[Paper Review] Long-term balanced allocation via thinning

Ohad N. Feldheim, Ori Gurel-Gurevich+1 more|arXiv (Cornell University)|Oct 11, 2021

Algorithms and Data Compression28 references4 citations

TL;DR

This paper studies the two-thinning variant of the balls-and-bins model, where an overseer can reject a uniformly random bin allocation and reassign the ball to a new random bin to balance load. It provides explicit strategies that achieve a typical maximum load of $(\log n)^{1/2+o(1)}$ with high probability, matching the optimal single-time load for large $m$, while showing that the all-time maximum load is $\Omega(\log n / \log\log n)$, which is tight and unavoidable due to trade-offs between maintaining low typical load and avoiding temporary high peaks.

ABSTRACT

We study the long-term behavior of the two-thinning variant of the classical balls-and-bins model. In this model, an overseer is provided with uniform random allocation of $m$ balls into $n$ bins in an on-line fashion. For each ball, the overseer could reject its allocation and place the ball into a new bin drawn independently at random. The purpose of the overseer is to reduce the maximum load of the bins, which is defined as the difference between the maximum number of balls in a single bin and $m/n$, i.e., the average number of balls among all bins. We provide tight estimates for three quantities: the lowest maximum load that could be achieved at time $m$, the lowest maximum load that could be achieved uniformly over the entire time interval $[m]:=\{1, 2, \cdots, m\}$, and the lowest \emph{typical} maximum load that could be achieved over the interval $[m]$, where the typicality means that the maximum load holds for $1-o(1)$ portion of the times in $[m]$. We show that when $m$ and $n$ are sufficiently large, a typical maximum load of $(\log n)^{1/2+o(1)}$ can be achieved with high probability, asymptotically the same as the optimal maximum load that could be achieved at time $m$. However, for any strategy, the maximal load among all times in the interval $[m]$ is $Ω\big(\frac{\log n}{\log\log n}\big)$ with high probability. A strategy achieving this bound is provided. An explanation for this gap is provided by our optimal strategies as follows. To control the typical load, we restrain the maximum load for some time, during which we accumulate more and more bins with relatively high load. After a while, we have to employ for a short time a different strategy to reduce the number of relatively heavily loaded bins, at the expanse of temporarily inducing high load in a few bins.

Motivation & Objective

• To analyze the long-term load balancing behavior in the two-thinning balls-and-bins model, where an overseer can reject a primary random allocation and reassign to a secondary bin.
• To characterize the minimal achievable maximum load over time, distinguishing between single-time, all-time, and typical maximum load regimes.
• To resolve the gap between optimal single-time load and unavoidable all-time maximum load by constructing strategies that minimize typical load while managing temporary load spikes.
• To provide explicit, implementable two-thinning strategies that achieve asymptotically optimal performance under high-probability guarantees.

Proposed method

• The authors define three load metrics: single-time maximum load $\mathrm{MaxLoad}^f(m)$, all-time maximum load $\mathrm{MaxLoad}^f([m])$, and $\varepsilon$-typical maximum load $\mathrm{MaxLoad}^f_\varepsilon([m])$.
• They design a two-thinning strategy that alternates between load-reduction phases (to reduce heavily loaded bins) and load-maintenance phases (to keep typical load low).
• The analysis uses martingale concentration and exponential moment bounds via the Chernoff method to control deviations in bin loads over time.
• Key technical tools include bounding the expected exponential moment of load deviations and applying recursive inequalities to track load evolution across phases.
• The strategy leverages a time-splitting approach: first a stabilization phase of $m_0 = \Theta(n \log n)$ steps, followed by a balancing phase of $m_2 = \Theta(n \log n)$ steps, with careful control of load fluctuations.
• Theoretical guarantees are derived using union bounds and tail estimates, showing that with high probability, the load remains within $O(\log n)^{1/2+o(1)}$ for $1-o(1)$ of the time.

Experimental results

Research questions

• RQ1 What is the minimal achievable typical maximum load over time in the two-thinning balls-and-bins model, where 'typical' means holding for $1-o(1)$ of the allocation steps?
• RQ2 How does the optimal all-time maximum load compare to the optimal single-time maximum load, and what causes the gap?
• RQ3 Can a strategy achieve a typical maximum load asymptotically matching the optimal single-time load, despite unavoidable temporary load spikes?
• RQ4 What is the fundamental lower bound on the all-time maximum load across all two-thinning strategies, and can it be matched by an explicit strategy?
• RQ5 How do load fluctuations evolve over time, and what mechanisms allow maintaining low typical load while controlling worst-case load peaks?

Key findings

• For $m = \omega(n\sqrt{\log n})$, the optimal single-time maximum load is $\Theta((\log n)^{1/2+o(1)})$, and this bound is tight up to lower-order terms.
• The all-time maximum load is $\Omega(\log n / \log\log n)$ with high probability for any two-thinning strategy, and this bound is tight.
• An explicit two-thinning strategy achieves an all-time maximum load of $\Theta(\log n / \log\log n)$ when $m = \Omega(n \log^2 n)$, matching the lower bound.
• For $m = \omega(n\sqrt{\log n})$, the typical maximum load is $O((\log n)^{1/2+o(1)})$ with high probability, matching the optimal single-time load.
• The gap between typical and all-time maximum load arises from unavoidable temporary load spikes during load-rebalancing phases, even when typical load is kept low.
• The strategy maintains $\max_i |L_i^{f}(k)| \leq 100d\log n$ with high probability after $m_0 = \Theta(n\log n)$ steps, ensuring stability and enabling further load control.