Balls into Bins via Local Search

TL;DR

This paper introduces a local search-based process for allocating balls into bins arranged on a graph, demonstrating near-optimal load balancing on expanders, grids, and regular graphs with different tie-breaking rules.

Contribution

It presents a natural local search process for ball allocation on graphs and analyzes its effectiveness across various graph classes, revealing new load bounds.

Findings

Maximum load is ( \, ext{log} \, ext{log} \, n) on expander graphs.

Maximum load on d-dimensional grids is (ig(rac{ ext{log} \, n}{ ext{log} \, ext{log} \, n}ig)^{rac{1}{d+1}}).

Maximum load is constant on almost regular graphs with ( ext{log} \, n) minimum degree.

Abstract

We propose a natural process for allocating n balls into n bins that are organized as the vertices of an undirected graph G. Each ball first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. In our main result, we prove that this process yields a maximum load of only \Theta(\log \log n) on expander graphs. In addition, we show that for d-dimensional grids the maximum load is \Theta\Big(\big(\frac{\log n}{\log \log n}\big)^{\frac{1}{d+1}}\Big). Finally, for almost regular graphs with minimum degree \Omega(\log n), we prove that the maximum load is constant and also reveal a fundamental difference between random and arbitrary tie-breaking rules.