14 research outputs found
Parallel Balanced Allocations: The Heavily Loaded Case
We study parallel algorithms for the classical balls-into-bins problem, in
which balls acting in parallel as separate agents are placed into bins.
Algorithms operate in synchronous rounds, in each of which balls and bins
exchange messages once. The goal is to minimize the maximal load over all bins
using a small number of rounds and few messages.
While the case of balls has been extensively studied, little is known
about the heavily loaded case. In this work, we consider parallel algorithms
for this somewhat neglected regime of . The naive solution of
allocating each ball to a bin chosen uniformly and independently at random
results in maximal load (for ) w.h.p. In contrast, for the sequential setting Berenbrink et al (SIAM J.
Comput 2006) showed that letting each ball join the least loaded bin of two
randomly selected bins reduces the maximal load to w.h.p.
To date, no parallel variant of such a result is known.
We present a simple parallel threshold algorithm that obtains a maximal load
of w.h.p. within rounds. The algorithm
is symmetric (balls and bins all "look the same"), and balls send
messages in expectation per round. The additive term of in the
complexity is known to be tight for such algorithms (Lenzen and Wattenhofer
Distributed Computing 2016). We also prove that our analysis is tight, i.e.,
algorithms of the type we provide must run for rounds w.h.p.
Finally, we give a simple asymmetric algorithm (i.e., balls are aware of a
common labeling of the bins) that achieves a maximal load of in a
constant number of rounds w.h.p. Again, balls send only a single message per
round, and bins receive messages w.h.p
On Bernoulli Decompositions for Random Variables, Concentration Bounds, and Spectral Localization
As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli
component. This observation provides a tool for the extension of results which
are known for Bernoulli random variables to arbitrary distributions. Two
applications are provided here: i. an anti-concentration bound for a class of
functions of independent random variables, where probabilistic bounds are
extracted from combinatorial results, and ii. a proof, based on the Bernoulli
case, of spectral localization for random Schroedinger operators with arbitrary
probability distributions for the single site coupling constants. For a general
random variable, the Bernoulli component may be defined so that its conditional
variance is uniformly positive. The natural maximization problem is an optimal
transport question which is also addressed here
Fractal iso-contours of passive scalar in smooth random flows
We consider a passive scalar field under the action of pumping, diffusion and
advection by a smooth flow with a Lagrangian chaos. We present theoretical
arguments showing that scalar statistics is not conformal invariant and
formulate new effective semi-analytic algorithm to model the scalar turbulence.
We then carry massive numerics of passive scalar turbulence with the focus on
the statistics of nodal lines. The distribution of contours over sizes and
perimeters is shown to depend neither on the flow realization nor on the
resolution (diffusion) scale for scales exceeding . The scalar
isolines are found fractal/smooth at the scales larger/smaller than the pumping
scale . We characterize the statistics of bending of a long isoline by the
driving function of the L\"owner map, show that it behaves like diffusion with
the diffusivity independent of resolution yet, most surprisingly, dependent on
the velocity realization and the time of scalar evolution