## Parallel Randomized Load Balancing (1995)

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Venue: | In Symposium on Theory of Computing. ACM |

Citations: | 56 - 8 self |

### BibTeX

@INPROCEEDINGS{Adler95parallelrandomized,

author = {Micah Adler and Soumen Chakrabarti and Michael Mitzenmacher and Lars Rasmussen},

title = {Parallel Randomized Load Balancing},

booktitle = {In Symposium on Theory of Computing. ACM},

year = {1995},

pages = {119--130}

}

### Years of Citing Articles

### OpenURL

### Abstract

It is well known that after placing n balls independently and uniformly at random into n bins, the fullest bin holds \Theta(log n= log log n) balls with high probability. Recently, Azar et al. analyzed the following: randomly choose d bins for each ball, and then sequentially place each ball in the least full of its chosen bins [2]. They show that the fullest bin contains only log log n= log d + \Theta(1) balls with high probability. We explore extensions of this result to parallel and distributed settings. Our results focus on the tradeoff between the amount of communication and the final load. Given r rounds of communication, we provide lower bounds on the maximum load of \Omega\Gamma r p log n= log log n) for a wide class of strategies. Our results extend to the case where the number of rounds is allowed to grow with n. We then demonstrate parallelizations of the sequential strategy presented in Azar et al. that achieve loads within a constant factor of the lower bound for two ...

### Citations

1789 | Random Graphs
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Citation Context ...are the children of v 1 , and so on, and let X = P #v X #v . We show that X > 0 with at least constant probability by determining the expectation and variance of X and applying the simple bound (from =-=[5-=-], equation (3) of I.1): Pr(X = 0) # 1 - E[X] 2 E[X 2 ] . We first compute E[X]. The multinomial coe#cient n 1;T ;T -1;...;T -1 , where (T - 1) occurs T times and the sum of the terms on the second l... |

249 | Balanced allocations
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- 1999
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Citation Context ...ith high probability. More recently, Azar et al. analyzed the following process: randomly choose d bins for each ball, and then place the balls, one by one, into the least full bin from its d choices =-=[2]-=-. They show that after all n balls have been placed, the fullest bin contains only log log n/ log d + #(1) balls with high probability. We explore extensions of this result to parallel and distributed... |

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Citation Context ... i=1 Pr(Yi =1| X1,...,Xi−1) ≥ p, n� Yi ≤ k) ≤ Pr(B(n, p) ≤ k). i=1 The second lemma presents some useful Chernoff-type bounds that will be used frequently throughout the paper; proofs may be found in =-=[11]-=-. Lemma 2 If Xi (1 ≤ i ≤ n) are independent binary random variables such that Pr[Xi =1]=p, then the following hold: � n� � � �t np For t ≥ np, Pr Xi ≥ t ≤ e t i=1 t−np . (1) � n� � For t ≤ np, Pr Xi ≤... |

199 | The power of two choices in randomized load balancing
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- 2001
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Citation Context ...ion asynchronously); he also analyzes the case where the number of balls m di#ers from the number of bins n for his algorithm, as well as considers other similar load balancing problems. Mitzenmacher =-=[18, 19]-=- studies load balancing using multiple choices in dynamic settings related to queueing networks. Czumaj and Stemann [8] provide a general framework that extends the model and the results of Azar, Brod... |

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144 |
Urn Models and Their Applications
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- 1977
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Citation Context .../n)) the maximum number of balls received by any bin is #( log n log log n ). (In this paper log is used for log 2 .) Occupancy results such as this have a long history in the mathematical literature =-=[12, 15]-=- with numerous applications in hashing [2, 7, 14], PRAM simulation [17, 9, 14, 16, 20] and load balancing [2, 7]. # A preliminary version of this work appeared in the Proceedings of the Twenty-Seventh... |

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Citation Context ...tween these two models has been observed and made use of previously, and tighter bounds on specific problems can often be obtained with more detailed analyses; see, for example [3, Chapter 6],[6], or =-=[13]-=-. Apart from enabling us to prove our bounds, the tool may be of independent interest. In Section 4 we describe an asynchronous parallelization of greedy for two rounds that matches the lower bound to... |

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Citation Context ...nship between these two models has been observed and made use of previously, and tighter bounds on specific problems can often be obtained with more detailed analyses; see, for example [3, Chapter 6],=-=[6]-=-, or [13]. Apart from enabling us to prove our bounds, the tool may be of independent interest. In Section 4 we describe an asynchronous parallelization of greedy for two rounds that matches the lower... |

85 |
auf der Heide, Efficient PRAM Simulation on a Distributed Memory Machine
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- 1994
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Citation Context ...log n any bin is Θ( log log n ). (In this paper log is used for log2.) Occupancy results such as this have a long history in the mathematical literature [12, 15] with numerous applications in hashing =-=[2, 7, 14]-=-, PRAM simulation [17, 9, 14, 16, 20] and load balancing [2, 7]. ∗ A preliminary version of this work appeared in the Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing,... |

78 |
Expected length of the longest probe sequence in hash code searching
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- 1981
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Citation Context ...nd bins scenario to an edge orientation problem on random graphs. The rest of the paper deals with upper bounds. Our analysis exploits a basic, general tool that we derive, based on results of Gonnet =-=[10]-=-, relating the distribution of the number of balls that land in a bin when balls are thrown independently and uniformly at random and the distribution of Poisson random variables. We note that the clo... |

46 |
Multilevel adaptive hashing
- Broder, Karlin
- 1990
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Citation Context ...y bin is #( log n log log n ). (In this paper log is used for log 2 .) Occupancy results such as this have a long history in the mathematical literature [12, 15] with numerous applications in hashing =-=[2, 7, 14]-=-, PRAM simulation [17, 9, 14, 16, 20] and load balancing [2, 7]. # A preliminary version of this work appeared in the Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing,... |

