The Power of Filling in Balanced Allocations

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 $m\ge 1$. 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).

This paper refines and extends the content on filling processes in arXiv:2110.10759 (see also arXiv:2308.05087). It consists of 36 pages, 6 figures, 2 tables

References in corpus (2)

• Quasirandom Load Balancing
• Averaging over fast variables in the fluid limit for Markov chains: Application to the supermarket model with memory