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## Randomized Routing on Fat-Trees (1996)

Venue: | Advances in Computing Research |

Citations: | 52 - 10 self |

### Citations

2364 |
An Introduction to Probability Theory and
- Feller
- 1968
(Show Context)
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 ... |

961 |
A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations
- Chernoff
- 1952
(Show Context)
Citation Context ...that we actually prove. We begin by stating two technical lemmas concerning basic probability. One is a combinatorial bound on the tail of the binomial distribution of the kind attributed to Chernoff =-=[4]-=-, and the other is a general, but weak, bound on the probability that a random variable takes on values smaller than the expectation. The first lemma is the Chernoff bound. Consider t independent Bern... |

332 | Parallel Prefix Computation
- Ladner, Fischer
- 1980
(Show Context)
Citation Context ...l. Such a concentrator can be built, for example, with a logarithmic-depth sorting network [1]. A somewhat more practical logarithmic-depth circuit can be built by combining a parallel prefix circuit =-=[7]-=- with a butterfly (i.e., FFT, Omega) network. With switches of logarithmic depth, the time to run each delivery cycle is O(lg 2 n) bit times, making the natural assumption that messages are O(lg n) bi... |

291 |
Universal schemes for parallel communication
- Valiant, Brebner
- 1981
(Show Context)
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 ... |

289 |
FAT-trees: universal networks for hardware-efficient supercomputing
- Leiserson
(Show Context)
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... |

241 |
A scheme for fast parallel communication
- Valiant
- 1982
(Show Context)
Citation Context ...ere each processor can send 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 ... |

178 | Mathematical Theory of Connecting Networks and Telephone Traffic - Benes - 1965 |

126 | A complexity theory for VLSI
- Thompson
- 1980
(Show Context)
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... |

124 |
Sorting in c log n parallel steps
- Ajtai, Komlós, et al.
- 1983
(Show Context)
Citation Context ...are lost if the number of messages destined for a channel is less than or equal to the capacity of the channel. Such a concentrator can be built, for example, with a logarithmic-depth sorting network =-=[1]-=-. A somewhat more practical logarithmic-depth circuit can be built by combining a parallel prefix circuit [7] with a butterfly (i.e., FFT, Omega) network. With switches of logarithmic depth, the time ... |

105 |
Complexity Issues in VLSI
- Leighton
- 1983
(Show Context)
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... |

88 | A permutation network - Waksman - 1968 |

67 |
Efficient schemes for parallel communication
- Upfal
- 1984
(Show Context)
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... |

48 |
Three-dimensional circuit layouts
- LEIGHTON, ROSENBERG
- 1986
(Show Context)
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... |

31 |
Provably good routing in graphs: regular arrays
- Raghavan, Thompson
- 1985
(Show Context)
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... |

29 |
Randomized parallel communication
- Aleliunas
- 1982
(Show Context)
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... |

25 |
Parallel Communication with Limited Buffers
- Pippenger
- 1984
(Show Context)
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... |

12 |
personal communication
- LEIGHTON, MAKEDON, et al.
- 1990
(Show Context)
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... |

4 |
Leiserson, Area-Efficient VLSI Computation
- E
- 1983
(Show Context)
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... |

1 |
Interprocessor Communication Issues in Fat-Tree Architectures, Bachelor's thesis
- Ishii
- 1985
(Show Context)
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 ... |

1 |
private communication
- Maley
- 1984
(Show Context)
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... |