Tight & Simple Load Balancing

Abstract

We consider the following load balancing process for m tokens distributed arbitrarily among n nodes connected by a complete graph. In each time step a pair of nodes is selected uniformly at random. Let ℓ_1 and ℓ_2 be their respective number of tokens. The two nodes exchange tokens such that they have ⌈(ℓ_1 + ℓ_2)/2⌉ and ⌈(ℓ_1 + ℓ_2)/2⌉ tokens, respectively. We provide a simple analysis showing that this process reaches almost perfect balance within O(n log n + n log Δ) steps with high probability, where Δ is the maximal initial load difference between any two nodes. This bound is asymptotically tight.