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• W3181326093 abstract "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 $textit{two}$ 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 $textit{binary queries}$ 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 $O(sqrt{log n/log log n})$ maximum load with one query per chosen bin using an oblivious strategy, as shown by Feldheim and Gurel-Gurevich (2018). For the case $m=Omega(n)$, the authors conjectured that the same strategy achieves a maximum load of $m/n+O(sqrt{log n/log log n})$. In this work, we disprove this conjecture by showing a lower bound of $m/n+Omega( sqrt{log n})$ for a fixed $m=Theta(n sqrt{log n})$, and a lower bound of $m/n+Omega(log n/loglog n)$ for some $m$ depending on the used strategy. Surprisingly, these lower bounds hold even for any $textit{adaptive strategy}$ 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 $m/n+O(sqrt{log n})$ whp for any $m geq 1$. For any $k=O(log log n)$ binary queries, the upper bound on the maximum load improves to $m/n+O(k(log n)^{1/k})$ whp for any $m geq 1$. Hence for $k=Theta(loglog n)$, we recover the two-choice result up to a constant multiplicative factor, including the heavily loaded case where $m=Omega(n)$. One novel aspect of our proof techniques is the use of multiple super-exponential potential functions, which might be of use in future work." @default.
• W3181326093 created "2021-07-19" @default.
• W3181326093 creator A5017390373 @default.
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• W3181326093 date "2021-07-08" @default.
• W3181326093 modified "2023-09-27" @default.
• W3181326093 title "Balanced Allocations with Incomplete Information: The Power of Two Queries." @default.
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