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Balanced Allocation on Graphs: A Random Walk Approach

  • Ali PourmiriEmail author
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

Local Search Maximum Load Allocation Algorithm Minimum Load Expander Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgment

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

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

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