Balanced Allocations with Incomplete Information: The Power of Two Queries

We consider the allocation of $m$ balls into $n$ bins with incomplete information. In the classical Two-Choice process a ball first queries the load of two randomly chosen bins and is then placed in the least loaded bin. In our setting, each ball also samples two random bins but can only estimate a bin's load by sending binary queries of the form "Is the load at least the median?" or "Is the load at least 100?". For the lightly loaded case $m=O(n)$, Feldheim and Gurel-Gurevich (2021) showed that with one query it is possible to achieve a maximum load of $O(\sqrt{\log n/\log \log n})$, and posed the question whether a maximum load of $m/n+O(\sqrt{\log n/\log \log n})$ is possible for any $m = \Omega(n)$. In this work, we resolve this open problem by proving a lower bound of $m/n+\Omega( \sqrt{\log n})$ for a fixed $m=\Theta(n \sqrt{\log n})$, and a lower bound of $m/n+\Omega(\log n/\log \log n)$ for some $m$ depending on the used strategy. We complement this negative result by proving a positive result for multiple queries. In particular, we show that with only two binary queries per chosen bin, there is an oblivious strategy which ensures a maximum load of $m/n+O(\sqrt{\log n})$ for any $m \geq 1$. Further, for any number of $k = O(\log \log n)$ binary queries, the upper bound on the maximum load improves to $m/n + O(k(\log n)^{1/k})$ for any $m \geq 1$. Further, this result for $k$ queries implies (i) new bounds for the $(1+\beta)$-process introduced by Peres et al (2015), (ii) new bounds for the graphical balanced allocation process on dense expander graphs, and (iii) the bound of $m/n+O(\log \log n)$ on the maximum load achieved by the Two-Choice process, including the heavily loaded case $m=\Omega(n)$ derived by Berenbrink et al. (2006). One novel aspect of our proofs is the use of multiple super-exponential potential functions, which might be of use in future work.