The Power of Filling in Balanced Allocations

TL;DR

This paper introduces a new class of balanced allocation processes characterized by filling underloaded bins, proving that the maximum load gap is logarithmic in the number of bins, and demonstrating the efficiency and extendability of these processes.

Contribution

The paper defines a new class of filling-based balanced allocation processes, proves a logarithmic bound on load imbalance, and extends results to related memory-based processes.

Findings

Maximum load gap is O(log n) with high probability.

Packing process is sample-efficient with expected balls per sample > 1.

The logarithmic gap bound extends to the Memory process.

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).