A Note On the Size of Largest Bins Using Placement With Linear Transformations

TL;DR

This paper improves the upper bound on the expected size of the largest bin when placing n balls into n bins using linear transformations over Z 2, reducing it from O(log n log log n) to O(log n).

Contribution

It provides a tighter analysis for the case of n balls into n bins, improving the known upper bound on the largest bin size.

Findings

Expected largest bin size is O(log n) for n balls into n bins.

The analysis tightens previous bounds by applying similar techniques with improved analysis.

The result applies to placement via linear transformations over Z 2.

Abstract

We study the placement of n balls into n bins where balls and bins are represented as two vector spaces over Z 2 . The placement is done according to a linear transformation between the two vector spaces. We analyze the expected size of a largest bin. The only currently known upper bound is O(log n log log n) by Alon et al. and holds for placing n log n balls into n bins. We show that this bound can be improved to O(log n) in the case when n balls are placed into n bins. We use the same basic technique as Alon et al. but give a tighter analysis for this case.