## Perfectly balanced allocation (2003)

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Venue: | in Proceedings of the 7th International Workshop on Randomization and Approximation Techniques in Computer Science, Princeton, NJ, 2003, Lecture Notes in Comput. Sci. 2764 |

Citations: | 17 - 1 self |

### BibTeX

@INPROCEEDINGS{Czumaj03perfectlybalanced,

author = {Artur Czumaj and Chris Riley and Christian Scheideler},

title = {Perfectly balanced allocation},

booktitle = {in Proceedings of the 7th International Workshop on Randomization and Approximation Techniques in Computer Science, Princeton, NJ, 2003, Lecture Notes in Comput. Sci. 2764},

year = {2003},

pages = {240--251},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

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### Abstract

Abstract. We investigate randomized processes underlying load balancing based on the multiple-choice paradigm: m balls have to be placed in n bins, and each ball can be placed into one out of 2 randomly selected bins. The aim is to distribute the balls as evenly as possible among the bins. Previously, it was known that a simple process that places the balls one by one in the least loaded bin can achieve a maximum load of m/n + Θ(log log n) with high probability. Furthermore, it was known that it is possible to achieve (with high probability) a maximum load of at most ⌈m/n ⌉ +1using maximum flow computations. In this paper, we extend these results in several aspects. First of all, we show that if m ≥ cn log n for some sufficiently large c, thenaperfect distribution of balls among the bins can be achieved (i.e., the maximum load is ⌈m/n⌉) with high probability. The bound for m is essentially optimal, because it is known that if m ≤ c ′ n log n for some sufficiently small constant c ′ , the best possible maximum load that can be achieved is ⌈m/n ⌉ +1with high probability. Next, we analyze a simple, randomized load balancing process based on a local search paradigm. Our first result here is that this process always converges to a best possible load distribution. Then, we study the convergence speed of the process. We show that if m is sufficiently large compared to n,thenno matter with which ball distribution the system starts, if the imbalance is ∆, then the process needs only ∆·n O(1) steps to reach a perfect distribution, with high probability. We also prove a similar result for m ≈ n, and show that if m = O(n log n / log log n), then an optimal load distribution (which has the maximum load of ⌈m/n ⌉ +1) is reached by the random process after a polynomial number of steps, with high probability.

### Citations

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(Show Context)
Citation Context ...l number of steps, with high probability. ∗ Research supported in part by NSF grant CCR-0105701.s1 Introduction The study of balls-into-bins games or occupancy problems has a long history (see, e.g., =-=[1, 2, 3, 4, 5, 9, 11, 13, 14, 20]-=-). These problems have numerous applications, e.g., in graph theory, queueing theory, hashing, and randomized rounding. In general, the goal of a balls-and-bins algorithm is to assign a set of indepen... |

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Citation Context ...q 2 ) ≥ q · m] ≤ exp(−2 q 2 (1 − q) 2 m). 3. For any 0 < q < 1, Pr[B(m, q 2 ) ≥ q · m] ≤ (q · (1 + 1/q) 1−q ) m . Proof : The first bound is a well known consequence of the Chernoff bound (see, e.g., =-=[7]-=-). The second inequality follows easily from the following well-known form of the Chernoff bound (see, e.g., [12, Lemma 2.3]): Let X1, . . .,Xm be identical independent 0–1 random variables. Let X = �... |

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Citation Context ...l number of steps, with high probability. ∗ Research supported in part by NSF grant CCR-0105701.s1 Introduction The study of balls-into-bins games or occupancy problems has a long history (see, e.g., =-=[1, 2, 3, 4, 5, 9, 11, 13, 14, 20]-=-). These problems have numerous applications, e.g., in graph theory, queueing theory, hashing, and randomized rounding. In general, the goal of a balls-and-bins algorithm is to assign a set of indepen... |

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Citation Context ...l number of steps, with high probability. ∗ Research supported in part by NSF grant CCR-0105701.s1 Introduction The study of balls-into-bins games or occupancy problems has a long history (see, e.g., =-=[1, 2, 3, 4, 5, 9, 11, 13, 14, 20]-=-). These problems have numerous applications, e.g., in graph theory, queueing theory, hashing, and randomized rounding. In general, the goal of a balls-and-bins algorithm is to assign a set of indepen... |

83 | How asymmetry helps load balancing
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Citation Context |

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Citation Context ...s of all balls are known to the algorithm (off-line case). This problem arises naturally in numerous applications, for example, in hashing, scheduling, load balancing, and video on demand (see, e.g., =-=[1, 8, 10, 16, 17, 18]-=-). (For example, Sanders et al. [18] discussed in depth applications to support fast parallel access to external memory systems with parallel disks and Karp [8] discussed applications in video on dema... |

