Balanced Allocations with the Choice of Noise

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 $\log_2 \log n + \Theta(1)$ (with high probability). In this work, we consider Two-Choice in different models with noisy load comparisons. One key model involves an adaptive adversary whose power is limited by some threshold $g \in \mathbb{N}$. In each round, 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 model, 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 many (or possibly all) load comparisons among "similarly loaded" bins are wrong. Finally, we complement these upper bounds with tight lower bounds, which establishes an interesting phase transition on how the parameter $g$ impacts the gap. We also apply a similar analysis to other noise models, including ones where bins only update their load information with delay. For example, for the model of [BCEFN12] where balls are allocated in consecutive batches of size $n$, we present an improved and tight gap bound of $\Theta(\log n/ \log \log n )$.