Balanced Allocation on Graphs with Random Walk Based Sampling

A well-known randomized algorithm to allocate n balls into n bins is the power-of-d choices policy: For each ball, d bins are sampled uniformly at random, and the ball is allocated to the least loaded bin. A classical result is that the maximum load under this policy is log log $n/\log d+O(1)$ with high probability. Many subsequent works have considered alternative ways to sample d bins.This paper considers two new policies to allocate n balls into n bins. Assuming that the bins are associated with vertices of a k-regular graph, the d sampled bins are given by d nonbacktracking random walk processes defined on the graph, where the positions of the random walkers can be reset to independent random positions when certain events happens. We show that, under certain assumptions of the graph, both schemes can achieve the same performance as power-of-d choices, i.e., the maximum load is bounded by log log $n/\log d+O(1)$ with high probability. Both policies can be considered as derandomized versions of power-of-d choices.