COCOON 2016: Computing and Combinatorics pp 330-341

# Balanced Allocation on Graphs: A Random Walk Approach

• Ali Pourmiri
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)

## 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

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## Notes

### Acknowledgment

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

## References

1. 1.
Alon, N., Benjamini, I., Lubetzky, E., Sodin, S.: Non-backtracking random walks mix faster. Commun. Contemp. Math. 9, 585–603 (2007)
2. 2.
Alon, N., Lubetzky, E.: Poisson approximation for non-backtracking random walks. Israel J. Math. 174(1), 227–252 (2009)
3. 3.
Azar, Y., Broder, A.Z., Karlin, A.R., Upfal, E.: Balanced allocations. SIAM J. Comput. 29(1), 180–200 (1999)
4. 4.
Berenbrink, P., Brinkmann, A., Friedetzky, T., Nagel, L.: Balls into bins with related random choices. J. Parallel Distrib. Comput. 72(2), 246–253 (2012)
5. 5.
Bogdan, P., Sauerwald, T., Stauffer, A., Sun, H.: Balls into bins via local search. In: Proceedings of the 24th Symposium Discrete Algorithms (SODA), pp. 16–34 (2013)Google Scholar
6. 6.
Byers, J.W., Considine, J., Mitzenmacher, M.: Geometric generalizations of the power of two choices. In: Proceedings of the 16th Symposium Parallelism in Algorithms and Architectures (SPAA), pp. 54–63 (2004)Google Scholar
7. 7.
Cooper, C., Frieze, A.M., Radzik, T.: Multiple random walks in random regular graphs. SIAM J. Discrete Math. 23(4), 1738–1761 (2009)
8. 8.
Dahan, X.: Regular graphs of large girth and arbitrary degree. Combinatorica 34(4), 407–426 (2014)
9. 9.
Godfrey, B.: Balls, bins with structure: balanced allocations on hypergraphs. In: Proceedings of the 19th Symposium Discrete Algorithms (SODA), pp. 511–517 (2008)Google Scholar
10. 10.
Kenthapadi, K., Panigrahy, R.: Balanced allocation on graphs. In: Proceedings of the 17th Symposium Discrete Algorithms (SODA), pp. 434–443 (2006)Google Scholar
11. 11.
Mitzenmacher, M., Richa, A.W., Sitaraman, R.: The power of two random choices: a survey of technique and results. Handb. Randomized Comput. 1, 255–312 (2001)
12. 12.
Peres, Y., Talwar, K., Wieder, U.: Graphical balanced allocations and the $$(1 + \beta )$$-choice process. Random Struct. Algorithms (2014). doi: Google Scholar
13. 13.
Vöcking, B.: How asymmetry helps load balancing. J. ACM 50(4), 568–589 (2003)

© Springer International Publishing Switzerland 2016

## Authors and Affiliations

1. 1.School of Computer ScienceInstitute for Research in Fundamental Sciences (IPM)TehranIran