Balanced Allocations with Incomplete Information: The Power of Two Queries
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We consider the problem of allocating 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), one can achieve an O(√(log n/loglog n)) maximum load with one query per chosen bin using an oblivious strategy, as shown by Feldheim and Gurel-Gurevich (2018). For the case m=Ω(n), the authors conjectured that the same strategy achieves a maximum load of m/n+O(√(log n/loglog n)). In this work, we disprove this conjecture by showing a lower bound of m/n+Ω( √(log n)) for a fixed m=Θ(n √(log n)), and a lower bound of m/n+Ω(log n/loglog n) for some m depending on the used strategy. Surprisingly, these lower bounds hold even for any adaptive strategy with one query, i.e., queries may depend on the full history of the process. 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(√(log n)) whp for any m ≥ 1. For any k=O(loglog n) binary queries, the upper bound on the maximum load improves to m/n+O(k(log n)^1/k) whp for any m ≥ 1. Hence for k=Θ(loglog n), we recover the two-choice result up to a constant multiplicative factor, including the heavily loaded case where m=Ω(n). One novel aspect of our proof techniques is the use of multiple super-exponential potential functions, which might be of use in future work.
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