The power of thinning in balanced allocation

Feldheim, Ohad N.; Gurel-Gurevich, Ori

doi:10.1214/21-ecp400

Cited by 12 publications

(3 citation statements)

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“…Another variant of Two-Choice that has received some attention recently is the family of Two-Thinning processes [9,10], where the ball is allocated to the second sample only if the first one does not meet a certain criterion, e.g., based on a threshold or quantile.…”

Section: Introductionmentioning

confidence: 99%

Balanced Allocation in Batches: The Tower of Two Choices

Los¹,

Sauerwald²

2023

Preprint

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In the balanced allocations framework, the goal is to allocate m balls into n bins, so as to minimize the gap (difference of maximum to average load). The One-Choice process allocates each ball to a randomly sampled bin and achieves w.h.p. a Θ( (m/n) • log n) gap. The Two-Choice process allocates to the least loaded of two randomly sampled bins, and achieves w.h.p. a log 2 log n + Θ(1) gap. Finally, the (1 + β) process mixes between these two processes with probability β ∈ (0, 1), and achieves w.h.p. an Θ(log n/β) gap.We focus on the outdated information setting of [5], where balls are allocated in batches of size b. For almost the entire range b ∈ [1, O(n log n)], it was shown in [18] that Two-Choice achieves w.h.p. the asymptotically optimal gap and for b = Ω(n log n) it was shown in [16] that it achieves w.h.p. a Θ(b/n) gap.In this work, we establish that the (1 + β) process for appropriately chosen β, achieves w.h.p. the asymptotically optimal gap of O( (b/n) • log n) for any b ∈ [2n log n, n 3 ]. This not only proves the surprising phenomenon that allocating greedily based on Two-Choice is not the best, but also that mixing two processes (One-Choice and Two-Choice) leads to a process with a gap that is better than both. Furthermore, the upper bound on the gap applies to a larger family of processes and continues to hold in the presence of weights sampled from distributions with bounded MGFs.

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Section: Introductionmentioning

confidence: 99%

Balanced Allocation in Batches: The Tower of Two Choices

Los¹,

Sauerwald²

2023

Preprint

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“…Another important process is 2-Thinning, which was studied in [8,12]. In this process, each ball first samples a bin uniformly and if its load is below some threshold, the ball is placed into that sample.…”

Section: Introductionmentioning

confidence: 99%

“…Otherwise, the ball is placed into a second bin sample, without inspecting its load. In [8], the authors proved that for m = n, there is a fixed threshold which achieves a gap of O( log n log log n ). This is a significant improvement over One-Choice, but also the total number of samples is greater than one per ball.…”

Section: Introductionmentioning

confidence: 99%

The Power of Filling in Balanced Allocations

Los¹,

Sauerwald²,

Sylvester³

2022

Preprint

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It is well known that if m balls (jobs) are placed sequentially into n bins (servers) according to the One-Choice protocol -choose a single bin in each round and allocate one ball to it -then, for m n, the gap between the maximum and average load diverges. Many refinements of the One-Choice protocol have been studied that achieve a gap that remains bounded by a function of n, for any m. However most of these variations, such as Two-Choice, are less sample-efficient than One-Choice, in the sense that for each allocated ball more than one sample is needed (in expectation).We introduce a new class of processes which are primarily characterized by "filling" underloaded bins. A prototypical example is the Packing process: At each round we only take one bin sample, if the load is below the average load, then we place as many balls until the average load is reached; otherwise, we place only one ball. We prove that for any process in this class the gap between the maximum and average load is O(log n) for any number of balls m. For the Packing process, we also prove a matching lower bound. We also prove that the Packing process is more sample-efficient than One-Choice, that is, it allocates on average more than one ball per sample. Finally, we also demonstrate that the upper bound of O(log n) on the gap can be extended to the Caching process (a.k.a. memory protocol) studied by Mitzenmacher, Prabhakar and Shah (2002).

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