42 research outputs found
The coalescing-branching random walk on expanders and the dual epidemic process
Information propagation on graphs is a fundamental topic in distributed
computing. One of the simplest models of information propagation is the push
protocol in which at each round each agent independently pushes the current
knowledge to a random neighbour. In this paper we study the so-called
coalescing-branching random walk (COBRA), in which each vertex pushes the
information to randomly selected neighbours and then stops passing
information until it receives the information again. The aim of COBRA is to
propagate information fast but with a limited number of transmissions per
vertex per step. In this paper we study the cover time of the COBRA process
defined as the minimum time until each vertex has received the information at
least once. Our main result says that if is an -vertex -regular graph
whose transition matrix has second eigenvalue , then the COBRA cover
time of is , if is greater than a positive
constant, and , if . These bounds are independent of and hold for . They improve the previous bound of for expander graphs.
Our main tool in analysing the COBRA process is a novel duality relation
between this process and a discrete epidemic process, which we call a biased
infection with persistent source (BIPS). A fixed vertex is the source of an
infection and remains permanently infected. At each step each vertex other
than selects neighbours, independently and uniformly, and is
infected in this step if and only if at least one of the selected neighbours
has been infected in the previous step. We show the duality between COBRA and
BIPS which says that the time to infect the whole graph in the BIPS process is
of the same order as the cover time of the COBRA proces
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
Balanced Allocation on Graphs: A Random Walk Approach
In this paper we propose algorithms for allocating sequential balls into
bins that are interconnected as a -regular -vertex graph , where
can be any integer.Let be a given positive integer. In each round
, , ball picks a node of uniformly at random and
performs a non-backtracking random walk of length from the chosen node.Then
it allocates itself on one of the visited nodes with minimum load (ties are
broken uniformly at random). Suppose that has a sufficiently large girth
and . Then we establish an upper bound for the maximum number
of balls at any bin after allocating balls by the algorithm, called {\it
maximum load}, in terms of with high probability. We also show that the
upper bound is at most an factor above the lower bound that is
proved for the algorithm. In particular, we show that if we set , for every constant , and
has girth at least , then the maximum load attained by the
algorithm is bounded by with high probability.Finally, we
slightly modify the algorithm to have similar results for balanced allocation
on -regular graph with and sufficiently large girth
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
Parallel Search with no Coordination
We consider a parallel version of a classical Bayesian search problem.
agents are looking for a treasure that is placed in one of the boxes indexed by
according to a known distribution . The aim is to minimize
the expected time until the first agent finds it. Searchers run in parallel
where at each time step each searcher can "peek" into a box. A basic family of
algorithms which are inherently robust is \emph{non-coordinating} algorithms.
Such algorithms act independently at each searcher, differing only by their
probabilistic choices. We are interested in the price incurred by employing
such algorithms when compared with the case of full coordination. We first show
that there exists a non-coordination algorithm, that knowing only the relative
likelihood of boxes according to , has expected running time of at most
, where is the expected running time of the best
fully coordinated algorithm. This result is obtained by applying a refined
version of the main algorithm suggested by Fraigniaud, Korman and Rodeh in
STOC'16, which was designed for the context of linear parallel search.We then
describe an optimal non-coordinating algorithm for the case where the
distribution is known. The running time of this algorithm is difficult to
analyse in general, but we calculate it for several examples. In the case where
is uniform over a finite set of boxes, then the algorithm just checks boxes
uniformly at random among all non-checked boxes and is essentially times
worse than the coordinating algorithm.We also show simple algorithms for Pareto
distributions over boxes. That is, in the case where for
, we suggest the following algorithm: at step choose uniformly
from the boxes unchecked in ,
where . It turns out this algorithm is asymptotically
optimal, and runs about times worse than the case of full coordination