the target.
This is biologically motivated by cooperative, central place foraging such as
performed by ants around their nest. In this type of search there is a strong
preference to locate nearby food sources before those that are further away.
Our focus is on trying to find what can be achieved if communication is limited
or altogether absent. Indeed, to avoid overlaps agents must be highly dispersed
making communication difficult. Furthermore, if agents do not commence the
search in synchrony then even initial communication is problematic. This holds,
in particular, with respect to the question of whether the agents can
communicate and conclude their total number, k. It turns out that the knowledge
of k by the individual agents is crucial for performance. Indeed, it is a
straightforward observation that the time required for finding the treasure is
(D + D 2 /k), and we show in this paper that this bound can be matched
if the agents have knowledge of k up to some constant approximation. We present
an almost tight bound for the competitive penalty that must be paid, in the
running time, if agents have no information about k. Specifically, on the
negative side, we show that in such a case, there is no algorithm whose
competitiveness is O(log k). On the other hand, we show that for every constant
\epsilon \textgreater{} 0, there exists a rather simple uniform search
algorithm which is -competitive. In addition, we give
a lower bound for the setting in which agents are given some estimation of k.
As a special case, this lower bound implies that for any constant \epsilon
\textgreater{} 0, if each agent is given a (one-sided)
-approximation to k, then the competitiveness is (log k).
Informally, our results imply that the agents can potentially perform well
without any knowledge of their total number k, however, to further improve,
they must be given a relatively good approximation of k. Finally, we propose a
uniform algorithm that is both efficient and extremely simple suggesting its
relevance for actual biological scenarios
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
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
Mobile agent rendezvous: A survey
Abstract. Recent results on the problem of mobile agent rendezvous on distributed networks are surveyed with an emphasis on outlining the various approaches taken by researchers in the theoretical computer science community.
Trade-offs between Selection Complexity and Performance when Searching the Plane without Communication
We consider the ANTS problem [Feinerman et al.] in which a group of agents
collaboratively search for a target in a two-dimensional plane. Because this
problem is inspired by the behavior of biological species, we argue that in
addition to studying the {\em time complexity} of solutions it is also
important to study the {\em selection complexity}, a measure of how likely a
given algorithmic strategy is to arise in nature due to selective pressures. In
more detail, we propose a new selection complexity metric , defined for
algorithm such that , where is
the number of memory bits used by each agent and bounds the fineness of
available probabilities (agents use probabilities of at least ). In
this paper, we study the trade-off between the standard performance metric of
speed-up, which measures how the expected time to find the target improves with
, and our new selection metric.
In particular, consider agents searching for a treasure located at
(unknown) distance from the origin (where is sub-exponential in ).
For this problem, we identify as a crucial threshold for our
selection complexity metric. We first prove a new upper bound that achieves a
near-optimal speed-up of for . In particular, for , the speed-up is
asymptotically optimal. By comparison, the existing results for this problem
[Feinerman et al.] that achieve similar speed-up require . We then show that this threshold is tight by describing a
lower bound showing that if , then
with high probability the target is not found within moves per
agent. Hence, there is a sizable gap to the straightforward
lower bound in this setting.Comment: appears in PODC 201