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Citation Context ...s exactly ⌈m/n⌉ with high probability. The Self-Balancing Algorithm is a simple example of a local search algorithm, similar to load balancing algorithms existing in the literature before, see, e.g., =-=[6, 15]-=-. Theorem 3 shows the nontrivial property that no matter with which state (i.e., assignment of balls to bins) the Self-Balancing Algorithm starts, it will always converge to a state in which the maxim... |

47 | Local divergence of Markov chains and the analysis of iterative load-balancing schemes
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- 1999
(Show Context)
Citation Context ...s exactly ⌈m/n⌉ with high probability. The Self-Balancing Algorithm is a simple example of a local search algorithm, similar to load balancing algorithms existing in the literature before, see, e.g., =-=[6, 15]-=-. Theorem 3 shows the nontrivial property that no matter with which state (i.e., assignment of balls to bins) the Self-Balancing Algorithm starts, it will always converge to a state in which the maxim... |

43 |
Load Balancing and Density Dependent Jump Markov
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Citation Context |

30 |
Random duplicate assignment: An alternative to striping in video servers
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Citation Context ...s of all balls are known to the algorithm (off-line case). This problem arises naturally in numerous applications, for example, in hashing, scheduling, load balancing, and video on demand (see, e.g., =-=[1, 8, 10, 16, 17, 18]-=-). (For example, Sanders et al. [18] discussed in depth applications to support fast parallel access to external memory systems with parallel disks and Karp [8] discussed applications in video on dema... |

20 | E.: On balls and bins with deletions
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Citation Context |

18 | Concentration - McDiarmid - 1998 |

16 | Reconciling simplicity and realism in parallel disk models
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Citation Context ...s of all balls are known to the algorithm (off-line case). This problem arises naturally in numerous applications, for example, in hashing, scheduling, load balancing, and video on demand (see, e.g., =-=[1, 8, 10, 16, 17, 18]-=-). (For example, Sanders et al. [18] discussed in depth applications to support fast parallel access to external memory systems with parallel disks and Karp [8] discussed applications in video on dema... |

15 | Randomized protocols for low-congestion circuit routing in multistage interconnection networks
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Citation Context |

13 | Asynchronous scheduling of redundant disk arrays
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12 | Reducing network congestion and blocking probability through balanced allocation
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9 |
Randomized allocation processes.” Random Structures and Algorithms 18(2001):297–331
- Czumaj, Stemann
(Show Context)
Citation Context |

7 | A new algorithm for the recognition of series parallel graphs - Schoenmakers - 1995 |

3 |
Random graphs, random walks, differential equations and the probabilistic analysis of algorithms
- Karp
- 1998
(Show Context)
Citation Context |

1 |
Balanced allocations: The heavily loaded case. STOC
- Berenbrink, Czumaj, et al.
- 2000
(Show Context)
Citation Context ...y. (We say that an event A occurs with high probability (w.h.p.) if Pr[A] ≥ 1 −n −α for an arbitrarily chosen constant α ≥ 1.) On the other hand, it was shown by Azar et al. [1] and Berenbrink et al. =-=[2]-=- that if the balls are placed in a sequential (on-line) fashion and each ball is assigned to the currently least loaded of the two locations (ties broken arbitrarily), then the maximum load of any bin... |

1 |
On balls and bins with deletions. RANDOM
- Cole, Frieze, et al.
- 1998
(Show Context)
Citation Context ...and hence by Lemma 2 (1) and by the inequality � � n k k ≤ n , we get (provided c is a large enough constant): � n k Pr[EU] ≤ 2 −t = 2 −m k/n ≤ e −(γ+1)ln n−k ln n = 1 nγ+1 ≤ · nk 1 n γ+1 · � n k � . =-=(3)-=- �Next, we consider 0.1 n ≤ k < 2/3 n. Then, by Lemma 2 (2) and by observing that n ≤ 2 , we have (again, if we set m = c n lnn for a large enough constant c) k (n−k) Pr[EU] ≤ exp(−2( n2 ) 2 m) ≤ e −0... |

1 |
How asymetry helps load balancing. FOCS
- Vöcking
- 1999
(Show Context)
Citation Context ...s, with high probability. Keywords: load balancing, local search algorithms, stochastic processes. 1 Introduction The study of balls-into-bins games or occupancy problems has a long history (see e.g. =-=[1,2,3,4,5,8,10,11,12,18]-=-). These problems have numerous applications, e.g., in graph theory, queueing theory, hashing, and randomized rounding. In general, the goal of a ⋆ Research supported in part by NSF grant CCR-0105701.... |