Balancing sums of random vectors

TL;DR

This paper investigates a high-dimensional 'balls-into-bins' problem where random vectors are allocated to bins to keep their sums balanced, with implications for vector partitioning and computer science applications.

Contribution

It introduces a new framework for balancing sums of random vectors in multiple dimensions and analyzes the limits of how close these sums can be kept almost surely.

Findings

Established bounds on the closeness of bin sums in high dimensions.

Connected the problem to classical vector partitioning and computer science applications.

Provided probabilistic methods for nearly optimal balancing strategies.

Abstract

We study a higher-dimensional 'balls-into-bins' problem. An infinite sequence of i.i.d. random vectors is revealed to us one vector at a time, and we are required to partition these vectors into a fixed number of bins in such a way as to keep the sums of the vectors in the different bins close together; how close can we keep these sums almost surely? This question, our primary focus in this paper, is closely related to the classical problem of partitioning a sequence of vectors into balanced subsequences, in addition to having applications to some problems in computer science.