## Randomized Routing on Fat-Trees (1996)

Venue: | Advances in Computing Research |

Citations: | 51 - 11 self |

### BibTeX

@INPROCEEDINGS{Greenberg96randomizedrouting,

author = {Ronald I. Greenberg and Charles E. Leiserson},

title = {Randomized Routing on Fat-Trees},

booktitle = {Advances in Computing Research},

year = {1996},

pages = {345--374},

publisher = {JAI Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Fat-trees are a class of routing networks for hardware-efficient parallel computation. This paper presents a randomized algorithm for routing messages on a fat-tree. The quality of the algorithm is measured in terms of the load factor of a set of messages to be routed, which is a lower bound on the time required to deliver the messages. We show that if a set of messages has load factor on a fat-tree with n processors, the number of delivery cycles (routing attempts) that the algorithm requires is O(+lg n lg lg n) with probability 1 \Gamma O(1=n). The best previous bound was O( lg n) for the off-line problem in which the set of messages is known in advance. In the context of a VLSI model that equates hardware cost with physical volume, the routing algorithm can be used to demonstrate that fat-trees are universal routing networks. Specifically, we prove that any routing network can be efficiently simulated by a fat-tree of comparable hardware cost. 1 Introduction Fat-trees constitute...

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Citation Context ...at a random variable takes on values smaller than the expectation. The first lemma is the Chernoff bound. Consider t independent Bernoulli trials, each with probability p of success. It is well known =-=[5]-=- that the probability that there are at least s successes out of the t trials is B(s; t; p) = t X k=s / t k ! p k (1 \Gamma p) t\Gammak : The lemma bounds the probability that the number of successes ... |

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Citation Context ...d and receive polynomially many messages. Our routing algorithm also differs from others in the literature in the way randomization is used. Unlike the algorithms of Valiant [18], Valiant and Brebner =-=[19]-=-, Aleliunas [2], Upfal [17] and Pippenger [14], for example, it does not randomize with respect to paths taken by messages. For example, Valiant's classic scheme for routing on a hypercube sends each ... |

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Citation Context ...sality result by showing, in a three-dimensional VLSI model, that any network can be efficiently simulated by a fat-tree of comparable volume. The result had been proved only for off-line simulations =-=[12]-=-, where the communication pattern is known in advance; this paper extends it to the more interesting on-line case, where messages are spontaneously generated by processors. As is illustrated in Figure... |

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Citation Context ...that requires no routing algorithm. Finally, we give a stronger universality theorem based on RANDOM. VLSI models provide a means of measuring hardware costs quantitatively in terms of area or volume =-=[8, 10, 11, 12, 16]. These mo-=-dels are interesting from an engineering point of view because "pin-boundedness" is modeled directly as the limitation on communication imposed by the perimeter of a two-dimensional region o... |

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Citation Context ...to the more interesting on-line case, where messages are spontaneously generated by processors. As is illustrated in Figure 1, a fat-tree is a routing network based on Leighton's treeof -meshes graph =-=[8]-=-. A set of n processors are located at the leaves of a complete binary This research was supported in part by the Defense Advanced Research Projects Agency under Contract N00014--80--C--0622 and in pa... |

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Citation Context ...many messages. Our routing algorithm also differs from others in the literature in the way randomization is used. Unlike the algorithms of Valiant [18], Valiant and Brebner [19], Aleliunas [2], Upfal =-=[17]-=- and Pippenger [14], for example, it does not randomize with respect to paths taken by messages. For example, Valiant's classic scheme for routing on a hypercube sends each message to a randomly chose... |

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Citation Context ...that requires no routing algorithm. Finally, we give a stronger universality theorem based on RANDOM. VLSI models provide a means of measuring hardware costs quantitatively in terms of area or volume =-=[8, 10, 11, 12, 16]. These mo-=-dels are interesting from an engineering point of view because "pin-boundedness" is modeled directly as the limitation on communication imposed by the perimeter of a two-dimensional region o... |

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Citation Context ...lynomially many messages. Our routing algorithm also differs from others in the literature in the way randomization is used. Unlike the algorithms of Valiant [18], Valiant and Brebner [19], Aleliunas =-=[2]-=-, Upfal [17] and Pippenger [14], for example, it does not randomize with respect to paths taken by messages. For example, Valiant's classic scheme for routing on a hypercube sends each message to a ra... |

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Citation Context ...xists that achieves our better bound is an open question. Perhaps the bound on off-line routing can be further improved (e.g., to O(+lg n)). The integer programming framework of Raghavan and Thompson =-=[15]-=- is one possible approach which might give a probabilistic construction that achieves this bound. On the other hand, it may be possible to apply more direct combinatorial techniques to yield an improv... |

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Citation Context ...routing algorithm also differs from others in the literature in the way randomization is used. Unlike the algorithms of Valiant [18], Valiant and Brebner [19], Aleliunas [2], Upfal [17] and Pippenger =-=[14]-=-, for example, it does not randomize with respect to paths taken by messages. For example, Valiant's classic scheme for routing on a hypercube sends each message to a randomly chosen intermediate dest... |

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personal communication
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Citation Context ... n times, where jM j = O(n k ). 4 Interestingly, a mesh-of-trees [8] can be directly embedded in this fat-tree. In fact, it can be shown using sorting arguments that a mesh-of-trees is area-universal =-=[9]-=-. Figure 8: Another fat-tree design. The switches in this structure have constant size. The proof that the algorithm works applies the analysis from Section 4 to individual wires, treating them as cha... |

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Citation Context ...that requires no routing algorithm. Finally, we give a stronger universality theorem based on RANDOM. VLSI models provide a means of measuring hardware costs quantitatively in terms of area or volume =-=[8, 10, 11, 12, 16]. These mo-=-dels are interesting from an engineering point of view because "pin-boundedness" is modeled directly as the limitation on communication imposed by the perimeter of a two-dimensional region o... |

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Citation Context ...al to implement. The logarithmic-factor overhead that we have been able to show is mitigated by a constant factor of 1 12 . Simulations indicate that a greedy algorithm might actually work quite well =-=[6]-=-, but we have been unable to prove a good upper bound on its performance. Despite the simplicity of control offered by GREEDY, it seems unwise to base the design of a large, parallel supercomputer on ... |

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private communication
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Citation Context ...dy strategy is worse than RANDOM, which essentially guarantees O(+lgn lg lg n) delivery cycles for any set of messages. The lower-bound proof for greedy routing is based on an idea due to F. M. Maley =-=[13]-=-. Figure 5 shows the greedy algorithm. The code for GREEDY does not completely specify the behavior of message routing on a fat-tree because the switches have a choice as to which messages to drop whe... |