118 research outputs found
A Nearly Tight Lower Bound for the -Dimensional Cow-Path Problem
In the -dimensional cow-path problem, a cow living in must
locate a -dimensional hyperplane whose location is unknown. The
only way that the cow can find is to roam until it
intersects . If the cow travels a total distance to locate a
hyperplane whose distance from the origin was , then the cow is
said to achieve competitive ratio .
It is a classic result that, in , the optimal (deterministic)
competitive ratio is . In , the optimal competitive ratio is
known to be at most . But in higher dimensions, the asymptotic
relationship between and the optimal competitive ratio remains an open
question. The best upper and lower bounds, due to Antoniadis et al., are
and , leaving a gap of roughly . In this
note, we achieve a stronger lower bound of
Searching in an Unknown Environment: An Optimal Randomized Algorithm for the Cow-Path Problem
AbstractSearching for a goal is a central and extensively studied problem in computer science. In classical searching problems, the cost of a search function is simply the number of queries made to an oracle that knows the position of the goal. In many robotics problems, as well as in problems from other areas, we want to charge a cost proportional to the distance between queries (e.g., the time required to travel between two query points). With this cost function in mind, the abstract problem known as thew-lane cow-path problem was designed. There are known optimal deterministic algorithms for the cow-path problem; we give the first randomized algorithm in this paper. We show that our algorithm is optimal for two paths (w=2) and give evidence that it is optimal for larger values ofw. Subsequent to the preliminary version of this paper, Kaoet al.(in“Proceedings, 5th ACM–SIAM Symposium on Discrete Algorithm,” pp. 372–381, 1994) have shown that our algorithm is indeed optimal for allw⩾2. Our randomized algorithm gives expected performance that is almost twice as good as is possible with a deterministic algorithm. For the performance of our algorithm, we also derive the asymptotic growth with respect tow—despite similar complexity results for related problems, it appears that this growth has never been analyzed
Searching in an Unknown Environment: An Optimal Randomized Algorithm for the Cow-Path Problem
Searching for a goal is a central and extensively studied problem in computer science. In classical searching problems, the cost of a search function is simply the number of queries made to an oracle that knows the position of the goal. In many robotics problems, as well as in problems from other areas, we want to charge a cost proportional to the distance between queries (e.g., the time required to travel between two query points). With this cost function in mind, the abstract problem known as the w-lane cow-path problem was designed. There are known optimal deterministic algorithms for the cow-path problem; we give the first randomized algorithm in this paper. We show that our algorithm is optimal for two paths (w = 2) and give evidence that it is optimal for larger values of w. Subsequent to the preliminary version of this paper, Kao et al. (in “Proceedings, 5th ACM–SIAM Symposium on Discrete Algorithm," pp. 372-381, 1994) have shown that our algorithm is indeed optimal for all w = 2. Our randomized algorithm gives expected performance that is almost twice as good as is possible with a deterministic algorithm. For the performance of our algorithm, we also derive the asymptotic growth with respect to w—despite similar complexity results for related problems, it appears that this growth has never been analyzed
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
Parallel Exhaustive Search without Coordination
We analyze parallel algorithms in the context of exhaustive search over
totally ordered sets. Imagine an infinite list of "boxes", with a "treasure"
hidden in one of them, where the boxes' order reflects the importance of
finding the treasure in a given box. At each time step, a search protocol
executed by a searcher has the ability to peek into one box, and see whether
the treasure is present or not. By equally dividing the workload between them,
searchers can find the treasure times faster than one searcher.
However, this straightforward strategy is very sensitive to failures (e.g.,
crashes of processors), and overcoming this issue seems to require a large
amount of communication. We therefore address the question of designing
parallel search algorithms maximizing their speed-up and maintaining high
levels of robustness, while minimizing the amount of resources for
coordination. Based on the observation that algorithms that avoid communication
are inherently robust, we analyze the best running time performance of
non-coordinating algorithms. Specifically, we devise non-coordinating
algorithms that achieve a speed-up of for two searchers, a speed-up of
for three searchers, and in general, a speed-up of
for any searchers. Thus, asymptotically, the speed-up is only four
times worse compared to the case of full-coordination, and our algorithms are
surprisingly simple and hence applicable. Moreover, these bounds are tight in a
strong sense as no non-coordinating search algorithm can achieve better
speed-ups. Overall, we highlight that, in faulty contexts in which coordination
between the searchers is technically difficult to implement, intrusive with
respect to privacy, and/or costly in term of resources, it might well be worth
giving up on coordination, and simply run our non-coordinating exhaustive
search algorithms
Lower Bounds for Shoreline Searching With 2 or More Robots
Searching for a line on the plane with unit speed robots is a classic
online problem that dates back to the 50's, and for which competitive ratio
upper bounds are known for every . In this work we improve the best
lower bound known for robots from 1.5993 to 3. Moreover we prove that the
competitive ratio is at least for robots, and at least
for robots. Our lower bounds match the best upper
bounds known for , hence resolving these cases. To the best of our
knowledge, these are the first lower bounds proven for the cases of
this several decades old problem.Comment: This is an updated version of the paper with the same title which
will appear in the proceedings of the 23rd International Conference on
Principles of Distributed Systems (OPODIS 2019) Neuchatel, Switzerland, July
17-19, 201
Fast Two-Robot Disk Evacuation with Wireless Communication
In the fast evacuation problem, we study the path planning problem for two
robots who want to minimize the worst-case evacuation time on the unit disk.
The robots are initially placed at the center of the disk. In order to
evacuate, they need to reach an unknown point, the exit, on the boundary of the
disk. Once one of the robots finds the exit, it will instantaneously notify the
other agent, who will make a beeline to it.
The problem has been studied for robots with the same speed~\cite{s1}. We
study a more general case where one robot has speed and the other has speed
. We provide optimal evacuation strategies in the case that by showing matching upper and lower bounds on the
worst-case evacuation time. For , we show (non-matching)
upper and lower bounds on the evacuation time with a ratio less than .
Moreover, we demonstrate that a generalization of the two-robot search strategy
from~\cite{s1} is outperformed by our proposed strategies for any .Comment: 18 pages, 10 figure
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