Balanced Allocations with the Choice of Noise

Abstract

We consider the allocation of m balls (jobs) into n bins (servers). In the standard Two-Choice process, at each step t =1,2,... ,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 ⩾ 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 ⩾ n the gap is \(\mathcal {O}(g+\log n)\) with high probability. Then through a refined analysis we prove that if g ⩽ log n , then for any m ⩾ n the gap is \(\mathcal {O}(\frac{g}{\log g} \cdot \log \log n)\) . For constant values of g , this generalizes the heavily loaded analysis of [ 19 , 61 ] 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 The analysis also applies to settings with outdated and delayed information.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 [ 18 ] where balls are allocated in consecutive batches of size \(b = n\) , we present an improved and tight gap bound of \(\Theta (\frac{\log n}{\log \log n})\) . 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.