Approximated Two Choices in Randomized Load Balancing

Abstract

This paper studies the maximum load in the approximated d-choice balls-and-bins game where the current load of each bin is available only approximately. In the model of this game, we have r thresholds T ₁,...,T _r (0 < T ₁ < ... < T _r ) for an integer r (≥ 1). For each ball, we select d bins and put the ball into the bin of the lowest range, i.e., the bin of load i such that T _k ≤ i ≤ T _k + 1–1 and no other selected bin has height less than T _k . If there are two or more bins in the lowest range (i.e., their height is between T _k and T _k + 1–1), then we assume that those bins cannot be distinguished and so one of them is selected uniformly at random. We then estimate the maximum load for n balls and n bins in this game. In particular, when we put the r thresholds at a regular interval of an appropriate Δ, i.e., T _r  − T _r − 1 = ...T ₂ − T ₁ = T ₁ = Δ, the maximum load L(r) is given as $(r+O(1))\sqrt[r+1]{\frac{r+1}{(d-1)^{r}}{\rm ln}{\it n}/{\rm ln}(\frac{r+1}{(d-1)^{r}}{\rm ln}{\it n})}$ The bound is also described as L(Δ) ≤ {(1 + o(1))ln ln n + O(1)}Δ/ln ((d – 1)Δ) using parameter Δ. Thus, if Δ is a constant, this bound matches the (tight) bound in the original d-choice model given by Azar et al., within a constant factor. The bound is also tight within a constant factor when r = 1.