The 1-2-3-Toolkit for Building Your Own Balls-into-Bins Algorithm

In this work, we examine a generic class of simple distributed balls-into-bins algorithms. Exploiting the strong concentration bounds that apply to balls-into-bins games, we provide an iterative method to compute accurate estimates of the remaining balls and the load distribution after each round. Each algorithm is classified by (i) the load that bins accept in a given round, (ii) the number of messages each ball sends in a given round, and (iii) whether each such message is given a rank expressing the sender's inclination to commit to the receiving bin (if feasible). This novel ranking mechanism results in notable improvements, in particular in the number of balls that may commit to a bin in the first round of the algorithm. Simulations independently verify the correctness of the results and confirm that our approximation is highly accurate even for a moderate number of $10^6$ balls and bins.