# Balanced Allocation on Graphs: A Random Walk Approach

## Abstract

The standard balls-into-bins model is a process which randomly allocates *m* balls into *n* bins where each ball picks *d* bins independently and uniformly at random and the ball is then allocated in a least loaded bin in the set of *d* choices. When \(m=n\) and \(d=1\), it is well known that at the end of process the maximum number of balls at any bin, the *maximum load*, is \((1+o(1))\frac{\log n}{\log \log n}\) with high probability (With high probability refers to an event that holds with probability \(1-1/n^c\), where *c* is a constant. For simplicity, we sometimes abbreviate it as whp). Azar et al. [3] showed that for the *d*-choice process, \(d\geqslant 2\), provided ties are broken randomly, the maximum load is \(\frac{\log \log n}{\log d}+\mathcal {O}(1)\).

In this paper we propose algorithms for allocating *n* sequential balls into *n* bins that are interconnected as a *d*-regular *n*-vertex graph *G*, where \(d\geqslant 3\) can be any integer. Let *l* be a given positive integer. In each round *t*, \(1\leqslant t\leqslant n\), ball *t* picks a node of *G* uniformly at random and performs a non-backtracking random walk of length *l* from the chosen node. Then it allocates itself on one of the visited nodes with minimum load (ties are broken uniformly at random). Suppose that *G* has a sufficiently large girth and \(d=\omega (\log n)\). Then we establish an upper bound for the maximum number of balls at any bin after allocating *n* balls by the algorithm, called *maximum load*, in terms of *l* with high probability. We also show that the upper bound is at most an \(\mathcal {O}(\log \log n)\) factor above the lower bound that is proved for the algorithm. In particular, we show that if we set \(l=\lfloor (\log n)^{\frac{1+\epsilon }{2}}\rfloor \), for every constant \(\epsilon \in (0, 1)\), and *G* has girth at least \(\omega (l)\), then the maximum load attained by the algorithm is bounded by \(\mathcal {O}(1/\epsilon )\) with high probability. Finally, we slightly modify the algorithm to have similar results for balanced allocation on *d*-regular graph with \(d\in [3, \mathcal {O}(\log n)]\) and sufficiently large girth.

## Keywords

Local Search Maximum Load Allocation Algorithm Minimum Load Expander Graph## Notes

### Acknowledgment

The author wants to thank Thomas Sauerwald for introducing the problem and several helpful discussions.

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