Derandomized Balanced Allocation

In this paper, we study the maximum loads of explicit hash families in the d-choice schemes when allocating sequentially n balls into n bins. We consider the Uniform-Greedy scheme [ABKU99], which provides d independent bins for each ball and places the ball into the bin with the least load, and its non-uniform variant — the Always-Go-Left scheme introduced by Vöcking [Vöc03]. We construct a hash family withO(log n log log n) random bits based on the previous work of Celis et al. [CRSW13] and show the following results. 1. This hash family has a maximum load of log logn log d +O(1) in the Uniform-Greedy scheme. 2. It has a maximum load of log logn d log φd + O(1) in the Always-Go-Left scheme for a constant φd > 1.61. The maximum loads of our hash family match the maximum loads of a perfectly random hash function [ABKU99, Vöc03] in the Uniform-Greedy and Always-Go-Left scheme separately. Previously, the best known hash families that guarantee the same maximum loads as a perfectly random hash function in any of these schemes were O(log n)-wise independent functions [Vöc03], which needs Θ(log n) random bits. ∗Supported by NSF Grant CCF-1526952 and a Simons Investigator Award (#409864, David Zuckerman), part of this work was done while the author was visiting the Simons Institute. ar X iv :1 70 2. 03 37 5v 2 [ cs .D S] 2 4 Ja n 20 18