37 |
auf der Heide, Simple, Efficient Shared Memory Simulations
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- 1993
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Citation Context ...n ). (In this paper log is used for log2.) Occupancy results such as this have a long history in the mathematical literature [12, 15] with numerous applications in hashing [2, 7, 14], PRAM simulation =-=[17, 9, 14, 16, 20]-=- and load balancing [2, 7]. ∗ A preliminary version of this work appeared in the Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing, May 1995, pp. 238-247. † U.C. Berkel... |

36 |
A guided tour of Cherno bounds
- Hagerup, Rub
- 1990
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Citation Context ..., . . . , X i-1 ) # p, then Pr( n X i=1 Y i # k) # Pr(B(n, p) # k). The second lemma presents some useful Cherno#-type bounds that will be used frequently throughout the paper; proofs may be found in =-=[1-=-1]. Lemma 2 If X i (1 # i # n) are independent binary random variables such that Pr[X i = 1] = p, then the following hold: For t # np, Pr n X i=1 X i # t ! # np t t e t-np . (1) For t # np, Pr n X i... |

33 | R.: On contention resolution protocols and associated probabilistic phenomena
- MacKenzie, Plaxton, et al.
- 1998
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Citation Context ...). (In this paper log is used for log 2 .) Occupancy results such as this have a long history in the mathematical literature [12, 15] with numerous applications in hashing [2, 7, 14], PRAM simulation =-=[17, 9, 14, 16, 20]-=- and load balancing [2, 7]. # A preliminary version of this work appeared in the Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing, May 1995, pp. 238-247. + U.C. Berkel... |

32 | Randomized allocation processes
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Citation Context ...m, as well as considers other similar load balancing problems. Mitzenmacher [18, 19] studies load balancing using multiple choices in dynamic settings related to queueing networks. Czumaj and Stemann =-=[8]-=- provide a general framework that extends the model and the results of Azar, Broder, Karlin and Upfal [2]; for example, they also consider the average allocation time in a threshold-based scheme simil... |

32 | Exploiting storage redundancy to speed up randomized shared memory simulations. Theoret - Heide, Scheideler, et al. - 1996 |

18 | Parallel Balanced Allocations
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Citation Context ...by greedy, and much better than that achieved by choosing one bin randomly for each ball. We note that since the writing of this paper a great deal of further work has been done in this area. Stemann =-=[21]-=- extends our work by analyzing an algorithm that asymptotically matches our lower bound in the case of n balls and n bins for any r (although his algorithm does not function asynchronously); he also a... |

13 |
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Citation Context ...ion asynchronously); he also analyzes the case where the number of balls m di#ers from the number of bins n for his algorithm, as well as considers other similar load balancing problems. Mitzenmacher =-=[18, 19]-=- studies load balancing using multiple choices in dynamic settings related to queueing networks. Czumaj and Stemann [8] provide a general framework that extends the model and the results of Azar, Brod... |

8 | Fairness in scheduling
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Citation Context ...or our bounds, we will rephrase the balls and bins problem in terms of a random graph orientation problem. The relationship between balls and bins problems and random graphs has been noted previously =-=[1, 14, 16]-=-. Here, we show that proving a lower bound for the balls and bins problem is equivalent to showing that, with su#ciently high probability, a specific subgraph appears in a random graph. These results ... |

8 |
auf der Heide. Ecient PRAM simulation on a distributed memory machine
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Citation Context ...y bin is #( log n log log n ). (In this paper log is used for log 2 .) Occupancy results such as this have a long history in the mathematical literature [12, 15] with numerous applications in hashing =-=[2, 7, 14]-=-, PRAM simulation [17, 9, 14, 16, 20] and load balancing [2, 7]. # A preliminary version of this work appeared in the Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing,... |

6 | auf der Heide, E cient PRAM Simulation on a Distributed Memory Machine - Karp, Luby, et al. - 1994 |

5 |
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Citation Context ... the average allocation time in a threshold-based scheme similar in spirit to the one we analyze here. Berenbrink, Meyer auf der Heide, and Schroder develop algorithms for dealing with weighted balls =-=[4]-=-. We note that these papers also contain many other results for further variations on the original model; because of the many interesting and useful variations of the basic balls and bins problem, we ... |

3 | Tail bounds for occupancy and the satis ability threshold conjecture - Kamath, Motwani, et al. - 1994 |

2 |
auf der Heide. Simple, ecient shared memory simulations
- Dietzfelbinger, Meyer
- 1993
(Show Context)
Citation Context ...). (In this paper log is used for log 2 .) Occupancy results such as this have a long history in the mathematical literature [12, 15] with numerous applications in hashing [2, 7, 14], PRAM simulation =-=[17, 9, 14, 16, 20]-=- and load balancing [2, 7]. # A preliminary version of this work appeared in the Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing, May 1995, pp. 238-247. + U.C. Berkel... |

2 |
Contention Resolution Protocols in Hashing Based Shared Memory Simulations
- Stemann
- 1995
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Citation Context ...). (In this paper log is used for log 2 .) Occupancy results such as this have a long history in the mathematical literature [12, 15] with numerous applications in hashing [2, 7, 14], PRAM simulation =-=[17, 9, 14, 16, 20]-=- and load balancing [2, 7]. # A preliminary version of this work appeared in the Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing, May 1995, pp. 238-247. + U.C. Berkel... |

1 | Random Allocations. V.H.Winston - Kolchin, Sevsat’yanov, et al. - 1978 |

1 | and Orli Waarts. Fairness in scheduling - Atjai, Aspnes, et al. - 1995 |