45,929 research outputs found
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
Extra heads and invariant allocations
Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm
version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists
a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y
of \Pi such that translating \Pi by -Y yields a configuration whose law is that
of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of
\Pi, and prove that there is no shift coupling in which \Pi is a function of
\Pi^*. The key ingredient is a deterministic translation-invariant rule to
allocate sets of equal volume (forming a partition of R^d) to the points of
\Pi. The construction is based on the Gale-Shapley stable marriage algorithm
[Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random
element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A
shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show
that there exists an extra head scheme which is a function of \Gamma if and
only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law
of \Gamma is product measure and d\geq3, we prove that there exists an extra
head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of
Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].Comment: Published at http://dx.doi.org/10.1214/009117904000000603 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Self-Stabilizing Repeated Balls-into-Bins
We study the following synchronous process that we call "repeated
balls-into-bins". The process is started by assigning balls to bins in
an arbitrary way. In every subsequent round, from each non-empty bin one ball
is chosen according to some fixed strategy (random, FIFO, etc), and re-assigned
to one of the bins uniformly at random.
We define a configuration "legitimate" if its maximum load is
. We prove that, starting from any configuration, the
process will converge to a legitimate configuration in linear time and then it
will only take on legitimate configurations over a period of length bounded by
any polynomial in , with high probability (w.h.p.). This implies that the
process is self-stabilizing and that every ball traverses all bins in
rounds, w.h.p
Networking - A Statistical Physics Perspective
Efficient networking has a substantial economic and societal impact in a
broad range of areas including transportation systems, wired and wireless
communications and a range of Internet applications. As transportation and
communication networks become increasingly more complex, the ever increasing
demand for congestion control, higher traffic capacity, quality of service,
robustness and reduced energy consumption require new tools and methods to meet
these conflicting requirements. The new methodology should serve for gaining
better understanding of the properties of networking systems at the macroscopic
level, as well as for the development of new principled optimization and
management algorithms at the microscopic level. Methods of statistical physics
seem best placed to provide new approaches as they have been developed
specifically to deal with non-linear large scale systems. This paper aims at
presenting an overview of tools and methods that have been developed within the
statistical physics community and that can be readily applied to address the
emerging problems in networking. These include diffusion processes, methods
from disordered systems and polymer physics, probabilistic inference, which
have direct relevance to network routing, file and frequency distribution, the
exploration of network structures and vulnerability, and various other
practical networking applications.Comment: (Review article) 71 pages, 14 figure
The Predictive Information Content of External Imbalances for Exchange Rate Returns: How Much Is It Worth?
This paper examines the exchange rate predictability stemming from the equilibrium model of international financial adjustment developed by Gourinchas and Rey (2007). Using predictive variables that measure cyclical external imbalances for country pairs, we assess the ability of this model to forecast out-of-sample four major US dollar exchange rates using various economic criteria of model evaluation. The analysis shows that the model provides economic value to a risk-averse investor, delivering substantial utility gains when switching from a portfolio strategy based on the random walk benchmark to one that conditions on cyclical external imbalances.foreign exchange; predictability; global imbalances; fundamentals.
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