7,988 research outputs found
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
Deterministic Random Walks on Regular Trees
Jim Propp's rotor router model is a deterministic analogue of a random walk
on a graph. Instead of distributing chips randomly, each vertex serves its
neighbors in a fixed order.
Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable
similarity of both models. If an (almost) arbitrary population of chips is
placed on the vertices of a grid and does a simultaneous walk in the
Propp model, then at all times and on each vertex, the number of chips on this
vertex deviates from the expected number the random walk would have gotten
there by at most a constant. This constant is independent of the starting
configuration and the order in which each vertex serves its neighbors.
This result raises the question if all graphs do have this property. With
quite some effort, we are now able to answer this question negatively. For the
graph being an infinite -ary tree (), we show that for any
deviation there is an initial configuration of chips such that after
running the Propp model for a certain time there is a vertex with at least
more chips than expected in the random walk model. However, to achieve a
deviation of it is necessary that at least vertices
contribute by being occupied by a number of chips not divisible by at a
certain time.Comment: 15 pages, to appear in Random Structures and Algorithm
Multiscale Finite-Difference-Diffusion-Monte-Carlo Method for Simulating Dendritic Solidification
We present a novel hybrid computational method to simulate accurately
dendritic solidification in the low undercooling limit where the dendrite tip
radius is one or more orders of magnitude smaller than the characteristic
spatial scale of variation of the surrounding thermal or solutal diffusion
field. The first key feature of this method is an efficient multiscale
diffusion Monte-Carlo (DMC) algorithm which allows off-lattice random walkers
to take longer and concomitantly rarer steps with increasing distance away from
the solid-liquid interface. As a result, the computational cost of evolving the
large scale diffusion field becomes insignificant when compared to that of
calculating the interface evolution. The second key feature is that random
walks are only permitted outside of a thin liquid layer surrounding the
interface. Inside this layer and in the solid, the diffusion equation is solved
using a standard finite-difference algorithm that is interfaced with the DMC
algorithm using the local conservation law for the diffusing quantity. Here we
combine this algorithm with a previously developed phase-field formulation of
the interface dynamics and demonstrate that it can accurately simulate
three-dimensional dendritic growth in a previously unreachable range of low
undercoolings that is of direct experimental relevance.Comment: RevTeX, 16 pages, 10 eps figures, submitted to J. Comp. Phy
Discrete analogue computing with rotor-routers
Rotor-routing is a procedure for routing tokens through a network that can
implement certain kinds of computation. These computations are inherently
asynchronous (the order in which tokens are routed makes no difference) and
distributed (information is spread throughout the system). It is also possible
to efficiently check that a computation has been carried out correctly in less
time than the computation itself required, provided one has a certificate that
can itself be computed by the rotor-router network. Rotor-router networks can
be viewed as both discrete analogues of continuous linear systems and
deterministic analogues of stochastic processes.Comment: To appear in Chaos Special Focus Issue on Intrinsic and Designed
Computatio
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
Constructing a Stochastic Model of Bumblebee Flights from Experimental Data
PMCID: PMC3592844This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited
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