Parallel Balanced Allocations: The Heavily Loaded Case

We study parallel algorithms for the classical balls-into-bins problem, in which $m$ balls acting in parallel as separate agents are placed into $n$ 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 $m=n$ 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 $m\gg n$. The naive solution of allocating each ball to a bin chosen uniformly and independently at random results in maximal load $m/n+\Theta(\sqrt{m/n\cdot \log n})$ (for $m\geq n \log n$) 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 $m/n+O(\log\log m)$ 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 $m/n+O(1)$ w.h.p. within $O(\log\log (m/n)+\log^* n)$ rounds. The algorithm is symmetric (balls and bins all "look the same"), and balls send $O(1)$ messages in expectation per round. The additive term of $O(\log^* n)$ 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 $\Omega(\min\{\log\log (m/n),n\})$ 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 $m/n + O(1)$ in a constant number of rounds w.h.p. Again, balls send only a single message per round, and bins receive $(1+o(1))m/n+O(\log n)$ 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 $r_d$ for scales exceeding $r_d$. The scalar isolines are found fractal/smooth at the scales larger/smaller than the pumping scale $L$. 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