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arXiv · 2204.04057

The Power of Filling in Balanced Allocations

Dimitrios Los·Thomas Sauerwald·John Sylvester
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Abstract

We introduce a new class of balanced allocation 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 $\mathcal{O}(\log n)$ w.h.p. for any number of balls $m\geq 1$. For the Packing process, we also provide a matching lower bound. Additionally, we prove that the Packing process is sample-efficient in the sense that the expected number of balls allocated per sample is strictly greater than one. Finally, we also demonstrate that the upper bound of $\mathcal{O}(\log n)$ on the gap can be extended to the Memory process studied by Mitzenmacher, Prabhakar and Shah (2002).

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Dimitrios Los, Thomas Sauerwald, John Sylvester. 2022-04-08. The Power of Filling in Balanced Allocations. https://doi.org/10.1137/23m1552231

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