Balls into bins via local search: cover time and maximum load

TL;DR

This paper analyzes a local search process for allocating balls into bins arranged as graph vertices, providing bounds on maximum load and cover time, with tight results for certain graph classes.

Contribution

It introduces a local search allocation process on graphs, establishing bounds for maximum load and cover time, and characterizes their tightness on transitive or homogeneous graphs.

Findings

Upper bounds for maximum load on bounded degree graphs.

Upper bounds for cover time on bounded degree graphs.

Tight bounds for transitive or homogeneous graphs.

Abstract

We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G. Balls arrive one at a time. When a ball arrives, it 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. Then the next ball arrives and this procedure is repeated. For the case m = n, we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m > n.