The Power of Filling in Balanced Allocations
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It is well known that if m balls (jobs) are placed sequentially into n bins (servers) according to the One-Choice protocol - choose a single bin in each round and allocate one ball to it - then, for m ≫ n, the gap between the maximum and average load diverges. Many refinements of the One-Choice protocol have been studied that achieve a gap that remains bounded by a function of n, for any m. However most of these variations, such as Two-Choice, are less sample-efficient than One-Choice, in the sense that for each allocated ball more than one sample is needed (in expectation). We introduce a new class of processes which are primarily characterized by "filling" underloaded bins. A prototypical example is the Packing process: At each round we only take one bin sample, if the load is below the average load, then we place as many balls until the average load is reached; otherwise, we place only one ball. We prove that for any process in this class the gap between the maximum and average load is 𝒪(log n) for any number of balls m. For the Packing process, we also prove a matching lower bound. We also prove that the Packing process is more sample-efficient than One-Choice, that is, it allocates on average more than one ball per sample. Finally, we also demonstrate that the upper bound of 𝒪(log n) on the gap can be extended to the Caching process (a.k.a. memory protocol) studied by Mitzenmacher, Prabhakar and Shah (2002).
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