论文标题
平衡分配与噪音的选择
Balanced Allocations with the Choice of Noise
论文作者
论文摘要
我们考虑将$ M $ Balls（Jobs）分配到$ n $ bins（服务器）中。在标准的两项选择过程中，在每个步骤中，$ t = 1,2，\ ldots，m $我们首先采样两个随机选择的垃圾箱，比较它们的两个负载，然后将球放入最少的垃圾箱中。众所周知，对于任何$ m \ geq n $，这会导致$ \ log_2 \ log_2 \ log n+θ（1）$（具有高概率）的差距（最大负载和平均负载之间的差异）。 在这项工作中，我们考虑在不同的设置中进行嘈杂的负载比较。一个关键设置涉及一个自适应对手，其功率受\ mathbb {n} $中的某种阈值$ g \的限制。在每个步骤中，此类对手都可以确定两个垃圾箱之间的任何负载比较的结果，其负载的负载最多差异为$ g $，而如果负载差大于$ g $，则比较正确。 对于这种对抗性设置，我们首先证明，对于任何$ m \ geq n $，差距为$ O（g+\ log n）$，具有很高的概率。然后，通过精致的分析，我们证明，如果$ g \ leq \ log n $，那么对于任何$ m \ geq n $，差距为$ o（\ frac {g} {\ log g} \ cdot \ cdot \ log \ log \ log \ log n）$。对于$ g $的恒定值，这将在两项选择过程中概括[BCSV06，TW14]的大量加载分析，并确定即使“类似加载”的垃圾箱之间的负载比较是错误的，即使负载比较是错误的。最后，我们通过紧密的下限对这些上限进行补充，这对参数$ g $如何影响差距建立了有趣的相变。 该分析还适用于具有过时和延迟信息的设置。例如，对于[BCEFN12]的设置，将球分配为$ b = n $的连续批次，我们提出了$θ（\ frac {\ log n} {\ log log \ log \ log \ log \ log \ log n}）的改进而紧密的间隙结合。该界限还扩展了一系列$ b $的范围，并适用于轻松的设置，其中报告的垃圾箱的负载可以是最后$ b $步骤的任何负载值。
We consider the allocation of $m$ balls (jobs) into $n$ bins (servers). In the standard Two-Choice process, at each step $t=1,2,\ldots,m$ we first sample two randomly chosen bins, compare their two loads and then place a ball in the least loaded bin. It is well-known that for any $m\geq n$, this results in a gap (difference between the maximum and average load) of $\log_2\log n+Θ(1)$ (with high probability). In this work, we consider Two-Choice in different settings with noisy load comparisons. One key setting involves an adaptive adversary whose power is limited by some threshold $g\in\mathbb{N}$. In each step, such adversary can determine the result of any load comparison between two bins whose loads differ by at most $g$, while if the load difference is greater than $g$, the comparison is correct. For this adversarial setting, we first prove that for any $m \geq n$ the gap is $O(g+\log n)$ with high probability. Then through a refined analysis we prove that if $g\leq\log n$, then for any $m \geq n$ the gap is $O(\frac{g}{\log g}\cdot\log\log n)$. For constant values of $g$, this generalizes the heavily loaded analysis of [BCSV06, TW14] for the Two-Choice process, and establishes that asymptotically the same gap bound holds even if load comparisons among "similarly loaded" bins are wrong. Finally, we complement these upper bounds with tight lower bounds, which establish an interesting phase transition on how the parameter $g$ impacts the gap. The analysis also applies to settings with outdated and delayed information. For example, for the setting of [BCEFN12] where balls are allocated in consecutive batches of size $b=n$, we present an improved and tight gap bound of $Θ(\frac{\log n}{\log\log n})$. This bound also extends for a range of values of $b$ and applies to a relaxed setting where the reported load of a bin can be any load value from the last $b$ steps.
