Balanced Allocation Through Random Walk

TL;DR

This paper analyzes a balanced allocation method using random walks, demonstrating that under certain conditions, the expected insertion time remains constant even with high load, improving understanding of load balancing efficiency.

Contribution

It introduces a random walk-based allocation procedure and proves constant expected insertion time under specific load conditions, advancing load balancing techniques.

Findings

Expected insertion time is constant under specified load conditions.

The method effectively balances load with high probability.

The analysis applies to large-scale distributed systems.

Abstract

We consider the allocation problem in which $m \leq (1-\epsilon) dn $ items are to be allocated to $n$ bins with capacity $d$. The items $x_1,x_2,\ldots,x_m$ arrive sequentially and when item $x_i$ arrives it is given two possible bin locations $p_i=h_1(x_i),q_i=h_2(x_i)$ via hash functions $h_1,h_2$. We consider a random walk procedure for inserting items and show that the expected time insertion time is constant provided $\epsilon = \Omega\left(\sqrt{ \frac{ \log d}{d}} \right).$