Chains-into-Bins Processes

TL;DR

This paper analyzes the maximum load in a chains-into-bins process where chains of balls are allocated to consecutive bins, showing that the maximum load scales with the double logarithm of the number of chains, depending on the number of choices.

Contribution

It introduces and analyzes the chains-into-bins model, extending classical balls-into-bins problems to chains with consecutive bin constraints and multiple choices.

Findings

Maximum load is approximately (ln ln m)/ln d for d ≥ 2.

Probability of exceeding this load decreases rapidly with m.

Results hold when total objects times chain length is proportional to n.

Abstract

The study of {\em balls-into-bins processes} or {\em occupancy problems} has a long history. These processes can be used to translate realistic problems into mathematical ones in a natural way. In general, the goal of a balls-into-bins process is to allocate a set of independent objects (tasks, jobs, balls) to a set of resources (servers, bins, urns) and, thereby, to minimize the maximum load. In this paper, we analyze the maximum load for the {\em chains-into-bins} problem, which is defined as follows. There are $n$ bins, and $m$ objects to be allocated. Each object consists of balls connected into a chain of length $\ell$, so that there are $m \ell$ balls in total. We assume the chains cannot be broken, and that the balls in one chain have to be allocated to $\ell$ consecutive bins. We allow each chain $d$ independent and uniformly random bin choices for its starting position. The chain is allocated using the rule that the maximum load of any bin receiving a ball of that chain is minimized. We show that, for $d \ge 2$ and $m\cdot\ell=O(n)$, the maximum load is $((\ln \ln m)/\ln d) +O(1)$ with probability $1-\tilde O(1/m^{d-1})$.